Central European Journal of Mathematics

DOI: 10.2478/s11533-006-0007-8 Research article CEJM 4(2) 2006 183–193 Some non-local problems for the parabolic-hyperb...

4 downloads 569 Views 2MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

DOI: 10.2478/s11533-006-0007-8 Research article CEJM 4(2) 2006 183–193

Some non-local problems for the parabolic-hyperbolic type equation with non-characteristic line of changing type A.S. Berdyshev∗, E.T. Karimov† Institute of Mathematics, Uzbek Academy of Sciences, 700125 Tashkent, Uzbekistan

Received 11 December 2005; accepted 9 February 2006 Abstract: In this work two non-local problems for the parabolic-hyperbolic type equation with noncharacteristic line of changing type are considered. Unique solvability of these problems is proven. The uniqueness of the solution is proven by the method of energy integrals and the existence is proven by the method of integral equations. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Parabolic-hyperbolic type equations, non-local problems,energy integrals, integral equations, spectral parameter, Greenfunction MSC (2000): 35M10, 35P05, 35A05

1

Introduction

The first fundamental research on the theory of mixed type equations are works of F. Tricomi, and S. Gellerstedt, which were published in the 1920’s. Due to the research of F.I. Frankl, I.N. Vekua, M.A. Lavrent’ev and A.V.Bitsadze, K.I. Babenko, P. Germain and R. Bader, M. Protter, K. Morawetz, M.S. Salakhitdinov, T.D. Djuraev, A.M. Nakhushev, V.N. Vragov and many other authors, this theory became one of the main directions of the modern theory of partial differential equations. The necessity of the consideration of the parabolic-hyperbolic type equation was specified in 1959 by I.M. Gel’fand [1]. He gave an example connected to the movement of the ∗ †

E-mail: [email protected] E-mail: [email protected]

184

A.S. Berdyshev, E.T. Karimov / Central European Journal of Mathematics 4(2) 2006 183–193

gas in a channel surrounded by a porous environment. Inside the channel the movement of the gas was described by the wave equation, outside by the diffusion equation. By the 1970’s interest in the consideration of non-local problems grew up. The definitions and classifications of non-local problems is given by A.M. Nakhushev [2]. Consideration of non-local problems is interesting on the theoretical side, because they consist of many local problems. On practical side, they are directly connected with studying the problems of mathematical biology [2], the dynamics of groundwater and sound[3, 4], and the mathematical models of a laser [5, 6], etc. In particular, non-local conditions, which are used in this paper for the first time are considered in the works of M.S. Salakhitdinov and A.K. Urinov [7]. It is not difficult to see, that from the form of the integral operator, which is used in the non-local condition, at μ = 0 implies simple non-local conditions. We note the following works that are connected with studying parabolic-hyperbolic type equations with non-characteristic line of changing type. Regular solvability of the Tricomi problem for the parabolic-hyperbolic type equation with non-characteristic line of changing type was proven by V.A. Eleev [8] and strong solvability of this problem was proven by N.Y. Kapustin [9]. M.A.Sadybekov and G.D. Tojzhanova [10] have shown the existence of the eigenvalues of the problem with directional derivatives on the right characteristic of the hyperbolic part of the mixed domain. In this work, two non-local problems for parabolic-hyperbolic type equation with noncharacteristic line of changing type with spectral parameter were considered. The unique solvability of these problems will be shown. We define conditions of spectral parameters that provide uniqueness of the solution.

2

Formulation of the problems and main functional relations

Consider the equation 0=

⎧ ⎪ ⎨ ux − uyy − λu, y > 0 ⎪ ⎩ uxx − uyy + μu, y < 0,

(1)

in the domain Ω , where λ, μ are given real numbers. Ω is a simple connected domain, located in the plane of independent variables x and y, bounded by y > 0 with segments AA0 , BB0 , A0 B0 (A(0, 0), B(1, 0), A0 (0, 1), B0 (1, 1)) and by y < 0 with characteristics AC : x + y = 0, BC : x − y = 1 of the equation (1). We use following designations: Ω1 = Ω ∩ (y > 0), Ω2 = Ω ∩ (y < 0), AB = {(x, y) : y = 0, 0 < x < 1}, θ0 and θ1 are the points of intersections of the characteristics outgoing from the point (x, 0) ∈ AB with characteristics AC and BC respectively, i.e. x x x+1 x−1 θ0 = ,− , θ1 = , . 2 2 2 2 We investigate the following problems:


185

Problem N S1 . Find a function u(x, y) with the following properties: 1,2 1) u(x, y) ∈ C(Ω) ∩ C 1 (Ω ∪ AC) ∩ Cx,y (Ω1 ) ∩ C 2 (Ω2 ); 2) u(x, y) satisfies equation (1) in Ω1 ∪ Ω2 ; u(x, y)|A

0 B0

= ψ(x), 0 ≤ x ≤ 1,

u(x, y)|AA = ϕ(y), 0 ≤ y ≤ 1, 0

√ d 1, μ u (θ0 ) + a1 (x)uy (x, 0) = b1 (x), 0 < x < 1. A0x dx Problem N S2 . Find a function u(x, y) with the following properties: 1,2 1) u(x, y) ∈ C(Ω) ∩ C 1 (Ω ∪ BC) ∩ Cx,y (Ω1 ) ∩ C 2 (Ω2 ); 2) u(x, y) satisfies equation (1) in Ω1 ∪ Ω2 ; 3) u(x, y) satisfies conditions (2), (3) and

√ d 1, μ u (θ1 ) + a2 (x)uy (x, 0) = b2 (x), 0 < x < 1. Ax1 dx

(2) (3) (4)

(5)

Here ai (x), bi (x) (i = 1, 2) and ϕ(y), ψ(x) are given real-valued functions, n,μ Amx

x [f (x)] = f (x) −

f (t) m

t−m x−m

n

∂ J0 μ (x − m)(x − t) dt, m, n = 0, 1. ∂t

Consider equation (1) in the domain Ω1 . The result is the following lemma. Lemma 2.1. If u(x, y) is a solution of the equation (1), satisfying homogeneous conditions (2), (3) and u(x, 0) = τ (x) , 0 ≤ x ≤ 1 , then in Ω1 , the following equality is true 1 2 uy (x, y) − λu2 (x, y) dxdy = 0. u(x, 0)uy (x, 0)dx + (6) 0

Ω1

Proof. First we multiply equation (1) by u(x, y) and integrate along the domain Ω1ε , bounded by segments Aε A0ε , A0ε B0ε , B0ε Bε , Aε Bε , when ε is enough small positive quantities, we get 2 1 2 uy (x, y) − λu2 (x, y) dxdy = 0. ux (x, y) − [u(x, y)uy (x, y)]y dxdy + 2 Ω1ε Ω1ε Applying Green’s formula

∂ ∂ P (x, y) − Q(x, y) dxdy = Q(x, y)dx + P (x, y)dy ∂y D ∂x ∂D

and taking homogeneous conditions (2) and (3) into account we have (6).

186


Now we consider equation (1) in the hyperbolic part of the given domain Ω. Lemma 2.2. If u(x, y) is the solution of the Cauchy problem for the equation (1) in Ω2 , satisfying condition (4) or (5), then in Ω2 the following equalities respectively are true [1 − 2a1 (x)] ν(x) = −2b1 (x) + τ (x) + μ

x τ (t)

J1

√ μ(x − t) dt, √ μ(x − t)

(7)

J1

√ μ(x − t) dt. √ μ(x − t)

(8)

0

[1 + 2a2 (x)] ν(x) = −2b2 (x) − τ (x) + μ

1 τ (t) x

Proof. It is known that every regular solution of the equation (1) in Ω2 is represented as [13] x+y

τ (x + y) + τ (x − y) 1 u(x, y) = ν(t)J0 μ ((x − t)2 − y 2 ) dt + 2 2 x−y

x+y √ 2 − y2) J μ ((x − t) 1 μy + τ (t) dt, 2 (x − t)2 − y 2 x−y

where τ (x) = u(x, 0), ν(x) = uy (x, 0). Using (9), we define u(θ0 ) and u(θ1 ): x

x τ (0) + τ (x) 1 ν(t)J0 μt(t − x) dt u(θ0 ) = u ,− = − 2 2 2 2

0 x √ J1 μt(t − x) μx 1 + dt = {τ (0) + τ (x) τ (t) 4 2 μt(t − x) x

x + 0

0

x

x ∂ J0 τ (t) · · μt(t − x) dt − ν(t)J0 μt(t − x) dt }. t ∂x 0

Taking √ 0, μ B0x

x τ (t) ·

[τ (x)] = τ (x) + 0

x

√ 1, μ B0t

x ∂ μt(t − x) dt, · J0 t ∂x

x [ν(t)] dt =

0

ν(t) J0

μt(t − x) dt,

0

where n,μ Bmx

x [f (x)] = f (x) +

f (t) m

x−m t−m

1−n

∂ J0 μ (t − m)(t − x) dt, ∂x

(9)


187

m, n = 0, 1 we get (see [14]) ⎫ ⎧ x ⎬ ⎨ √ √ 1 0, μ 1, μ u(θ0 ) = τ (0) + B0x [τ (x)] − B0t [ν(t)] dt . ⎭ 2⎩ 0

Similarly we get u(θ1 ) =

⎧ 1⎨ 2⎩

τ (1) +

√ 0, μ Bx1

1 [τ (x)] −

√ 1, μ Bt1

x

⎫ ⎬ [ν(t)] dt . ⎭

Differentiating with respect to x we obtain the following equalities √ 1 d 0,√μ d 1, μ [τ (x)] − B0x [ν(x)] , u(θ0 ) = B dx 2 dx 0x √ 1 d 0,√μ d 1, μ [τ (x)] + Bx1 [ν(x)] , u(θ1 ) = B dx 2 dx x1 √ 1, μ

(10) (11)

√ 1, μ

Applying integral operators A0x , Ax1 to both side of equalities (10), (11) and taking √ √ 1, μ 1, μ Amx Bmx [f (x)] = f (x), √ 1, μ Amx

d dx

√ 0, μ Bmx [f (t)]

⎛ = sign(x − m) ⎝f (x) + μ

x m

√ ⎞ J1 μ(x − t) f (t) √ dt⎠ , m = 0, 1 μ(x − t)

into account we have √ 1, μ

A0x

√ 1, μ Ax1

⎧ ⎫ √ x ⎬ ⎨ J1 μ(x − t) d 1 u(θ0 ) = τ (x) + μ τ (t) √ dt − ν(x) , ⎭ dx 2⎩ μ(x − t) ⎧ 1⎨

0

⎫ √ 1 ⎬ J1 μ(x − t) d u(θ1 ) = −τ (x) + μ τ (t) √ dt + ν(x) . ⎭ dx 2⎩ μ(x − t) x

Substituting the equalities into (4) and (5) we get (7) and (8) respectively.

3

Main results

First we consider the problem N S1 . Theorem 3.1. Let a1 (x) = 1/2 . If ϕ(0) = 0, ϕ(x), ψ(x) ∈ C 1 [0, 1], a1 (x), b1 (x) ∈ C 1 [0, 1] ∩ C 2 (0, 1), λ ≤ 0, a1 (1) ≤ 0, a1 (x) ≤ 0, then the unique solution of the problem N S1 exists.

(12)

188


Proof. First we prove nonnegativity of the following integral 1 I=

τ (x)ν(x)dx.

(13)

0

Let u(x, y) be a solution of the homogeneous problem N S1 , then from (7) we have x

τ (x) + μ

τ (t)

J1

0

√ μ(x − t) dt = [1 − 2a1 (x)] ν(x). √ μ(x − t)

From here let τ (0) = 0 we get (see [14]) x τ (x) =

√ ν ∗ (t)J0 [ μ(x − t)] dt,

(14)

0

where ν ∗ (x) = [1 − 2a1 (x)] ν(x). Taking u(x, y) ∈ C(Ω) into account, substitute (14) to (13) ⎛ x ⎞ 1 √ I = ν(x) ⎝ ν ∗ (t)J0 [ μ(x − t)] dt⎠ dx. 0

0

Replacing the function J0 [•] by formula [15] 1 √ J0 [ μ(x − t)] = π

1

1 − ξ2

− 21

cos

√

μξ(x − t)dξ,

−1

we have 1 I=

1−ξ

1 2 −2

1 dξ

−1

0

1 dx 1 − 2a1 (x)

x

√ ν ∗ (x)ν ∗ (t) cos μξ(x − t)dt.

0

Considering x

∗

∗

ν (x)ν (t) cos

√

x μξ(x − t)dt =

0

[ν ∗ (x) cos

√

√ μξx ν ∗ (t) cos μξt +

0 ∗

+ν (x) sin

√

∗

μξx ν (t) sin

√

1 μξt ] dt = 2

d 2 d 2 M (x, ξ) + N (x, ξ) , dx dx

we get 1 I= −1

1−ξ

1 2 −2

1 dξ 0

1 1 − 2a1 (x)

d 2 d 2 M (x, ξ) + N (x, ξ) dx, dx dx

(15)


where

x

∗

ν (t) cos

M (x, ξ) =

√

x μξt dt, N (x, ξ) =

0

ν ∗ (t) sin

√

189

μξt dt.

