Properties of compact complex manifolds carrying closed ...shiffman/publications/34 closed positive...

The Journal of Geometric Analysis Volume 3, Number 1, 1993

Properties of Compact Complex Manifolds Carrying Closed Positive Currents

By Shanyu Ji and Bernard Shiffman

ABSTRACT. We show that a compact complex manifold is Moishezon if and only if it carries a strictly positive, integral (1, l)-current. We then study holomorphic line bundles carrying singular hermitian metrics with semi-positive curvature currents, and we give some cases in which these line bundles are big. We use these cases to provide sufficient conditions for a compact complex manifold to be Moishezon in terms of the existence of certain semi-positive, integral ( I, 1)-currents. We also show that the intersection number of two closed semi-positive currents of complementary degrees on a compact complex manifold is positive when the intersection of their singular supports is contained in a Stein domain.

1. Introduction

In complex geometry, positive smooth forms provide information of both an algebraic and an analytic nature. For example, a K~hler form on a compact complex manifold M is a closed, strictly positive (1, 1)-form on M. If the de Rham class of this form is integral, then M is projective- algebraic (Kodaira embedding theorem). If L is a holomorphic line bundle with a hermitian metric h on M , then its curvature, which we denote by el(L, h), is a closed smooth (1, 1)-form in the Chem class of L. If cl (L, h) is strictly positive, then L is ample (Kodaira). If cl (L, h) is only semi-positive and M is algebraic, then of course L must be nef (i.e., (cl (L), C) >_ 0 for all complex curves C C M). However, if cl (L, h) is semi-positive on M and also strictly positive at some point of M , then L is big (i.e., ~(L) = dim M ) and M is Moishezon (Demailly [2]). In this paper, we study the information provided by positive currents that are not assumed to be smooth. For example, we define a K~ihler current to be a strictly positive, closed (1, 1)-current. (A (p,p)-current T on M is said to be strictly positive if there is a strictly positive (1,1)-form w on M so that T - w p is semi-positive.) A second example is a holomorphic line bundle L on M that is given a "singular metric" h, where we assume only that log h is locally integrable; then its curvature Cl (L, h) is a (1, 1)-current on M. (See Definition 2.3 for details.)

Although closed (19, p)-currents can be approximated by closed smooth (/9, p)-forms, positivity may not be preserved. In fact, there are many examples of closed strictly positive ( 1, 1 )-currents

Math Subject Classification 32C30, 32C40, 32J20, 32J25. Key Words and Phrases Cbem class, compact complex manifold, singular hermitian metric, holomorphic line

bundle, Iitaka dimension, intersection number, Kahler current, Lelong number, Moisbezon manifold, positive current, Stein manifold.

The first author was partially supported by National Science Foundation Grant Nos. DMS-8922760 and DMS- 9204273. The second author was partially supported by National Science Foundation Grant Nos. DMS-9001365 and DMS-9204037.

(~)1993 CRC Press, Inc. ISSN 1050-6926

38 Shanyu Ji and Bernard Shiffman

on compact complex manifolds that cannot be approximated by K/thler forms. For example, let X be the blow-up of the complex projective plane obtained by replacing a point of the projective plane by a projective curve C C X , and consider the closed strictly positive (1, 1)-current u = [C] + ew on X, where w is a K/ihler form on X and ~ is a sufficiently small positive number such that the square of the de Rham class [u] E H2(X , R) is a negative class in H4(X, ~). Since [u] 2 < 0, smooth approximations of u cannot be positive. Another example is given by a Moishezon manifold M that is not projective-algebraic. Let

7r : M-.-~ M

be a modification of M where M is projective and hence carries an integral K~ler form 9. Then the current 7r.~ is strictly positive but cannot be approximated by Ktihler forms, since K/ihler Moishezon manifolds are projective-algebraic. In fact we shall prove the following singular analogues of Demailly's characterization of Moishezon manifolds in terms of the existence of such currents:

Theorem 1.1. Let M be a compact complex manifold. Then the following statements are equivalent:

(1) M is Moishezon.

(2) There is an integral Kiihler current on M .

(3) There is a holomorphic line bundle L over M with a singular hermitian metric h such that the curvature current Cl ( L, h) is strictly positive.

Theorem 1.2. Let M be a compact complex manifold. Suppose that there is a d-closed, integral, semi-positive (I, 1)-current T on M such that sing supp (T) is contained in a Stein open subset of M and T is strictly positive at some point Xo E M . Then M is Moishezon.

Theorem 1.3. Let M be a compact complex surface or a compact Kiihler manifold of arbitrary dimension. Suppose that there is a d-closed, integral, semi-positive ( 1, 1 )-current T on M , strictly positive at some point Xo E M , with zero Lelong numbers outside of a countable subset of M. Then M is projective-algebraic.

The hypothesis on the singularities of T in Theorem 1.3 is weaker than that of Theorem 1.2. Indeed, if sing supp (T) is contained in a Stein open set U C M , then for each c > 0, Ec(T) is a compact subvariety of U by a fundamental theorem of Siu [12] and therefore is finite. Thus the set of points where T has nonzero Lelong number is at most countable. It is an open question whether Theorem 1.2 is valid with the weaker singularity hypothesis of Theorem 1.3 for non-K/thler manifolds of dimension greater than 2.

