A balanced proper modification of P 3

Comment. Math. Helvetici 66 (1991) 505-511

0010-2571/91/040505-0751.50 + 0.20/0 9 1991 Birkh~iuserVerlag, Basel

A balanced proper modification of/)3 LUCIA ALESSANDRINI AND GIOVANNI BASSANELLI*

I. Introduction Since a p r o p e r modification M o f a K~ihler m a n i f o l d is not necessarily K~hler, it is n a t u r a l to ask what kind of weaker geometrical properties are preserved in this situation. In particular we are interested in the existence o f a balanced metric on M, i.e. a metric such that the trace o f the torsion o f the Chern connection vanishes. We p o i n t out that balanced manifolds are, in some sense, d u a l to Kfihlerian ones: in fact, a metric is balanced if its K/ihler form is " c o - c l o s e d " (see Definition 3.1); moreover, balanced n-dimensional manifolds are characterized by means o f their (n - 1, n - 1)-currents as well as K~ihler manifolds can be characterized by (1, 1)-currents (see [Mi] and [HL]). The classical example o f H i r o n a k a ([Hi] or IS]) o f a M o i s h e z o n non algebraic manifold X is given by a p r o p e r modification o f P3; in this note we prove that X is balanced. F r o m a geometrical point o f view, the obstruction for X to be K/ihler is a curve h o m o l o g o u s to zero on the exceptional divisor E: in a dual manner, our p r o o f lies in showing that if E is not h o m o l o g o u s to zero then M is balanced. This result is a first c o n f i r m a t i o n o f the conjecture: " E v e r y p r o p e r modification o f a K/ihler manifold is b a l a n c e d . " A b o u t techniques, since positive currents which are ( p , p ) - c o m p o n e n t s o f b o u n d a r i e s are not necessarily closed (see Question on p. 170 in [HL]), we c a n n o t apply the well k n o w n results on (locally) flat currents, and therefore we must develope a parallel technique for positive d0--closed currents.

*This work is partially supported by MPI, 40%. The second author thanks the Department of Mathematics of the University of Trento for its kind hospitality during the preparation of this paper.

506

LUCIA ALESSANDRINI A N D GIOVANNI BASSANELLI

2. Preliminaries

Let M be a connected complex manifold of dimension n; we denote as usually by ~P the sheaf of germs of holomorphic p-forms and by gP the sheaf of germs of ~ p-forms. Moreover, let ~P'P(M)R denote the Fr6chet space of real (p, p)-forms, while 8~.p(M)R denotes its dual space of real currents of bidimension (p, p). We will say that T ~ g'p.p(M)R is the (p, p)-component o f a boundary if there exists a current S such that for every ~o e gP'P(M), T(~o) = S(d~o), i.e. there exists a (p, p + 1)-current S such that T = ffS + OS. Let V be a complex vector space of dimension n, p an integer and ~p.'= iP22 P. An element of A p'~ that can be expressed as v~ ^ 9 9 9 ^ vp with vj~ Vis called simple. 2.1 D E F I N I T I O N . ([AA] and [AB]) ~ E o~P'P(M)R is called transverse if Vx e M, Vv ~ AP(T'~M), v r 0 and simple, it holds O x ( a i l v ^ ~) > O. T ~ g'p.p(M)a is called positive if T(O) > 0 for every transverse (p, p)-form Q. 2.2 R E M A R K . ([H1] p. 325-326) A positive (p,p)-current T has measure coefficients, and moreover, if [tTt[ denotes the mass of T, there exists a HT[[-measurable function ir such that ][iPx [I = 1 for HTI[-a.e. x ~ M, and

= f (L)II TH for all test forms r Let us prove now our main technical tool. 2.3 T H E O R E M . Let M be a complex n-dimensional manifold, and let T be a positive (p, p)-current on M such that d J T = 0 and the Hausdorff 2p-measure o f supp T, H2p(T), vanishes. Then T = O. 2.4 R E M A R K S . In this statement, M is not assumed to be compact. If moreover T is closed, the result is well known (see for instance [F]). P r o o f o f Theorem 2.3. Let x0~ supp T; as in [H2] p. 72 we can choose a coordinate neighbourhood (U, Z l , . . . , zn) centered at x0 such that for every increasing multiindex/, [I I = p, it holds locally: (a) if nz is the projection defined by T~I(Z 1 . . . . . Z n ) = (Zil . . . . . Zip), supp TC~ Ker n, = {Xo} (b) there exist two balls A'I~CP, A ~ C n-p such that ( A ' , x b A T ) c ~ supp T = ~ . This implies that n~ [suppr : supp T c~(d) x A 7 ) ~ A ) is a proper map.

A balanced proper modification of P~

507

Fix an I, I I [ = p, and call T,,= (z~, Isupp T). ( r ) ; 7", is a OJ-closed (p, p)-current on A), hence it can be identified with a OJ-closed distribution fr on A); since J) is pluriharmonic, it is smooth. But 7zI is a Lipschitzian map, and therefore H2p(suppf~) = 0, i.e. J) = 0. Let

c o , = a ~ d z ~ AdZ~ in B . ' = ~ (A} x AT). I

Since

T is positive, in order to prove

T=0

it is enough

to verify that

TO~B(e)P/p!)) = 0. But in B o) p

p! - ~, ap dz, ^ dS, = ~, (Ts

•I

A

de, ),

therefore

T Zo