0

Using integration by parts and choosing the function a1 (x) as needed, i.e. it satisfies the second and third conditions of the condition (12), we get 1 τ (x)ν(x)dx ≥ 0.

I=

(16)

0

Setting λ ≤ 0 from (6) we have u(x, 0) = τ (x) ≡ 0. Now consider the homogeneous conditions (2), (3) we get u(x, y) ≡ 0 in Ω1 . Since, u(x, y) ∈ C(Ω) , we can assert, that u(x, y) ≡ 0 in Ω . Now we prove existence of the solution. The solution of the first boundary problem for the equation (1) at y > 0,represented as [16] 1 u(x, y) =

x ϕ(y1 )G(x, y, 0, y1 , λ)dy1 +

0

0

x

+

ψ(x1 )Gy1 (x, y, x1 , 1, λ)dx1

τ (x1 )Gy1 (x, y, x1 , 0, λ)dx1 ,

(17)

0

where 1

∞

(2n+y−y )2 (2n+y+y )2 λ(x−x1 )− 4(x−x 1) λ(x−x1 )− 4(x−x 1) 1 1 e , −e

G(x, y, x1 , y1 , λ) = 2 π(x − x1 ) n=−∞

G(x, y, x1 , y1 , λ)– Green’s function of the first boundary problem for the equation (1) at y > 0. Differentiating (17) with respect to y once, we have 1 uy (x, y) =

x ϕ(y1 )Gy (x, y, 0, y1 , λ)dy1 +

0

+

d dy

ψ(x1 )Gy1 y (x, y, x1 , 1, λ)dx1 0

x

τ (x1 )Gy1 (x, y, x1 , 0, λ)dx1 . 0

Let’s consider the last integral in detail. x I=

τ (x1 )Gy1 (x, y, x1 , 0, λ)dx1 0

x = 0

∞ 2 τ (x1 ) y − 2n λ(x−x1 )− (y+2n) 4(x−x1 ) e dx1 , 2 π(x − x1 ) n=−∞ x − x1

(18)

190


d I= dy

x 0

∞

(y+2n)2 τ (x1 ) y + 2n λ(x−x1 )− 4(x−x ) 1 dx . √ e 1 − 1 2(x − x1 ) 2 π(x − x1 )3/2 n=−∞

(19)

Easily it can be seen, that the kernel has singularity more than 1 at y = 0, n = 0. To get the kernel with singularity less than 1 we integrate (19) by parts at x1 . Bearing in mind τ (0) = ϕ(0) = 0 , we get d lim I = − y→0 dy

x 0

∞ n2 τ (x1 ) − λτ (x1 ) λ(x−x1 )− x−x 1 dx . e 1 π(x − x1 ) n=−∞

(20)

Finally, evaluating the limit as y → 0 in (17) and taking (20) into account, after some transformations we get x

+

ν (x) = −

τ (t)K1 (x, t, λ)dt + Φ0 (x, λ),

(21)

0

Let u(x, y) ∈ C 1 (Ω) , from (7) and (21) we have x

τ (x) +

τ (t)K2 (x, t, λ, μ)dt = Φ(x, λ).

0

The solution of this equation is represented by τ (x) = Φ(x, λ) +

x Φ(t, λ) Γ(x, t, λ, μ)dt. 0

Now let τ (0) = ϕ(0) = 0 we get x τ (x) =

x Φ(t, λ) dt +

0

⎛ ξ ⎞ ⎝ Φ(t, λ) Γ(ξ, t, λ, μ)dt⎠ dξ.

0

0

where Γ(x, t, λ, μ) is resolvent of the kernel K2 (x, t, λ, μ), K1 (x, t, λ) − K2 (x, t, λ, μ) = 1 − 2a1 (x)

√ z √ μJ0 μ(t − z) dz, t−z t

z K1 (x, t, λ) = K(x − t, λ) − λ

K(t − z, λ)dz, t

∞

n2 1 K(x, λ) = √ e−λx− x , πx n=−∞

Φ(x, λ) = 2b1 (x) +

1 Φ0 (x, λ) =

x ϕ(y1 )Gy (x, 0, 0, y1 )dy1 +

0

Φ0 (x, λ) , 1 − 2a1 (x)

ψ(x1 )Gy1 y (x, 0, x1 , 1)dx1 , 0

(22)


191

∞

2n − y1 λx− (2n−y1 )2 1 2n + y1 λx− (2n+y1 )2 4x 4x − , Gy (x, 0, 0, y1 , λ) = √ − e e 2x 2x 2 πx n=−∞ ∞

1

(2n − 1)2 1− 2(x − x1 )

Gy1 y (x, 0, x1 , 1, λ) = 4 π(x − x1 ) n=−∞ (2n+1)2 (2n + 1)2 λ(x−x1 )− 4(x−x ) 1 . − 1− e 2(x − x1 )

(2n−1)2 1)

λ(x−x1 )− 4(x−x

e

The solution of the problem defined in Ω1 by formula (17) and in Ω2 by formula (9). Now consider the problem N S2 . Theorem 3.2. Let a2 (x) = − 12 . If 1 1 λ ≤ 0, μ ≤ 0, a2 (0) < − , a2 (1) < − , a2 (x) ≤ 0, 2 2 ϕ(0) = 0, ϕ(x), ψ(x) ∈ C 1 [0, 1], a2 (x), b2 (x) ∈ C 1 [0, 1] ∩ C 2 (0, 1), then the unique solution of the problem N S2 exists. Proof. As in the proof of the Theorem 3.1, first we prove the inequality (16). Let u(x, y) be a solution of a homogeneous problem N S2 , then from (8) we get ⎛ ν(x) =

1 ⎝−τ (x) + μ 1 + 2a2 (x)

1 x

⎞ J1 μ(x − t) τ (t) √ dt⎠ . μ(x − t) √

(23)

Bearing in mind u(x, y) ∈ C 1 (Ω), (23) substitute to (13) ⎧ 1 ⎛ 1 √ ⎞ ⎫ 1 ⎨ τ (x)τ (x) ⎬ J1 μ(x − t) τ (t) ⎝ ⎠ I=− dx − μ τ (x) dt dx = I1 + μI2 √ ⎩ 1 + 2a2 (x) ⎭ 1 + 2a2 (x) μ(x − t) 0

0

0

First we prove I1 > 0. Using integration by parts and taking τ (0) = ϕ(0) = 0 into account,we have τ 2 (1) 2I1 = − + 1 + 2a2 (1)

1 0

1 1 + 2a2 (x)

τ 2 (x)dx.

1 1 Setting ≥ 0 , we get I1 > 0. We prove I2 < 0 . For this < 0, 1 + 2a2 (1) 1 + 2a2 (x) we use the following formula √ J1 [ μ(x − t)] =

√

μ(x − t) 2π

1 −1

1 − ξ2

12

cos

√

μξ(x − t)dξ.

192


Denoting by 1 M (x, ξ) =

1

√

τ (t) cos μξt dt, N (x, ξ) = x

τ (t) sin

√

μξt dt

x

and using integration by parts we have 1 I2 = − 4π 1 + 0

1

1−ξ

2

12

dξ

−1

2

2

2 1 M (x, ξ) + N 2 (x, ξ) 1 + 2a2 (x)

M (x, ξ) + N (x, ξ)

1 1 + 2a2 (x)

1 0

+

dx

.

Taking M (1, ξ) = 0, N (1, ξ) = 0 into account and setting 1+2a12 (0) < 0, 1+2a12 (x) ≥ 0, we have I2 < 0. Finally setting μ ≤ 0 , we get I ≥ 0. Using Lemma 2.1 and setting λ ≤ 0 we have u(x, 0) = τ (x) ≡ 0. From here using (2) and (3), we obtain u(x, y) ≡ 0 in Ω1 . Since u(x, y) ∈ C(Ω) , we can assert, that u(x, y) ≡ 0 in Ω. The existence of the solution is proven similarly to the proof of existence in Theorem 3.1.

References [1] Gel’fand I.M.: “Some questions of analysis and differential equations”, UMN, Ser. 3(87), Vol. XIV, (1959), pp. 3–19. [2] Nakhushev A.M.: The equations of mathematical biology, Vishsaya shkola, Moscow, 1995, p. 301. [3] Nakhushev A.M.: “On one approximate method of solving boundary value problems for differential equations and its application to the dynamics of land and ground waters”, Differensialniye Uravneniya, Vol. 18(1), (1982), pp. 72–81. [4] S.V. Nerpin and A.F. Chudnovskij: Energy and mass exchange in the system flora – soil – air, Gidromechizdat, Leningrad, 1975. [5] P. Bassani and M. Calaverni: “Cantrazioni multi sistemi iperbolica,iproblema del lazer”, Atti.Semin.mat.e.fis., Univ.Modena, Vol. 35(2), (1982), pp. 32–50. [6] L. Bers: Mathematical questions of subsonic and transonic gas dynamics, IL, Moscow, 1961, p. 208. [7] M.S. Salakhitdinov and A.K. Urinov: “About one boundary value problem for the mixed type equation with non-smooth line of degeneracy”, Doklady AN SSSR, Vol. 262(3), (1982), pp. 539–541. [8] V.A. Eleev: “Analogue of the Tricomi problem for the mixed parabolic-hyperbolic equations with non-characteristic line of changing type”, Differensialniye Uravneniya, Vol. 13(1), (1977), pp. 56–63.


193

[9] N.Y. Kapustin : “The Tricomi problem for the parabolic-hyperbolic equation with degenerating hyperbolic part”, Differensialniye Uravneniya, Vol. 24(8), (1988), pp. 1379–1386. [10] G.D. Tojzhanova and M.A. Sadybekov: “About spectral properties of one analogue of the Tricomi problem for the mixed parabolic-hyperbolictype equation”, Izvestija AN KazSSR, ser.phys.-math.nauk, Vol. 3, (1989), pp. 48–52. [11] A.S. Berdyshev: “Nonlocal boundary problems for the mixed type equation in the domain with deviation from the characteristic”, Differensilaniye Uravneniya, Vol. 29(12), (1993), pp. 2118–2125. [12] A.S. Berdyshev: “On uniqueness of the solution of general Tricomi problem for the parabolic-hyperbolic equation”, Doklady ANRUz., Vol. 10, (1994), pp. 5–7. [13] A.N. Tikhonov and A.A. Samarskij: Equations of mathematical physics, Nauka, Moscow, 1977, p. 736. [14] M.S. Salakhitdinov and A.K. Urinov: Boundary value problems for the mixed type equations with spectral parameter, Fan, Tashkent, 1997, p. 166. [15] G. Bateman and A. Erdelji: Higher transcendent functions, Nauka, Moscow, 1965, p. 296. [16] A. Fridman: Partial differential equations of parabolic type, Izdatelstvo Mir, Moscow, 1968, p. 428.

DOI: 10.2478/s11533-006-0005-x Research article CEJM 4(2) 2006 194–208

Singularities on complete algebraic varieties∗ Fedor A. Bogomolov1† , Paolo Cascini2‡ , Bruno de Oliveira3§ 1

Courant Institute of Mathematical Sciences, New York, NY 10012, USA 2

Department of Mathematics, UC Santa Barbara, Santa Barbara CA 93106, USA 3

Department of mathematics, University of Miami, Coral Gables, FL 33124, USA

Received 6 October 2005; accepted 12 December 2005 Abstract: We prove that any finite set of n-dimensional isolated algebraic singularities can be afforded on a simply connected projective variety. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Isolated singularities, deformation of singularities MSC (2000): 14B05, 32S05

1

Introduction

It is a classical question in algebraic geometry to understand the constraints imposed on the singularities that can be afforded on a given class of algebraic varieties. A general result in this direction appeared in [4]. There it was shown that for any algebraic family of algebraic varieties there are isolated singular points which can not be afforded on any variety which is birationally equivalent to any member of this family. Our aim is to prove that any set of isolated algebraic n-dimensional singularities can be afforded on a simply connected projective variety. ∗

The first author was partially supported by NSF grant DMS-01591 and the third author was partially supported by NSF grant DMS-03693 † E-mail: [email protected] ‡ E-mail: [email protected] § E-mail: [email protected]

F.A. Bogomolov et al. / Central European Journal of Mathematics 4(2) 2006 194–208

195

More precisely we are going to prove the following result: Theorem A. Let (Y, y) be an isolated singularity. There exists a simply connected projective variety X having a unique singular point x ∈ X such that the singularities (X, x) and (Y, y) are isomorphic. The variety X we are constructing is of general type and we believe that general type condition is necessary in order to afford arbitrary isolated singularity. We will give an interesting example of the result in [4]. We will describe which sets of rational double points can be afforded on rational surfaces (with the surprising fact that two E6 can not be afforded). This paper is also devoted to what we consider to be a useful description of singularities. We describe the germ of a reduced and irreducible analytic space as a finite cover of a polydisc Δn branched along smooth divisors of Δn . In particular, this gives a new description for the deformations of isolated singularities and provides with a simple proof of the fact that an irreducible and reduced germ of an analytic surface is algebraic. Another motivation for theorem A was the work of C. Epstein and G. Henkin on the stability of the embeddability property of a strictly pseudo-convex 3-dimensional CR-structure [EH]. More precisely, C Epstein asked the third author if one can always embed an embeddable strictly pseudo-convex 3-dimensional CR-structure inside a regular variety. The methods used in [EH] view the embeddable CR-manifold M as the boundary of a pseudo-concave surface Y which can be attached to the Stein filling S of M to give a projective variety X = Y M S. The properties of X, and especially the regularity, played an important role in their results.

2

Analytic Singularities

This section introduces a local description of analytic spaces that we think is very useful to the analysis of a spectrum of problems about singularities. We describe the germ of a reduced and irreducible analytic space as a finite cover of a polydisc Δn branched along smooth divisors of Δn . We give then a new description of the deformation space of an isolated singularity. Another application is a simple proof of the algebraicity of isolated surface singularities.

2.1 Local Parameterization The following result is a simple modification of the lemma from [3] which extends Belyi’s argument to the case of arbitrary field of characteristic zero. Lemma 2.1. Let Y be an n-dimensional affine variety. Then there exists a proper map f : Y → Cn and a linear projection p : Cn → Cn−1 such that f is ramified over a finite set of sections Si of p.

196


Proof. Consider an arbitrary finite surjective map g : Y → Cn . Let D be the ramification divisor of g in Cn and let p : Cn → Cn−1 be a linear projection whose restriction to D is proper and surjective. The projection p is defined by a point x ∈ Pn−1 ∞ . To guarantee n ¯ properness take x outside of the intersection of the closure D ∈ P with Pn−1 ∞ . After a n n linear change of coordinates, φ : C → C , the projection p can be seen as the standard projection onto the last coordinate. Hence, we have a linear parameter zn on all the fibers of p and p(z1 , ..., zn ) = (z1 , ..., zn−1 ). By Noether normalization, the ramification divisor D0 of g0 = φ ◦ g is given as the set of zeroes of a monic polynomial f0 (zn ) = znd + ad−1 znd−1 + ... + a0 = 0 with coefficients ai ∈ C[z1 , ..., zn−1 ], for any i < n. Let F0 : Cn → Cn be the branch cover of degree d defined by F0 (z1 , ..., zn ) = (z1 , ..., zn−1 , f0 (zn )) and denote by g1 the composition g1 = F0 ◦ φ ◦ g0 . The ramification divisor of g1 is the union of the divisor zn = 0 (corresponding to F0 (D0 )) and the divisor D1 = F0 (R0 ), with R0 = {(f0 )zn (zn ) = 0} where (f0 )zn (zn ) = dznd−1 + (d − 1)an−1 znd−2 + ... + a1 . The projection p maps the divisor D1 properly onto Cn−1 . The divisor R0 has degree d − 1 with respect to zn and hence it’s image F0 (R0 ) = D1 is defined by a monic polynomial f1 (zn ) of degree d1 ≤ (d − 1) in zn . Let F1 : Cn → Cn be the branch cover of degree d1 defined by F1 (z1 , ..., zn ) = (z1 , ..., zn−1 , f1 (zn )) and g2 = F1 ◦ g1 . The ramification divisor is the union of two sections of p, F1 ({zn = 0}) and F1 (D1 ) = {zn = 0}, and the divisor D2 = F1 (R1 ) which is defined by a monic polynomial on zn of degree ≤ (d1 − 1). In conclusion, after i-step we have the map gi = Fi−1 ◦ gi−1 with ramification divisor consisting of the union of i sections of p and a divisor Di = Fi−1 (Ri−1 ) which is defined by a monic polynomial on zn of degree ≤ (d − i). Therefore, we obtain the lemma after l ≤ d steps. Remark 2.2. The proof of lemma 2.1 also works for a pair (X, D), where X is an arbitrary affine variety of dimension n and D is a divisor of X. In this case, the result would be that there is a finite map f : X → Cn such that the ramification divisor of f and f (D) are a set sections of a projection p : Cn → Cn−1 . The previous result can be reformulated in the category of complex analytic spaces to give local results. One such reformulation is a refinement of the Local Parameterization Theorem. Proposition 2.3 (Local Parameterization). Let x be a point in a complex analytic space X of dimension n and suppose that X is locally irreducible and reduced at x. Then x has neighborhood U ⊂ X with a finite map f : U → Δn onto an n-polydisc Δn = Δn−1 ×Δ ramified over a finite collection of sections Si over Δn−1 .


197

Proof. The standard Local Parameterization Theorem states that all x ∈ X have a neighborhood U ⊂ X admitting a finite map g : U → Δn onto a n-polydisc with g(x) = (0, ..., 0). The refinement consists of showing that one can make the ramification divisor of the finite map very well behaved, which provides us with a tool to better understand singularities. First, we remark that there is nothing to prove if x is not a singular point of X. Let D ⊂ Δn be the ramification divisor of the previously described finite map g : U → Δn . We can shrink U and choose a decomposition of the n-polydisc Δn = Δn−1 × Δ such that the projection of D onto Δn−1 is a finite mapping. The proof of the standard LP theorem also gives that D is given by a Weierstrass polynomial f0 (zn ) = znd + ad−1 znd−1 + ... + a0 with ai ∈ O(Δn−1 ) with ai = O(|(z1 , ..., zn−1 )|d−i ). The previous paragraph provides the setup to apply the method used in the previous lemma. We describe one of the steps to make clear the slight modifications. Using the Weierstrass polynomial f0 (zn ) we construct the map F0 : Δn−1 ×Δ → Δn−1 ×Δ , where Δ is some disc, given by F0 (z1 , ..., zn ) = (z1 , ..., zn−1 , f0 (zn )). The map F0 ◦g : U → Δn−1 ×Δ might not be surjective. But by picking a smaller disc Δ1 ⊂ Δ and shrinking U to U = (F0 ◦ g)−1 (Δn−1 × Δ1 ) we get a finite mapping g1 = F0 ◦ g : U → Δn−1 × Δ1 ramified at F0 (D) = {zn = 0} and D1 = F0 (R) where R = {(f0 )zn = 0}. The divisor D1 is given by the zero set of a Weierstrass polynomial f1 (zn ) = znd1 + ad1 −1 znd1 −1 + ... + a0 of degree d1 ≤ d − 1. Use f1 (zn ) to construct F1 and do the necessary shrinking of U , as before, and obtain a finite map g2 = F1 ◦ g1 : U → Δn−1 × Δ2 . The desired finite map f will be the map gl : U → Δn−1 × Δl obtained after some l ≤ d steps.

2.2

Applications

In this section we show how to apply proposition 2.1 to obtain the algebraicity of the germs of normal 2-dimensional complex spaces and give a description of isolated singularities that might prove to be useful for the description of their deformations. The Local Parameterization theorem presented in section 2.1 provides a simple proof of the algebraicity of any germ of an analytic surface. For isolated singularities this is a well known result due to Artin [A2] and later extended to a global result by Lempert [Le]. More precisely, Lempert showed that any reduced Stein space S with boundary ∂S = M a smooth CR-manifold can be embedded in an algebraic variety. On the other hand, recall that in [13] (examples 14.1 and 14.2) Whitney shows that analytic singularities in dimensions n ≥ 3 are, in general, not locally algebraic. Whitney constructs an example of a normal analytic variety V of dimension 3 and with a singular point p ∈ V , such that there exists no algebraic variety that is locally (in an analytic sense) biholomorphic to any open neighboorhood of p in V . We proceed to show that all the analytic singularities in dimension 2 are locally algebraic. Let p be a point of a complex analytic surface S, and suppose that S is normal at p. By Proposition 2.3, there exists an open neighborhood U ⊆ S of p, admitting a finite map g : U → Δ2 , where Δ2 is a polydisc in C2 and such that g(p) = (0, 0).

198


If the ramification divisor of g, D ⊂ Δ2 , was an algebraic curve on Δ2 (i.e. given by the zero locus of a polynomial), then U would be an open subset of an algebraic surface. But D is possibly reducible to an analytic curve in Δ2 . To deal with this case, we have: Lemma 2.4. Let D ⊆ Δ2 be a reduced analytic divisor, such that (0, 0) ∈ D. Then, up to shrinking Δ2 , there exists a biholomorphic map from Δ2 onto an open neighborhood V of (0, 0) in C2 such that the image D of D is an algebraic divisor passing through (0, 0). Proof. Levinson proves a more general result in [10] (see also [13], remark 14.3). But, for the sake of completeness, we show an easy proof of the lemma. Let D be a union of irreducible components Di , with i = 1, . . . , N , passing through (0, 0). By choosing a suitable system of coordinates z1 , z2 and after shrinking the polydisc Δ2 , we can suppose that if p1 : Δ2 → Δ is the projection with respect to the first coordinate, then each Di is a section of p1 . In other words, we can write each Di as the zero set of the function Fi (z1 , z2 ) = z2 − fi (z1 ), with fi analytic. We want to prove the lemma by induction on N (the number of irreduible components of D). Suppose that f1 , . . . , fk are polynomials, with k < N . We want to construct a biholomorphism of the form Φ(z1 , z2 ) = (z1 , z2 + g(z1 , z2 )

k

Fi (z1 , z2 ))

i=1

where g is an analytic function such that Φ(Dk+1 ) is algebraic. In fact, by construction it follows that Φ(Di ) = Di for any i ≤ k. In order to reach our aim, we have to choose g such that the analytic function P (z1 ) = fk+1 (z1 ) + g(z1 , fk+1 (z1 ))

k (fk+1 (z1 ) − fi (z1 ))

(1)

i=1

is indeed a polynomial. By shrinking Δ2 again, if necessary, we can suppose that k (fk+1 (z1 ) − fi (z1 )) = z1M φ(z1 ) i=1

for some M > 0 and φ analytic function such that φ(0) = 0. Therefore we can find a holomorphic function g, satisfying (1), for any polynomial P such that P (z1 ) − fk+1 (z1 ) is divisible by z1M . Our claim follows from the lemma. Choose U defined by h−1 (Δ2 ), where h = Φ ◦ g : U → V and (0, 0) ∈ Δ2 ⊂ V , as the neighborhood of p. The open set U is a branched covering of Δ2 branched over an algebraic curve. The Local Parameterization result described in proposition 2.1 gives directly the following description of isolated singularities.


199

Proposition 2.5. Let s be a normal isolated singularity in a n-dimensional complex analytic space Y . Then: a) There is an open neighborhood of s, U ⊂ Y , admitting a finite map f : U → Δn , onto an n-polydisc, which is unramified outside of a finite set of smooth subvarieties Si ⊂ Δn . b) The germ of the singularity s ∈ Y is determined by the pair (Δn − ki Si , Γ), where Γ is the subgroup of finite index of π1 (Δn − ki Si ) defining the covering.

The above picture of a singularity can be quite useful to determine the structure of the deformation space for many isolated singularities. Let (Y, s) be the germ of a normal n-dimensional singularity corresponding to the pair (Δn \ ki Si , Γ). Denote by si ∈ π1 (Δn \ ki Si ) the simple loops around the irreducible components Si . The s1 ,...,sk generate π1 (Δn \ ki Si ) ∼ = Zm , where m is the multiplicity of the irreducible holomorphic function germ g with g −1 (0) = ki Si . Let Γ be the maximal normal subgroup of π1 (Δn \ k i Si ) contained in Γ. The next short exact sequence holds:

1 → Γ → π1 (Δ \ n

k

Si ) → G → 1

i

where G is the Galois group of the cover induced by Γ . Consider a deformation ki Sit of ki Si . Let T be a tubular neighborhood of ki Si . The complement Δn \ T is homotopically equivalent to Δn \ ki Si and it is immersed in 1 Δn \ ki Sit for |t| 0. Hence there is a natural homomorphism jt : π1 (Δn \ ki Si ) → 1 π1 (Δn \ ki Sit ) for |t| 0. Assume that a surjection rt : π1 (Δn \ ki Sit ) G holds and moreover that rt ojt : 1 π1 (Δn \ ki Si ) G is constant for |t| 0. This implies that a Galois cover, associated k t n with G, of Δ \ i Si persists for small t and the induced covering of Δn \ T ⊂ Δn \ ki Sit is constant along the family. In turn, this implies that an intermediate covering associated with Γ inducing a constant covering of Δn \ T ⊂ Δn \ ki Sit also persists for small t. The end result is that from a family of divisors ki Sit ⊂ Δn for which rt ojt : π1 (Δn \ k 1 i Si ) G is constant for |t| 0 one obtains a family of singularities Yt associated with the pairs (Δn \ ki Sit , Γ). The singularities Yt all have a finite map ft : Yt → Δn of the same degree branched at ki Sit . Moreover, the Yt have an arbitrarily large open subset ft−1 (Δn \ T ) ⊂ ft−1 (Δn \ ki Sit ) which is biholomorphic to f0−1 (Δn \ T ) for all sufficiently small t. The conditions to impose on the Sit to guarantee the constancy of rt ojt : π1 (Δn \ ki Si ) G will be investigated in future work.

3

Singularities inside Projective varieties

Any collection of isolated singularities can be afforded in some projective variety (see paragraph below). On the other hand, a collection of singularities, or even one single

200


singularity, does impose global constraints on the type of the variety that possesses it (see the next subsection). The main goal of this section, theorem A, is to show that the property of being simply connected is not one of the properties which is conditioned by the presence of singularities. Along the same lines we would like to conjecture a stronger result: Conjecture 3.1. Let (Y, y) be the germ of a given isolated singularity. There exists a ˆ is simply projective variety X containing Y and with X \ {y} smooth whose resolution X connected. The following lemma shows that every finite set of isolated singularities, can be afforded in a unique projective variety. Lemma 3.2. Let Γ = {(Yi , yi )}i=1,...,k be any collection of germs of algebraic isolated singularities of dimension n. There exists a projective variety Y having Γ as its singular locus. Proof. Let Xi be a variety with only one singular point and the germ of the singularity is equivalent to (Yi , yi ). The lemma follows from induction on k. Let us assume that we with Γk−1 = {(Yi , yi )}i=1...k−1 as its singular locus. constructed a projective variety Yk−1 Let Hn be a general n-codimensional plane in the product variety Yk−1 × Xk and let us consider Yk = (Yk−1 × Xk ) ∩ Hn .

We can choose Hn so that it intersects transversely the singular locus {yi } × Xk or Yk−1 × {yk } and avoids the points yi × yk . Therefore Yk is a n-dimensional projective variety whose singularities are isomorphic to the singularities (Yi , yi )i=1...k . In order to have exactly one copy of each singularity, it is enough to resolve the possible extra copies of the singularities (Yi , yi )i=1...k that might occur.

3.1 An example of constraints imposed by singularities In the introduction we recalled a recent result of Ciliberto and Greco stating that for any algebraic family of algebraic varieties there are isolated singular points which can not be afforded on any variety birational to a member of this family. We proceed to give a concrete example of this result. More precisely, we describe all the sets of rational double points, RDP’s, that a rational surface can contain (the same result holds for all surfaces of Kodaira dimension −∞). Notation. Let X and Y be analytic normal complex surfaces and f : Y → X be a birational morphism with exceptional set E = Ei . The negative definiteness of the intersection matrix (Ei , Ej ) allow the existence of a unique solution to:


KY ≡ f ∗ KX +

201

ai Ei

The numbers ai are called the discrepancy of Ei with respect to X, discrep(Ei , X) = ai . The birational morphism will be called totally discrepant if E = ∅ and the ai > 0, for all i. It was shown in Sakai [S1] that given a normal surface X, there is a sequence X → X1 → X2 → ... → Xn of contractions of exceptional curves of the first kind, i.e C 2 < 0 and KXi .C < 0, such that Xn has no such curves. Xn is then called a minimal model of X and the morphism f : X → Xn is totally discrepant. Let X be the minimal model of the normal surface X and f : X → X be the totally discreptant birational morphism. Let π : Y → X and π : Y → X be respectively the minimal resolutions of X and X, with KY = π ∗ KX + Δ and KY = π ∗ KX + Δ. Then f induces a birational morphism g : Y → Y such that g∗ Δ ≥ Δ (this result supports the statement that going to the minimal model does not make singularities worse). A normal surface singularity (X, x0 ) is an RDP (rational double point) iff KY .Ei = 0 for every exceptional curve Ei of the minimal resolution f : Y → X or equivalently f ∗ KX = KY + Δ with Δ = 0. From the definition of an RDP singularity follows that the negative configuration of curves that form the exceptional set of f is composed of smooth rational curves with self intersection -2 in one of formations of the Dinkin diagrams An (n = 1, ...), Dn (n = 4, ...), En (n = 6, 7, 8). The observation of the previous paragraph implies that if a normal surface X has only RDP singularities then the same is true for its minimal model. Theorem 3.3. The collection of rational double points that can be in a rational surface X are the following: 1. Arbitrary collections of An and Dn singularities 2. An En singularity and an arbitrary collections of An singularities. Proof. First, we give the positive results. By blowing up over a point, one can get an An configuration of negative curves. Hence all birational classes of surfaces can have as many An singularities as desired. A Dn configuration of negative curves can be obtained by blowing up over a smooth rational curve C with C 2 = 0. Hence one can get as many Dn singularities as desired in all birational classes of ruled surfaces. An En configuration can not be obtained by blowing up over a smooth rational curve C with C 2 = 0. This is the reason behind the asymmetry of the theorem. On the other hand, one En configuration of negative curves can be obtained by blowing up over two lines in P2 . Hence one can get one En singularity in the birational class of rational surfaces. The minimal model program for normal singular surfaces developed by Sakai will give the negative results. Let Y be a normal surface with two En singularities or an En and

202


one Dm singularity. Resolve all the other singularities and still name that surface Y . The minimal model Ym of Y is a surface with only rational double points, as explained above. Moreover, the singularities En and Dm of Y will still exist in the minimal model Ym [Mo]. Let f : X → Ym be the minimal resolution of Ym (i.e. no (-1)-curves on the exceptional locus). Assume, as in the hypothesis of the theorem, that X is rational, then since the singularities of Ym are rational, KX = f ∗ KYm , one has Kod(Ym ) = −∞ and KYm is not nef. Sakai [S2] proved that if W is a minimal normal surface whose canonical bundle KW is not nef then W is projective, Kod(W ) = −∞ and either: 2 i) ρ(W ) = 1 and −KW is numerically ample, i.e. KW > 0 and KW .C < 0 for all curves C ⊂ W , or

ii) W has a P1 -fibration. So according to Sakai’s result Ym must be one of the two cases described above. We will show that both cases are not possible. 2 = Suppose Ym is as in i). The minimal resolution X of Ym is rational and has KX ∗ 2 2 (f KYm ) > 0. Hence X is P blown up at most 8 times or one of the Hirzebruch surfaces is Fn blown up at most 7 points. In both cases b− 2 (X) ≤ 8. But on the other hand the − minimal resolution of Ym must have b2 (X) ≥ n + m ≥ 10 and we obtain a contradiction. The inequality is just a consequence of the linear independence of homology classes of the curves in the exceptional locus. Suppose Ym is as in ii). The P1 -fibration of Ym induces a ruled-fibration, π : X → C, of X. The configuration of (-2)-curves coming from the resolution of the En singularity lies in one of the fibers. The surface resulting from contracting the (-1)-curves in the fibers of π is an Hirzebruch surface Fn . But an En configuration of (-2)-curves can not be obtained by blowing up over a smooth rational curve C with C 2 = 0 and the desired contradiction follows. Corollary 3.4. A surface X which is a resolution of a surface Y containing a En and a Dn singularity must have its Kodaira dimension Kod(X) ≥ 0. Proof. The last theorem states that X is not a rational surface. On the other hand, an En configuration of negative curves does not lie entirely in the blow up pre-image of a fiber of a ruled surface. This would force one of the (-2)-curves to surject to the base of the ruled surface imposing that X is rational. Corollary 3.5. There is a singularity that can not be afforded in a projective surface X with Kod(X) = −∞. Proof. Let X be a smooth projective surface with a En and a Dm configurations of −2curves which are disjoint. Let H be an ample divisor on X , blow up X at a sufficiently large number of points on H but not on the configurations En or Dm . We obtain a new surface X with a negative configuration of curves consisting of H (the strict transform


203

of H) plus the curves coming from En and Dm (the negative definiteness is guaranteed by making H 2 1 (2) H i (S l X, R) = H i (S m X, R) for m, l > i Proof. (1) The fundamental group of S n X is generated by the fundamental group of X. In particular, given the map sn : X n → X n with n > 1, every two elements in π1 (S n X) can be thought as induced by the first and second factor respectively. Thus, their commutator is trivial in π1 (S n X). (2) The cochains of S m X are symmetrizations of the cochains in the product of m copies of X. Thus for i < n symmetric cochains are generated by cochains in the product of ≤ i copies of X multiplied by 0-dimensional cochains. It implies that the i-skeletons of S n X and S m X are isomorphic for i < min(m, n). If X is an algebraic or projective variety then S m X is respectively an algebraic or projective variety. The variety S m X is singular unless X is a non-singular curve. Let us consider the case where X is an algebraic variety of dimension n with a finite collection of singular points Γ = {s1 , ..., sk }. Denote U = Xreg = X \ XSing and any of the i-diagonals of U m (the entries of a fixed set of i places of U m are identical) by Δi . We have the following stratification S of S m X: (1) (S m X)reg .

204


(2) Pi [m] = sm (Δi+1 ) \ ∪m j=i+2 sm (Δj ), 1 ≤ i ≤ m − 1. i m−i (3) Σi = S (Γ) × (S U )reg 1 ≤ i ≤ m. i (4) ΣPij = S (Γ) × Pj [m − i], 1 ≤ i ≤ m and 1 ≤ j ≤ m − i. We denote the complement of the union of of all strata of codimension ≥ (i + 1)n by (S m X)i . The (S m X)i are Zariski open subsets of S m X. For example, (S m X)0 = (S m X)reg and (S m X)1 = S m X)reg ∪ P1 [m] ∪ Σ1 . The following dimensional properties hold for the strata: (1) codimPi [m] = in, the singularities along Pi [m] are simple quotient singularities. (2) codimΣi = in. (3) codimΣPij = (i + j)n. We are now ready to state the main result of this section: Theorem 3.7. Let Γ = {(Yi , yi )}i=1,...,k be any collection of germs of equidimensional isolated singularities. There exists a projective variety X with abelian fundamental group whose collection of singular points coincide with Γ. Moreover if Y is a projective variety with YSing = Γ, then X can be made such that π1 (X \ Γ) = H1 (Y \ Γ, Z). Proof. Let Y be a projective variety whose collection of singular points coincide with Γ (lemma 3.2). From lemma 3.6, it follows that the fundamental group of the symmetric product of any algebraic variety is abelian, and therefore we would like to take a generic complete intersection Z in S 2 Y of the same dimension of Y and that contains the same singularities of Y in such a way that π1 (Z) = π1 (S 2 Y ). That would imply that Z has an abelian fundamental group. Lefschetz theorem on hyperplane sections states that if W is an algebraic variety with dim W > 2 and H ⊂ W is an hyperplane section such that W \ H is smooth then π1 (W ) = π1 (H). For the variety S 2 Y that we are considering, Lefshetz theorem cannot be applied directly, because we want to study complete intersection subvarieties that are transverse to the singular locus of S 2 Y . Hence, let us consider n generic hyperplane sections H1 , . . . , Hn of Y , passing through the singular points of Y . On the product Y 2 = Y × Y , let pi : Y 2 → Y with i = 1, 2, be the respective −1 projections and let H be the intersection of the divisors p−1 1 (Hj )∪p2 (Hj ) with j = 1, . . . n. Then H is a complete intersection of very ample divisors on Y 2 that is invariant with respect of the natural action of the group Z2 on Y 2 . We denote its quotient by Z = H/Z2 ⊆ S 2 Y. Let U be the smooth locus of Y , i.e. U = Y \ Γ, and let HU = H ∩ (U × U ). By applying Lefshetz theorem on HU . it follows that π1 (HU ) = π(U × U ) and since the action induced by Z2 on those groups is the same, we have that if ZU is the quotient of HU by Z2 , then ZU has the same fundamental group of S 2 U and in particular this is abelian.


205

On the other hand ZU is also an open set of Z such that its complement is a union of a finite number of point and therefore the fundamental group π1 (Z) is abelian since surjection π1 (ZU ) → π1 (Z) holds. From the construction of H, it follows easily that the singularities of Z are isolated and decompose into ZSing = (Z ∩Σ1 )∪(Z ∩P1 ). The singularities in Z ∩Σ1 are equivalent to the isolated singularities of Y and the singularities in Z ∩ P1 are double points. Let X be the projective variety obtained from Z by resolving the double points. Then X has abelian fundamental group and its singular locus coincides with Γ, as desired. Corollary 3.8. Let Y be a projective variety with a given collection Γ of isolated singular points such that H1 (Y, Z) = 0. Then there exists a projective variety X with XSing = Γ which is simply connected. If additionally Y is such that H1 (Y \ Γ, Z) = 0 then X can be made so that X \ Γ is also simply connected.

4

Reducing the abelian fundamental group

We are now ready to prove theorem A. Let S be a given isolated singolarity. By the results in the previous section, we can suppose that there exists a variety X such that XSing S, and, if U = X \ XSing , then π1 (U ) is abelian and the imbedding U → X defines a surjection π1 (U ) → π1 (X). Let us consider the infinite part of the fundamental group of X. If it is trivial, then the group is finite and hence there is a finite nonramified covering of X which is simply connected. Thus, we can suppose that the Albanese map f : X → A := Alb(X) is not trivial. The torsion subgroup π1 (X)tors is a direct summand of π1 (X), and therefore if π1 (X)F is the complementary subgroup, the induced morphism f∗ : π1 (X)F → π1 (A) is an isomorphism. For every n > 0, we can consider the iteration map fn : S n X → A, given by f (xi ). Since f (X) generates A, there exists n0 such that if n ≥ n0 , fn (x1 . . . xn ) = then fn is surjective. Moreover, we have Lemma 4.1. There exists a positive integer m > 0 such that the map fm : S m X → A admits a topological section s : A → S m X. Proof. Let g = dim A, and let [γ1 ], . . . , [γ2g ] be generators of π1 (A), given by considering 2g 1 1 the i-th component of Π2g i=1 S , for some homeomorphism Πi=1 S A. Since the induced map f∗ : π1 (X) → π1 (A) is surjective, for each i = 1, . . . , 2g, there exists an injective map ri : S 1 → X, so that f∗ (ri ) = γi . In particular, these maps induce an isomorphism π1 (A) → π1 (X)F .

206


The map f2g ◦ r : ΠSi1 → A, given by f2g ◦ r(z1 , . . . , z2g ) = morphism and therefore the continuous map

2g i=1

γi (zi ) is a homeo-

r = Πri : A ΠS 1 → S 2g X defines a topological section for f2g : S 2g X → A (and in fact, more generally for X 2g → A). Remark 4.2. The homotopy class of the section s above, is defined by the homotopy class of the corresponding subgroup π1 (X)F , generated by the elements ri : S 1 → X. Lemma 4.3. If dim X = n, then for N ≥ i, the natural imbedding S i X → S N X is a homotopy equivalence up to dimension i − 1. Proof. By lemma 3.6, the statement is true for the fundamental group of X. Moreover, it also holds for the homotopy groups H i (S i X), since it is generated by products of elements in H is (X) with i1 , . . . , ik so that their sum is equal to i and hence all such product are represented on S i X. In fact this is true on the level of complexes. Indeed the cells of dimension i in S N X are obtained from the cells i1 , ...., ik with sum equal to i in X. Thus each cell is the image of a product of at most i simplices from X and hence it comes from S i X. In particular the imbedding S i X → S N X is an homotopy equivalence up to dimension i − 1. By lemma 4.1, there exists m > 0 and a topological section s : A → S m X. In particular, fn : S n X → A is surjective for any n ≥ m. Let Rxn ⊆ S n X be the fiber of fn : S n X → A over a point x ∈ A. Moreover, let Sñ X → S n X be the abelian cover induced by the universal cover Cg → A, and let fñ : Sñ X → Cg . In particular, f˜m admits a topogical section s˜ : Cg → S˜m X, obtained as a cover of s(A). Moreover the natural map S n X × S k X → S n+k X, can be lifted to the map Sñ X × S˜k X → Sñ+k X. Lemma 4.4. The natural imbedding in : Rxn → Sñ X is an homotopy equivalence up to dimension n − m − 1. Proof. By lemma 4.3, the k-skeleton of S˜m X can be contracted to any subvariety S˜k+1 X × c ⊆ S˜m X where c ∈ S m−k−1 X is any cycle. Consider the map φk : S˜k+1 X → S˜k+1+m X which maps p ∈ S˜k+1 to p · s˜(−f˜k+1 (p)) ∈ S˜k+1+m .


207

Thus, we have s(−f˜k+1 (p))) = 0. f˜k+m+1 (φk (p)) = f˜k+1 (p) + f˜m (˜ ˜ 0k+m+1 . Therefore φk maps S k+1 X inside R Moreover it is homotopy equivalent to the standard imbedding. In fact the map (t, p) → p · s(−tf˜k+1 (p))

for 0 ≤ t ≤ 1

defines the homotopy equivalence. Thus S˜k+1 X ⊆ S˜k+m+1 X is homotopy equivalent to its image in R0k+m+1 , and therefore any nontrivial homotopy in S˜k+m+1 is the same as in R0k+m+1 , up to dimension k. This implies the lemma. Remark 4.5. The same result and proof applies for any continuous map a : S → T from a topological space S to a torus T , provided that the induced map a# : π(S) → π(T ) is surjective and the map π(S)ab → π(T )ab is an isomorphism. Let U ⊆ X be an open smooth subvariety of X, so that the natural map π1 (U ) → π1 (X) is a surjection. Let RUxn ⊆ Rxn be the fiber for the induced map S n U → A. In particular, RUxn is quasi-smooth, i.e. it has only quotient singularities and, by the same arguments used in lemma 4.4, the fibers RUxn are homotopically equivalent up to dimension n − m − 1. Thus, if n > m + 1 then the fundamental group of RUxn is abelian and equal to the kernel of the map π1 (U ) → π1 (A). Fixed x ∈ A, let M be the union of RUxn with the intersection of Rxn and the image of the map S n−1 U × XSing → S n X then the resulting variety M ⊂ Rn (X) has the following properties: (1) π1 (M ) = π1 (Rxn ); (2) codim(Rxn \ M ) > dim X; (3) M is quasi-smooth outside a singular subset which is locally isomorphic to SingX×D, where D is a polydisk. Thus if we take now a complete intersection of M of dimension equal to the dimension of X then the resulting variety X will have isolated singularities which are the same as X and π1 (X ) = π1 (M ) and hence it is a finite abelian group. This finishes the construction and the proof.

References [1] M. Artin: “On the solutions of analytic equations”, Invent. Math., Vol. 5, (1968), pp. 277–291. [2] M. Artin: “Algebraic approximation of structures over complete local rings”, Publ. Math. I.H.E.S., Vol. 36, (1969), pp. 23–58.

208


[3] F.A. Bogomolov and T. Pantev: “Weak Hironaka Theorem”, Math. Res. Let., Vol. 3, (1996), pp. 299–307. [4] C. Ciliberto and S. Greco: “On normal surface singularities and a problem of Enriques”, Commun. Algebra, Vol. 28(12), (2000), pp. 5891–5913. [5] C. Epstein and G. Henkin: “Stability of embeddings for pseudoconcave surfaces and their boundaries”, Acta Math., Vol. 185(2), (2000), pp. 161–237. [6] M J. Mather: Notes on Topological Stability, Mimeographed Notes, Harvard University, 1970. [7] D. Morrison: “The birational geometry of surfaces with rational double points”, Math. Ann., Vol. 271, (1985), pp. 415–438. [8] R. Hartdt: “Topological Properties of subanalytic sets”, Trans. Amer. Math. Soc., Vol. 211, (1975), pp. 193–208. [9] L. Lempert: “Algebraic approximations in analytic geometry”, Inv. Math., Vol. 121(2), (1995), pp. 335–353. [10] N. Levinson: “A polynomial canonical form for certain analytic functions of two variables at a critical point”, Bull. Am. Math. Soc., Vol. 66, (1960), 366–368. [11] F. Sakai: “Weil divisors on normal surfaces”, Duke Math., Vol. 51, (1984), pp. 877– 887. [12] F. Sakai: “The structure of normal surfaces”, Duke Math., Vol. 52, (1985), pp. 627– 648. [13] H. Whitney: Local Properties of Analytic Varieties, Differential and Combinatorial Topology, Princeton Univ. Press, Princeton N.J., 1965, pp. 205–244.

DOI: 10.2478/s11533-006-0003-z Research article CEJM 4(2) 2006 209–224

Subsheaves of the cotangent bundle Paolo Cascini∗ Department of Mathematics, University of California at Santa Barbara, Santa Barbara, CA 93106, USA

Received 19 January 2005; accepted 1 December 2005 Abstract: For any smooth projective variety, we study a birational invariant, defined by Campana which depends on the Kodaira dimension of the subsheaves of the cotangent bundle of the variety and its exterior powers. We provide new bounds for a related invariant in any dimension and in particular we show that it is equal to the Kodaira dimension of the variety, in dimension up to 4, if this is not negative. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Kodaira dimension, birational geometry MSC (2000): 14E05 14J35

1

Introduction

The attempt at classifying algebraic varieties has always been based on the study of the positivity (or negativity) of their cotangent bundles. One of the most important invariants of a smooth algebraic variety X defined over the complex numbers is its Kodaira dimension, which measures the number of global pluricanonical forms on X (see [12] for more details). A decade or so ago, Campana defined another important invariant for any smooth projective variety X (see e.g. [5]) : k + (X) = maxp {kod(X, det(F))} F ⊆ΩX

where the maximum is taken after considering all coherent subsheaves F ⊆ ΩpX of rank r for any r, p > 0. ∗

E-mail: [email protected]

210

P. Cascini / Central European Journal of Mathematics 4(2) 2006 209–224

Clearly k + (X) is strictly related to the Kodaira dimension of X and, in particular, we always have k + (X) ≥ kod(X). The equality does not hold in general (consider, for example, the surface X = P1 × C where C is any curve of genus g ≥ 1). On the other hand, it is natural to conjecture that for any non-uniruled projective variety, this invariant coincides with the Kodaira dimension of the variety. More precisely: Conjecture (Cn ). Let X be a smooth projective variety of dimension n and let r : X Z be the Maximal Rationally Connected (MRC) fibration associated to X. Then k + (X) = kod(Z). (e.g. see [4], for a construction of the MRC fibration) In [5], Campana showed that (Cn ) is a consequence of the minimal model program and the following Conjecture (Rn ). A smooth algebraic variety of dimension n is uniruled if and only if kod(X) = −∞. We recall that a smooth variety is said to be uniruled if, for any generic point, there exists a rational curve passing through it. At the moment the conjecture (Rn ) (and therefore (Cn )) is known to be true for n ≤ 3 [19]. A weaker version of the conjecture (Cn ) is obtained by considering only the line bundles L ⊆ ΩpX and the invariant k1+ (X) = maxp {kod(X, L))} L⊆ΩX

where L is any line bundle and p > 0 is arbitrary. Therefore, we have: Conjecture (Cn ). : Let X, Z be as above then k1+ (X) = kod(Z). The main result of this paper is Theorem 1.1. Let X be a projective variety of dimension ≤ 4 and with non-negative Kodaira dimension. Then k1+ (X) = kod(X).

Theorem 1.1 will be a consequence of the following results: Theorem 1.2. Let X be a smooth and non-uniruled projective variety and let L ⊆ Ωn−1 X be any invertible subsheaf such that kod(X, KX + L) ≥ dim(X) − 2. Then kod(X, L) ≤ max{0, kod(X)}. The requirement kod(X, KX +L) ≥ dim(X)−2 is justified by the fact that the relative


211

minimal model is known only for relative dimension ≤ 2. The theorem will follow from the generic semi-positivity of the cotangent bundle of X. On the other hand, we have: Theorem 1.3. Let X be a smooth projective variety with kod(X) = 0 and let L be an invertible subsheaf of ΩiX for some i > 0. If kod(X, L + KX ) = kod(X, L), then kod(X, L) ≤ dim X − 4. The last result is a generalization of Campana’s theorem on the “speciality” of varieties of zero Kodaira dimension (see [6]): Theorem 1.4. Let X be a smooth projective variety of dimension 4 and zero Kodaira dimension, and let L ⊆ ΩiX . Then kod(X, L) ≤ i − 2. The above bounds immediately imply the conjecture (C4 ) for varieties with zero Kodaira dimension. In fact, if X is a a 4-fold with kod(X) = 0 and k1+ (X) > 0, then theorem 1.4 rules out all the cases, except a possible sub-line bundle L ⊆ Ω3X of Kodaira dimension 1. By theorem 1.2, it follows that kod(X, KX + L) = 1, but that contradicts theorem 1.3. In section 6, we will see that it suffices to consider varieties with zero Kodaira dimension in order to prove (Cn ) for varieties with positive Kodaira dimension (see also [5]).

Acknowledgment The content of this paper forms a portion of the author’s PhD Thesis at New York University. I am very grateful to my advisor, prof. F.A. Bogomolov, for his guidance and constant support. I would also like to thank G. La Nave for an uncountable number of helpful discussions. A very special thanks goes to prof. F. Campana for his inspiring preprint in November 2001 and to the referee for his very helpful advice and for pointing out many inaccuracies in the first version of this paper. I would also like to point out that many of the results of this paper were obtained independently by Campana and Peternell [7].

2

A Positivity Result

The aim of this section is to prove: Theorem 2.1. Let X be a non-uniruled variety and let L be a line bundle that is a quotient of Ω1X . Then L is pseudo-effective. We recall that a line bundle is said to be pseudo-effective if it is a limit of line bundles with non-zero sections. In order to prove the theorem, we will use a recent characterization of pseudo-effective

212


line bundles [2]: Theorem 2.2. [Boucksom - Demailly - Paun - Peternell] A line bundle L is pseudo-effective if and only if L · C ≥ 0, for every moving curve C, i.e. for every curve C that is a member of a family Ct that covers X. Theorem 2.1 can be seen as a variation of Miyaoka’s generic semi-positivity theorem [19]: Theorem 2.3. Let X be a normal projective variety of dimension n that is not uniruled, and let H1 , . . . , Hn−2 be ample Cartier divisors on X. If m1 , . . . , mn−1 are any sufficiently large integers, then the restriction of Ω1X to any general complete intersection curve C cut out by the linear systems |mi Hi |, is a semi-positive vector bundle. Miyaoka’s theorem implies that on a non-uniruled projective variety X, every line bundle that is a quotient of Ω1X is non-negative on a generic complete intersection curve. Unfortunately, this alone does not imply the pseudo-effectiveness of the line bundle. In fact, Demailly, Peternell and Schneider [8] constructed an example of a non-pseudoeffective line bundle that has non-negative degree on any generic complete intersection curve, by considering X = P(TK ) to be the projectivization of the tangent bundle of a generic quartic surface K ⊆ P3 and L = O(1) be the associated line bundle on X. On the other hand, the proof of theorem 2.1 closely follows the proof of Miyaoka’s theorem. In fact, we will reduce X modulo p so that the line bundle L in theorem 2.1, defines a foliation on X, and thereafter apply Ekedahl’s theory of foliations over a field in positive characteristic [9]. In order to do that, we recall some basic definitions. Let X be a normal projective variety defined over a field k. A saturated subsheaf of the tangent bundle of X, F ⊆ TX , defines a foliation, denoted by (X, F), if it is closed under Lie brackets. For any foliation, there exists an immersion: j : (X0 , F0 ) → (X, F) such that X \ X0 has codimension 2, X0 is non-singular, and F0 = F|X0 is locally free. Moreover we define KF = j∗ (c1 (F0∗ )) If k is a field of characteristic p > 0, there exists a natural map: F0p → TX0 /F0 . We say that (X, F) is p-closed if such a map is zero. Theorem 2.4 (Ekedahl). Given a normal variety X in characteristic p > 0, there is a one-to-one correspondence between: (1) p-closed foliations with rank r (2) Factorizations ρ X → Y → X (1) (1)


213

of the geometric Frobenius map F : X → X (1) with deg ρ = pr . In particular, if F ⊆ TX is a saturated subsheaf such that HomOX0 (∧2 F, TX /F) = 0

and

HomOX0 (F ∗ F, TX /F) = 0

then (X, F) defines a p-closed foliation and therefore a factorization as in (1). We will write the quotient Y as X/F and we have: ρ∗ KX/F = (p − 1)KF + KX .

(2)

To proceed with the proof of 2.1, let us reduce X modulo p. Any line bundle L that is a quotient of Ω1X , defines a foliation given by F = L∗ ⊆ TX . Suppose that there exists a moving curve C such that LC < 0. Then HomOC (F ∗ F, TX /F) = HomOC (L−p , TX /F) = H0 (C, TX /F ⊗ Lp ⊗ OC ) = 0 provided that p >> 0. Since C is a moving curve, we have: Hom(F ∗ F, TX /F) = 0 and therefore F is a p−closed foliation for any sufficiently large p that defines a map ρ : X → Y = X/F as in (1). By (2), it follows that ρ∗ KY C = (p − 1)LC + KX C. Since LC < 0, by bend-andbreak we have that for any p sufficiently large, there exists a rational curve L through a general point of ρ(C), and since ρ is purely inseparable, L pulls back to a rational curve passing through a generic point of X. Therefore, X must be uniruled and this completes the proof of theorem 2.1.

3

Proof of theorem 1.2

Let X be a smooth projective variety of dimension n that is non-uniruled and let L be an invertible subsheaf of Ωn−1 X . ∗ From the isomorphism Ωn−1 TX ⊗ ωX , it follows that L ⊗ ωX ⊆ TX defines a X foliation on X and therefore, by theorem 2.1, KX − L is a pseudo-effective divisor, i.e. L · C ≤ KX · C, for any moving curve C (theorem 2.2). Lemma 3.1. Suppose that X is a non-uniruled smooth variety that is not of general type and let L ⊂ Ωn−1 X . Then L + KX is not big. Proof. If L + KX is big, then by the pseudo-effectiveness of KX − L, it follows that 2KX = (KX − L) + (KX + L) is big, for the cone of big divisors is the interior of the cone of pseudo-effective divisors on X. Thus, X is of general type. Let us suppose now, as in the hypothesis of theorem 1.2, that kod(X, KX + L) ≥ dim(X) − 2.

214


We need to show that kod(X, L) ≤ max{0, kod(X)}. Let φ : X → Z be the Iitaka fibration of KX + L. Without loss of generality, by taking a suitable modification of X, we may suppose that φ is a regular morphism such that for any generic fiber Xz of φ, we have kod(Xz , Lz + KXz ) = 0. where Lz is the restriction of L to the generic fiber Xz . We may also suppose that X is not of general type and that kod(X, L) ≥ 0, otherwise there would be nothing to prove. By lemma 3.1, the fibration φ is not an isomorphism. Moreover, by the assumption with regard to the Kodaira dimension of KX + L (i.e. kod(X, KX + L) ≥ dim(X) − 2), we have that the generic fiber of φ is either a non-rational curve or a non-uniruled surface, otherwise X would be uniruled. Moreover, kod(L) ≥ 0 implies that kod(Xz , Lz ) ≥ 0 for the generic point z ∈ Z, and therefore kod(Xz ) ≤ kod(Xz , Lz + KXz ) = 0 and by the classification of curves and surfaces, it follows immediately that the generic fiber of φ has zero Kodaira dimension. Thus, since kod(Xz , Lz +KXz ) = kod(Xz ) = 0, it follows that kod(Xz , Lz ) ≤ kod(Xz , Lz + KXz ) = 0, and since kod(X, L) ≥ 0, it follows that kod(Xz , Lz ) must vanish for the generic point z ∈ Z. This implies that the Iitaka fibration associated to L must factor through φ. Hence, without loss of generality we may suppose that L = φ∗ L1 where L1 is a Q−divisor on Z. Let us first assume that Xz is a curve, then φ : X → Z is an elliptic fibration. Consequently, the canonical bundle of X is contained in the fibers of φ, i.e. there exists a divisor D on Z such that φ∗ D − KX is effective and kod(X) = kod(Z, D). We wish to show that D is a big divisor on Z. Theorem 1.2 will follow immediately. In fact, kod(X, L) = kod(Z, L1 ) ≤ dim Z = kod(Z, D) = kod(X). To prove the claim, we first notice that, since φ∗ (D − L1 ) = (φ∗ D − KX ) + (KX − L) is a sum of an effective divisor with a pseudo-effective one, φ∗ (D − L1 ) and therefore, D − L1 , must be pseudo-effective. Moreover, since kod(X, KX + L) = dim Z, we have that D + L1 is big on Z, and therefore 2D = (D − L1 ) + (D + L1 ) is a sum of a pseudo-effective divisor with a big divisor. Thus D is big. Now, let Xz be a surface. The situation is very similar. In fact, we have the following relative version of the minimal model program for algebraic surfaces (e.g. see [18], pag. 161):


215

Lemma 3.2. Let φ : S → B a family of algebraic surfaces such that a generic fiber Sb is a surface with zero Kodaira dimension. φ Then, there exists a rational map S → S to a smooth variety S → B over B, such that the generic fiber of φ is a surface with numerically trivial canonical bundle. From the lemma it follows that we may suppose, without loss of generality, that the generic fiber of f has numerically trivial canonical line bundle. In fact, let π : X → X , be a rational map over Z as in the lemma, and let φ : X → Z, be the associated map. As in the case (?) of the elliptic fibration, there exists a divisor D on Z and a positive integer m, such that φ ∗ (D) − mKX is effective, and kod(X) = kod(X ) = kod(Z, D). If L = φ ∗ (L1 ), as in the previous claim, it follows that D is big and therefore kod(X, L) = kod(X, L ) ≤ kod(X).

4


Let X be a smooth variety with zero Kodaira dimension. We may assume, without loss of generality, that Pg (X) = dim H0 (X, KX ) = 1, in fact for any smooth variety X with kod(X) ≥ 0 , there exists a cyclic cover π : X → X, such that Pg (X ) > 0, and such that KX is a multiple of π ∗ (?)KX . Therefore, kod(X ) = kod(X) (see [13], pag.263). Before proceeding with the proof of theorem 1.3, we need some preliminary results. The following is a theorem of Griffiths [11] (see also [25]): Theorem 4.1. Let f : Z → C be an algebraic fibration from a smooth projective variety Z to a smooth curve C. Let B ⊂ C be such that f is smooth over C \ B and D = f −1 (B) is a reduced normal crossing divisor on Z. If m is the dimension of the generic fiber of f , then, for any q = 1, . . . , m, any invertible sub-sheaf of Rm f∗ ΩqZ|C (log D) has non-positive degree on C. Let Z be a smooth projective variety of dimension n. We will say that Z admits a perfect complex Poincaré pairing, if the natural map 0 H0 (Z, ΩiZ ) × H0 (Z, Ωn−i Z ) → H (Z, ωZ )

(3)

is a perfect pairing, i.e. for any non-zero η ∈ H0 (Z, ΩiZ ), the linear map ·∧η : H0 (Z, Ωn−i Z ) → 0 H (Z, ωZ ) is not identically zero. Lemma 4.2. [Complex Poincaré Pairing] Any smooth variety Z of dimension ≤ 3, with zero Kodaira dimension and with Pg (X) = 1, admits a perfect complex Poincaré pairing. This lemma was proved by Bogomolov in the special case KZ = 0, and it should hold, at least conjecturally, for any variety of zero Kodaira dimension, as a consequence of the minimal model program. On the other hand, by taking a generic hyper-surface of K × K, where K is a K3 surface, it can be easily shown that such a statement does not hold for

216


varieties of higher Kodaira dimension, even in the case Pg (X) > 0. Proof. The only interesting case is when X is a three-fold. Let us first review the proof in the case KZ = 0 (the same proof would hold in general for any dimension). Let ω ∈ H0 (Z, ωZ ) be a non-zero canonical form, that is, by assumption, nowhere zero. Any non-zero (i, 0)-form s ∈ H0 (Z, ΩiZ ) corresponds, by the Hodge theorem, to a non-zero (3 − i, i)-form s ∈ H3 (Z, Ω3−i ¯ , it can Z ). On dividing s by ω ω is a (3 − i)-holomorphic form that is not homologous to zero. be easily shown that s /¯ Therefore, we have defined an isomorphism H0 (Z, ΩiZ )−→ ˜ H0 (Z, Ω3−i Z ) which implies the exactness of the pairing (3). Let us consider now the more general case of a three-fold Z with kod(Z) = 0 and with Pg (Z) = 1. The minimal model program implies that Z is birational to a product of an abelian variety with a simply-connected variety Z with only terminal singularities and with numerically trivial canonical bundle (see e.g. [17]). Let us suppose that Z is a three-fold. By Namikawa’s theorem [21], Z is smoothable, i.e. there exists a family Z over a disc Δ such that the central fiber is isomorphic to Z and with Z smooth. Since terminal singularities are rational, by Steenbrink’s theorem [22] it follows that Hi (OZ ) is naturally isomorphic to Hi (OZη ), where Zη is a generic member of the family Z. Since Zη is smooth, the claim follows from the previous case. We are now ready to proceed with the proof of theorem 1.3. Let us consider a line bundle L ⊆ ΩiX of Kodaira dimension k > 0 such that kod(X, L+ KX ) = kod(L) as per the hypothesis of the theorem. Let φ : X → Y be the Iitaka map associated to L, with dim Y = k. We may assume once again that φ is a regular morphism and that there exists a big Q-divisor H on Y ˆ → X be a modification, for which the Iitaka such that φ∗ (H) = L. In fact, let π : X ˆ → Y of π ∗ L is a regular morphism. Then we can find an effective divisor fibration φˆ : X ˆ and a big divisor H on Y , such that L := π ∗ L − A = φˆ∗ (H). We want to show A on X ˆ L + K ˆ ) = kod(L ). Obviously we have that kod(X, ˆ L + K ˆ ) ≥ kod(L ). that kod(X, X X ˆ → X, then On the other hand, if E is the exceptional divisor for π : X ˆ L ) = kod(X, ˆ π ∗ L) = kod(X, ˆ π ∗ (L + KX )) = kod(X, ˆ π ∗ (L + KX ) + E) ≥ kod(X, ˆ L + K ˆ ). = kod(X, X

ˆ = X and L = L. Thus, we can suppose that X Lemma 4.3. Under the assumptions of theorem 1.3, the generic fiber Xy of φ has zero Kodaira dimension. Proof. Let η : X → Y be the Iitaka fibration associated to the divisor KX + L defined on a smooth modification X of X. We have, by assumption, dim(Y ) = kod(X, KX + L) = kod(X, L) = dim(Y ). The restriction of KX + L to the generic fiber has zero Kodaira dimension and since that restriction of KX to such a fiber is Q-effective, it follows that the restriction of L to


217

the generic fiber of η has zero Kodaira dimension. Thus, by theorem 10.6 in [12], it follows that η coincides with the fibration φ. In particular, the generic fiber of φ will have zero Kodaira dimension.

By lemma 4.2, Theorem 1.3 will be a consequence of the following: Lemma 4.4. Let f : X → Y be an algebraic fibration such that its generic fiber Xy has zero Kodaira dimension and Pg (Xy ) = 1 and let H ∈ Pic Y be a big divisor, such that if L = f ∗ H, there exists an embedding L ⊆ ΩiX . Then dim(Y ) ≤ kod(X). Proof. Let k = dim Y and let C0 be the generic member of a family of moving curves on Y (if Y is a curve, we just take C0 = Y ). In particular, we may assume that C0 is not contained in the singular locus of f . Moreover, by theorem 1.5 in [2], we may suppose that the curve C0 is strongly movable, i.e. there exists a modification μ : Y˜ → Y , and very ample divisors A˜1 , . . . , A˜k−1 such that μ∗ C = C0 , where C = A˜1 ∩ · · · ∩ A˜k−1 . Let g : Z → C be the fibration over C, obtained as the pull-back of the fibration f on C. By taking a semi-stable reduction of g, we can construct a ramified cover v : C → C, such that if Z is a desingularization of Z ×C C , then the induced map g : Z → C does not have multiple fibers [14]. Moreover, if u : Z → Z is the induced cover, then it is possible to check with a local computation, that there exists an isomorphism in codimension 1 u∗ ΩqZ|C (log D) ΩqZ |C (log D ). where C0 is the locus on which g is smooth, D is the inverse image of C \ C0 and D = u−1 (D). Let ψ : Z → X be the map induced by μ ◦ v. If M = ωX ⊗ L∗ on X, and M = ψ ∗ M on Z , we want to show Claim: There exists an embedding ∗ M ∗ → Ωi−k Z |C (log D ) ⊗ ωZ |C . ∗ and therefore M is a quotient of Ωi−k Z |C (log D) ⊗ ωZ |C . Let B be a divisor on Y such that f is smooth on Y \B, and let S = f −1 (B) = ai Si . By Hironaka’s theorem, we can suppose that B and S are normal crossing(?). Let S = (ai − 1)Si . We will first show that there exists an embedding ∗ L → Ωi−k X|Y (log S) ⊗ f ωY ⊗ O(S ),

(see also lemma 5.1).

(4)

218


Over X \ S, we choose local coordinates z1 , . . . , zn , and so we are enabled to write f (z1 , . . . , zn ) = (z1 , . . . , zk ). Moreover, by assumption on L, we can suppose that there exist k + 1 analytically independent local sections of L, s0 , . . . sk such that si = zi s0 . Since Z is a Kähler manifold, by the “covering trick” (see [1]) the holomorphic forms si are closed and therefore we have: 0 = dsi = dzi ∧ s0 + zi ds0 = dzi ∧ s0 , i.e. dzi divides s0 and locally we have s0 = dz1 ∧ . . . dzk ∧ w, where w is a i − k form, that is a local section of Ωi−k X|Y (log S). The same local computation shows, more in general, that the embedding in (4) holds in an open set of codimension 1 in X. Thus, we have ∗ M ∗ → Ωi−k X|Y (log S) ⊗ ωX|Y ⊗ O(S ).

From the exact sequence 0 → f ∗ Ω1Y (log B) → Ω1X (log S) → Ω1X|Y (log S) → 0, it follows det Ω1X|Y (log S) ωX|Y ⊗ O(−S ). Hence, the isomorphism in codimension 1 ψ ∗ (Ω1X|Y (log S)) Ω1Z |C (log D ) implies the claim. The generic fiber F of g has zero Kodaira dimension and Pg (F ) = 1. Therefore by lemma 4.2, it admits a perfect complex Poincaré pairing. Thus, by considering the restriction to the generic fiber of g , it follows that the map ∗ g∗ (Ωi−k Z |C (log D ) ⊗ ωZ |C ) → g∗ M

induced by the quotient map, given by the claim above, is not trivial. By Steenbrink’s theorem [22], the sheaves Rq g∗ Ωi−k Z |C (log D ) are locally free, and therefore, by Grothendieck-Serre duality, there exists an isomorphism n−k i−k ∗ g∗ (Ωi−k g∗ ΩZ |C (log D ))∗ Z |C (log D ) ⊗ ωZ |C ) (R

Thus, the map ∗ (Rn−k g∗ Ωi−k Z |C (log D )) → g∗ M

is generically surjective. Since n − k is the dimension of the generic fiber of g , it follows, by theorem 4.1, that Rn−k g∗ ΩqZ |C (log(D ))∗ is a semi-positive vector bundle, and in particular g∗ M has positive degree on C . Thus, by theorem 2.2, M is a pseudo-effective divisor on X. Since H is big on Y , it follows that if E = M + 12 L, then 1 1 kod(X, E) = kod(X, M + L) = kod(Y, f∗ M + H) ≥ 0, 2 2


219

and therefore kod(X) = kod(X, E + 12 L) ≥ kod(X, L) = dim Y .

In the proof of lemma 4.4, we used only the fact that the generic fiber of φ admits a perfect complex Poincaré parity. Therefore, lemma 4.4 can be stated in a more general way: Proposition 4.5. Let φ : X → Y be an algebraic fibration such that its generic fiber admits a perfect complex Poincaré pairing and let H ∈ Pic Y be a big divisor, such that if L = φ∗ H, there exists an embedding L ⊆ ΩiX . Then dim(Y ) ≤ kod(X). We will use this statement again later on.

5


In [1], Bogomolov proved that in a smooth projective variety X, any sub-line bundle L ⊆ ΩiX has Kodaira dimension less than i (this statement is known to be false in characteristic p > 0, see [16]). The idea of the proof was to consider the Iitaka map φ : X → Y associated to L and show that any global form of L is monomial (see definition below) and induced by a form in Y . Campana [6] improved the inequality for a variety X with zero Kodaira dimension (or, in greater generality, when X is special), by showing that for any line bundle L ⊆ ΩiX , we have kod(X, L) < i. Inspired by the above results, we are going to prove that under the same assumption on the Kodaira dimension of X, we have kod(X, L) ≤ i − 2 (Campana’s conjecture would imply kod(X, L) ≤ 0). For the sake of completeness, we also sketch the proof of the above results. A line bundle L ⊆ ΩiX is said to be monomial if any global section of L⊗k can be k written in a neighborhood of a generic point p ∈ X, as φ(z)(dz1 ∧ . . . dzi )⊗ , where φ is a holomorphic function. Lemma 5.1. Let L ⊆ ΩiX such that kod(X, L) ≥ i − 1. Then L is monomial. Proof. Let φ : X → Y be the Iitaka fibration associated to L. We can suppose, without loss of generality, that φ is a regular map. Moreover, for the covering trick [1], we need only consider the global sections of L; in fact for any section s ∈ H0 (X, L⊗k ), there exists a cyclic covering π : X → X and a section t ∈ H0 (X , π ∗ L) such that π ∗ s = t⊗k . For the same reason, we can suppose that φ is the map defined by the global sections of L. On an analytic neighborhood of a generic point p ∈ X, we choose local coordinates so that the map φ has the form φ(x1 , . . . , xn ) = (x1 , . . . , xp ), where p = dim Y = kod(X, L). Thus, for any section s0 of L and for any j = 1, . . . , p, sj = xj s0 is also a section of L. Since

220


any holomorphic p−form on X is closed, we have that 0 = dsj = dxj ∧s0 +xj ds0 = dxj ∧s0 , i.e. dxj divides s0 and since p ≥ i − 1 we can write s0 = dx0 ∧ dx1 ∧ · · · ∧ dxi−1 ∧ ω, for some holomorphic 1−form ω on X. Thus L is monomial.

From the proof of the previous lemma, it follows immediately that for any line bundle L ⊆ ΩiX , we have kod(X, L) ≤ i. Let us suppose now that the equality holds and let φ : X → Y be its Iitaka fibration. By Hironaka’s theorem, we can suppose that φ is a prepared morphism, i.e. it is a regular map and, the locus S where φ is not smooth and its inverse image φ−1 (S) are contained inside simple normal crossing divisors in Y and X respectively. By lemma 5.1 (see also [6], thm 2.25), L = φ∗ (KY ) at a generic point of Y . By [13] and [24] (see also prop. 4.15 in [6]), there exists a finite and flat covering v : Y → Y , with Y smooth and such that if φ : X → Y is the map obtained by v, smoothing the base change of φ, then there exists an injection φ∗ (KX /Y ) → v ∗ (φ∗ (KX − L)). It follows that φ∗ (KX − L) is semi-positive, and in particular (lemma 4.10 in [6]) we have kod(X) ≥ kod(X, L) = i > 0 . Let us suppose now that kod(L) = i − 1 and let us still denote by φ : X → Y the fibration associated to L. As in the previous case, we can suppose that φ is a prepared morphism. Lemma 5.1 implies that any section s of L can be locally written as s = φ∗ θ ∧ ω, where θ is an (i − 1)-form of Y , i.e. a section of ωY and ω is a 1-form on X that defines a global 1−form on a generic fiber of φ. In particular, it follows that L ⊆ φ∗ ωY ⊗ Ω1X/Y , at the generic point of Y . Every generic fiber of φ admits a global 1-form, and therefore it admits a non-trivial β γ relative Albanese map α : X → A (see [3]). Let α : X → Z → A be its Stein Factorization, where β is an algebraic fibration and γ is a finite map onto its image. We have the following diagram: α β

/Z ~ ~ ~ φ ~~π ~~ ~

X

γ

/A

Y

We will distinguish two different cases: First Case: dim Z < dim X, i.e. β is not an isomorphism. 1 Since L ⊆ φ∗ Ωi−1 Y ⊗ ΩX/Y , from the exact sequence of sheaves: 0 → β ∗ Ω1Z/Y → Ω1X|Y → Ω1X/Z → 0


221

and by the definition of Albanese map, we have that L ⊆ φ∗ ωY ⊗ β ∗ Ω1Z/Y = β ∗ (π ∗ ωY ⊗ Ω1Z/Y ).

(5)

Since φ is the Iitaka fibration associated to L, we can suppose that L = φ∗ L0 , where L0 is a big Q-divisor on Y . Therefore, if L1 = π ∗ L0 , then L = β ∗ L1 and, by (5), it follows that L1 ⊆ π ∗ ωY ⊗ Ω1Z/Y → ΩiZ . Moreover, kod(Z, L1 ) = kod(X, L) = i − 1. Let us consider now the maximal rationally connected fibration η : Z Z , such that its generic fiber is rationally connected. By the results in [10], it follows that Z is non-uniruled. Since m = dim Z < dim X, it follows that (Rm ) holds. Moreover, Iitaka’s conjecture (Cn,m ) holds true for any n ≤ 4 (e.g. see [20]). Therefore, since there exists a fibration from X onto Z , we have: 0 = kod(X) ≥ kod(Z ) ≥ 0, i.e. Z has zero Kodaira dimension. If we can show that there exists a line bundle L2 ⊆ ΩjZ for some j > 0 such that kod(Z , L2 ) = kod(X, L), then we can conclude by induction that kod(X, L) = kod(Z , L1 ) = 0. The claim follows immediately by considering the exact sequence 0 → NZ∗t |Z → i∗ Ω1Z → Ω1Zt → 0 where NZt |Z OZ⊕dt is the normal bundle of i : Zt → Z, and by the fact that the generic fiber of η is rationally connected and therefore H0 (Zt , (ΩiZt )k ) = 0 for any i, m > 0 (e.g. see [17]).

Second Case: dim Z = dim X, i.e. α is a finite map, We want to show that this implies that a generic fiber of φ admits a perfect complex Poincaré pairing as in lemma 4.2. Lemma 5.2. Let W be a smooth variety such that its Albanese map α : W → Alb(W ) is finite onto its image α(W ). Then W admits a perfect complex Poincaré pairing. Proof. Let 0 = η ∈ H0 (ΩiW ). We want to show that the linear map 0 · ∧ η : H0 (W, Ωm−i W ) → H (W, ωW )

is not zero, where m is the dimension of W . Since the Albanese map α is finite onto its image, we have that Ω1W is generically globally generated, i.e. the natural map H0 (W, Ω1W ) ⊗ OW −→ Ω1W

222


is surjective on an open set of W . In fact, if W = α(W ) is the image of the Albanese map of W and A = Alb W , then the map OW ⊗ H0 (W , Ω1W ) = Ω1A |W −→ Ω1W is surjective and therefore the sheaf Ω1W is globally generated. Since α is by assumption a finite map, the claim follows. In particular, it follows that the map ∧j H0 (W, Ω1W ) ⊗ OW −→ ΩjW is surjective for any i = 1, . . . , m. Therefore, if p ∈ W is a generic point, we can suppose that η is locally monomial, i.e. there exist local coordinates z = (z1 , . . . , zm ) and an holomorphic function φ, around a point p ∈ W , such that η = φ(z)dz1 ∧ · · · ∧ dzi . Thus there exist linearly independent global 1-forms ω1 , . . . , ωm ∈ H0 (Ω1W ), such that, in a neighborhood of p, ωj = dzj , for any j = 1, . . . , m, and such that the m−form ω1 ∧ · · · ∧ ωm is not zero. In particular, there exists η ∈ H0 (Ωm−i ), defined by η = ωi+1 ∧ · · · ∧ ωm , such that η ∧ η is not zero. The proof of theorem 1.4 in this second case, follows therefore from prop. 4.5.

6

Conclusions

As explained in the introduction, theorems 1.2, 1.3 and 1.4 imply Lemma 6.1. Let X be a projective variety of dimension 4 and with zero Kodaira dimension. Then k1+ (X) = 0, i.e. any invertible subsheaf L ⊆ ΩiX has non-positive Kodaira dimension. We can now proceed with the proof of the main result of this paper. Proof (Proof of theorem 1.1). It follows immediately from the definition that k1+ (X) ≥ kod(X). Let φ : X → IX be the Iitaka fibration of X. Then dim IX = kod(X) > 0. Let Xz be a generic fiber of φ, then Xz is connected and kod(Xz ) = 0. By theorem 6.1, we have k+ (Xz ) = 0. Let L ⊆ ΩiX (for some i > 0) be a line bundle with positive Kodaira dimension. We want to show Claim: kod(Xz , L|Xz ) ≤ 0. The theorem will be a consequence of the claim. In fact, it implies that h0 (Xz , Ln|Xz ) ≤ 1 for any n and therefore φ∗ Ln is a sheaf of rank at most 1. Thus, h0 (X, Ln ) = h0 (IX , φ∗ Ln ) ≤ ank + b, with a, b positive integers, and k = dim(IX ) = kod(X). Thus kod(X, L) ≤ kod(X) that concludes the proof.


223

In order to prove the claim, we consider the exact sequence 0 → NX∗ z |X → i∗ Ω1X → Ω1Xz → 0. ⊕d is the normal bundle of i : Xz → X, with d = kod(X). where NXz |X OX z By taking the i−th exterior power, we get a long exact sequence: ⊕d ⊕d → S i−1 OX ⊗ i∗ Ω1X → · · · → i∗ ΩiX → ΩiXz → 0, 0 → S i OX z z

where S • denotes the symmetric power. Since L is contained in ΩiX , it follows that L|Xz must be contained in ΩjXz for some j ≤ i, and therefore, by lemma 6.1, kod(Xz , L|Xz ) ≤ k1+ (Xz ) = 0.

References [1] F. Bogomolov: “Holomorphic Tensors and Vector Bundles on Projective Varieties”, Math. USSR Izv., Vol. 13, (1979), pp. 499–555. [2] S. Boucksom, J.P. Demailly, M. Paun and T. Peternell: “The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension”, math.AG/0405285. [3] F. Campana: “Réduction d’Albanèse d’un morphisme propre et faiblement kählérien. II. Groupes d’automorphismes relatifs”, Compositio Math., Vol. 54(3), (1985), pp. 399–416. [4] F. Campana: “Connexité rationelle des variétés de Fano”, Ann. Sci. E.N.S., Vol. 25, (1992), pp. 539–545. [5] F. Campana: “Fundamental Group and Positivity of Cotangent Bundles of Compact Kähler Manifolds”, J. Algebraic Geom., Vol. 4, (1995), pp. 487–502. [6] F. Campana: “Orbifolds, Special Varieties and Classification Theory”, Ann. Inst. Fourier, Grenoble, Vol. 54(3), (2004), pp. 499–630. [7] F. Campana and T. Peternell: “Geometric Stability of the Cotangent Bundle and the Universal Cover of a Projective Manifold”, math.AG/0405093. [8] J.P. Demailly, T. Peternell and M.Schneider: “Pseudo-effective Line Bundles on compact Kähler Manifolds”, Intern. J. Math., Vol. 12(6), (2001), pp. 689–741. [9] T. Ekedahl: T. Ekedahl: “Foliations and inseparable morphisms” (english), In: Algebraic geometry, Proc. Summer Res. Inst., (Brunswick/Maine 1985), Proc. Symp. Pure Math., Vol. 46(2), Amer. Math. Soc., Providence, RI, 1987, pp. 139–149. [10] T. Graber, J. Harris and J. Starr: “Families of rationally connected varieties”, J. Amer. Math. Soc., Vol. 16, (2003), pp. 57–67. [11] P. Griffiths: “Periods of Integrals on Algebraic Manifolds III”, Publ. Math. I.H.E.S., Vol. 38, (1970), pp. 125–180. [12] S. Iitaka: Algebraic Geometry, Graduate Texts in Math., Vol. 76, Springer, 1982. [13] Y. Kawamata: “Characterization of Abelian Varieties”, Comp. Math., Vol. 43, (1981), pp. 253–276.

224


[14] G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat: Toroidal Embeddings I, Lectures Notes in Math., Vol. 339, Springer Verlag, 1973. [15] J. Kollár: “Higher Direct Images of Dualizing Sheaves II”, Ann. Math., Vol. 124, (1986), pp. 171–202. [16] J. Kollár: “Nonrational Hypersurfaces”, J. Am. Math. Soc., Vol. 8(1), (1995), pp. 241–249. [17] J. Kollár: Shafarevich maps and automorphic forms, Princeton University Press, 1995. [18] K. Matsuki, Introduction to the Mori program, Springer-Verlag, New York, 2002. [19] Y. Miyaoka: “The Chern classes and Kodaira dimension of a minimal variety”, In: Proc. Sympos. Alg. Geom., Sendai 1985, Adv. Stud. Pure Math, Vol. 10, Kynokuniya, Tokyo, 1985, pp. 449–476. [20] S. Mori: “Classification of higher-dimensional varieties”, In: Algebraic geometry, Bowdoin 1985 (Brunswick/Maine 1985), Proc. Symp. Pure Math., Vol. 46(1), Amer. Math. Soc., Providence, RI, 1987, pp. 269–331. [21] Y. Namikawa: “On deformations of Calabi-Yau 3-folds with terminal singularities”, Topology, Vol. 33(3), (1994), pp. 429–446. [22] J.H.M. Steenbrink: Mixed Hodge Structure on the Vanishing Cohomology, Real and Complex Singularities, Nordic Summer School, Oslo, 1976, pp. 525–563. [23] K. Ueno: Classification Theory of Algebraic Varieties and Compact Complex Spaces, Lectures Notes in Math., Vol. 439, Springer Verlag, 1975. [24] E. Viehweg: “Die Additivität der Kodaira Dimension f¨ ur projektive Faserräume u ¨ber Varietäten des allgemeinen Typs”, J. Reine Angew. Math., Vol. 330, (1982), pp. 132–142. [25] E. Viehweg and K. Zuo: “On the isotriviality of families of projective manifolds over curves Complex Spaces”, J. Alg. Geom., Vol. 10, (2001), pp. 781–799.

DOI: 10.1007/s11533-006-0001-1 Research article CEJM 4(2) 2006 225–241

A poset hierarchy∗ Mirna Dˇzamonja1† , Katherine Thompson2‡ 1

School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK 2

University of Vienna, W¨ ahringerstrasse 25, 1090 Wien, Austria

Received 7 March 2005; accepted 5 December 2005 Abstract: This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of scattered linear orders. They gave an inductively defined hierarchy that characterised the class of scattered posets which do not have infinite incomparability antichains (i.e. have the FAC). We define a larger inductive hierarchy κ H∗ which characterises the closure of the class of all κ-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. This includes a broader class of “scattered” posets that we call κ-scattered. These posets cannot embed any order such that for every two subsets of size < κ, one being strictly less than the other, there is an element in between. If a linear order has this property and has size κ it is unique and called Q(κ). Partial orders such that for every a < b the set {x : a < x < b} has size ≥ κ are called weakly κ-dense, and posets that do not have a weakly κ-dense subset are called strongly κ-scattered. We prove that κ H∗ includes all strongly κ-scattered FAC posets and is included in the class of all FAC κ-scattered posets. For κ = ℵ0 the notions of scattered and strongly scattered coincide and our hierarchy is exactly aug(H) from the Abraham-Bonnet theorem. c Versita Warsaw and Springer-Verlag Berlin Heidelberg. All rights reserved. Keywords: Set theory, ordered sets, κ-dense, ηα -orderings MSC (2000): 03E04, 06A05, 06A06

∗

The authors warmly thank Uri Abraham for his many useful suggestions and comments. Mirna Dˇzamonja thanks EPSRC for their support on an EPSRC Advanced Fellowship. † E-mail: [email protected] ‡ E-mail: aleph [email protected]

226

1

M. Dˇzamonja and K. Thompson / Central European Journal of Mathematics 4(2) 2006 225–241

Introduction

A scattered order is one which does not embed the rationals. Hausdorff ([2], or see [7]) proved that the class of scattered linear orders is the least family of linear orderings which includes the ordinals and is closed under lexicographic sums and inversions. The paper [1] by Abraham and Bonnet proved that the class of scattered posets satisfying FAC (the finite antichain condition) is the least family of posets satisfying FAC which includes the well-founded posets and is closed under inversions, lexicographic sums and augmentations. There are several routes for expansion on these results which centre around a generalisation of the concept of scattered to higher cardinalities. To this effect, one would consider a κ-scattered poset (or linear order) to be one which does not embed a κ-dense set. There are two definitions that one could give of a κ-dense set. The first was introduced by Hausdorff in 1908 as an ηα -ordering for κ = ℵα . This is an order such that between any two subsets of size < κ, one being strictly less than the other, there is an element in between. Orders with this property are here called strongly κ-dense. When an ηα -ordering is linear and also has size κ, we call it Q(κ). Such an ordering is easily seen to be unique up to isomorphism. The other definition of κ-dense is a strictly weaker one in which between every two elements there is a subset of size κ. We call this notion weakly κ-dense. Using either definition of κ-scattered, namely weakly κ-scattered (not embedding a strongly κ-dense set) and strongly κ-scattered (not embedding even a weakly κ-dense set) orders, we can attempt to expand the characterisation results on linear orders or FAC posets. Note that the class of strongly κ-scattered is included in the class of weakly κ-scattered orders. This paper builds on [1] and extends its results. As in [1], a class of posets is built in a hierarchical way such that for any regular κ we have that κ H is the least family of posets satisfying FAC which includes the κ-well founded posets and is closed under inversions, lexicographic sums and augmentations. We then close this class under FAC weakenings (the dual notion to augmentations, but retaining the FAC) to obtain the class κ H∗ . We prove that the class κ H∗ contains all strongly κ-scattered posets and is contained in the class of all weakly κ-scattered FAC posets. For κ = ℵ0 where the two notions of scattered agree the two hierarchies agree and both are equal to the class of FAC scattered posets. This follows by the Abraham-Bonnet theorem. It is also shown that the class κ H∗ can be constructed in a simpler way. We may start with the κ-well founded linear orders and close under inversions, lexicographic sums and FAC weakenings. It is proved that this is exactly the same class as the one constructed by posets. So in particular augmentations are not needed in our hierarchy. A reader familiar with [1] may at this point wonder why it is that for κ > ℵ0 we do not obtain the complete analogue of the Abraham-Bonnet theorem. There are two main difficulties, apart from the fact that the notions of weakly and strongly κ-scattered for κ > ℵ0 are distinct, as opposed to what happens at κ = ℵ0 . The first one is that it is not necessarily the case that if all augmentations of a poset are weakly or strongly κ-scattered

M. Dˇzamonja and K. Thompson / Central European Journal of Mathematics 4(2) 2006 225–241

227

then the poset has the FAC. The other difficulty is that we do not know how to prove that FAC posets which are not in the hierarchy defined above actually embed a strongly κ-dense set, although we can prove that they embed a weakly κ-dense subset. It remains unknown whether every weakly κ-scattered poset is in the hierarchy κ H∗ or if κ H∗ and κ H are in general equal. However, κ H does contain examples of weakly κ-dense posets (as we will show in the final section), so it cannot be the case that κ H only contains strongly κ-scattered posets.

2

Background on κ-scattered posets

We start by explaining how Abraham and Bonnet’s theorem extends Hausdorff’s theorem. We first need several definitions. In this paper, we use ‘order’ to denote a ‘partial order’, and whenever we deal with linear orders we specify this. A (partial) order P embeds an order Q iff there is an order preserving one-to-one function from Q to P . An order is said to be scattered iff it does not embed the rationals, Q, with their usual ordering. If (I, ≤I ) is a partial order and P¯ = (Pi , ≤i ) : i ∈ I is a sequence of partial orders, the lexicographic sum of P¯ is the order whose universe is i∈I Pi , ordered by letting p ≤ q if and only if p, q ∈ Pi and p ≤i q for some i ∈ I or there exists i

Central European Journal of Mathematics

Central European Journal of Mathematics

Central European Journal of Mathematics