A compact complex manifold is Moishezon if and only if it carries a big line bundle. This well-known fact enables us to prove the above three theorems by showing that holomorphic line

Properties of Compact Complex Manifolds Carrying Closed Positive Currents 39

bundles with curvature currents satisfying the hypotheses of the theorems are big (Theorems 4.6, 4.7, and 4.12), and thus the global sections of sufficiently high powers of these bundles give a birneromorphic map to a projective variety. A key argument that we use is the following result of Demailly [4] on smoothing semi-positive (1, 1)-currents. Suppose that T is a closed, semi-positive (1 ~ 1 )-current on a compact complex manifold M of dimension n. Demailly's smoothing theorem (Lemma 4.1) allows the approximation of T by currents in the de Rham class of T that are smooth outside the subvariety Ec (T) consisting of points of M where the Lelong numbers of T are at least c (for any c > 0). A corollary of Demailly's result is that if T has zero Lelong numbers at all points outside of a countable subset of M, then

(T. S) > 0

for all closed, semi-positive (n - 1, n - 1)-currents S on M, with strict inequality holding if there is a point of M where both S and T are strictly positive (Theorem 4.3). For d-closed currents Tx and T2 of complementary degrees on M, the intersection number (T1 �9 T2) is defined by

(T1. T2) = ([T1] U [Tz], M ) , (1.1)

where [Tj] denotes the de Rham cohomology class of Tj. We shall prove the following positivity theorem for the intersection numbers of closed, semi-positive currents:

Theorem 1.4. Let M be a compact complex manifold of dimension n. Let T1 and T2 be d-closed, semi-positive currents of bidegrees (p, p) and (n - p, n - p), respectively, on M (0 < p < n). If (sing supp 7"1) 71 (sing supp T2) is contained in a Stein open subset of M , then

(T1. T2) _> 0;

if in addition there is a point x E M such that Tx and T2 are strictly positive at x, then

. > o .

For example, let C be an analytic curve in a compact complex surface M , let U be an open subset of M that intersects each irreducible component of C, and suppose that T1 and T2 are semi-positive currents of bidegree (1,1) on M such that T1 is smooth on M - C and either T1 or T2 is smooth on U. Then (sing supp TI) N (sing supp T2) C C - U. By a theorem of Sin [13], C - U has a Stein open neighborhood in M, and thus we can apply Theorem 1.4 to conclude that (7'1" T2) _> 0.

We would like to thank Jean-Pierre Demailly and Reese Harvey for their helpful conversations and suggestions.


2. Smooth ing of cur rents on Stein manifolds

We first recall some basic notation and definitions. Let V~ denote a real vector space of dimension 2n with an almost-complex structure J : V~ ~ V~. We write

V=V~|

where V' and V -'7 are the ~ and - ~ eigenspaces of J. (We shall consider the case where V~ is the real cotangent space at a point x of a complex manifold M; then V ' is the holomorphic cotangent space T~,~). We also write

AP'qV = A P V ' | A q v 7 C AP+qV,

A F v = {A �9 A = A )

Definition 2.1. A vector A �9 A~'vV is called strictly positive [semi-positive] if for all r l �9 A " - v V ' - {0}, we can write

A A crpr I A ~ = a crne~ A �9 . . A en A ~] A �9 �9 �9 A ~.~ ,

where crv = (~Z ' I ) v2' { e , , . . . , e , } is a basis of V', and a > 0 [a > 0].

We recall the following facts: (See [6, Corollary 1.3]) If A and B are strictly positive vectors, then A A B is a strictly positive vector. Furthermore, A �9 A~'vV is strictly positive if and only if A A B is strictly positive for all strictly positive B �9 A ~ - V ' n - v V . (The same statements are valid with "strictly positive" replaced by "semi-positive.")

Now let M be an n-dimensional complex manifold. We let 73v,q(M) denote the space of C ~176 (p, q)-forms on M with compact support, and we let 7) 'v 'q(M) = 7 ) n - v ' n - q ( M ) ' denote the space of (p~ q)-currents on M. We also let ~v 'q (M) denote the space of C~ q)-forms on M and E,v,q ( M ) = E , - v , n - q ( M ) ' denote the space of compactly supported (2, q)-currents on M. For a compact set K C M , we write

~t~,q = { T ~ E'P'q(M) : suppT C K}.

Suppose T 6 1) 'v 'q(M). We let sing supp T denote the smallest closed subset A of M such that T is a smooth form on M - A. For ~b 6 7 ) " - v ' n - q ( M ) , we let (T, if) = T(~b) denote the pairing of T and ~b. We note that if M is compact and T, r are closed, then (T, ~) = (T . if), where ( T . if) is the intersection number given by (1.1).

We say that 7r : M ---+ M is a modification (or blow-up) of M if M is a complex manifold, 7r is a proper holomorphic map, and there is a proper analytic sub.variety S of M such that 7r [ (M - 7r-1(S)) is biholomorphic. We note that if T fi l ) ' v ' q (M) such that the coefficients of T are in L~o r then 7r.T also has coefficients in Llo r From this we conclude that if a 6 Cv 'q (M) , then 7r.(Tr*o 0 = o~.


Definition 2.2. (a) A real (p,p)-form on M is strictly positive [semi-positive] if it is strictly positive [semi-positive] at each point. A real (p~p)-current T on M is semi-positive if (T, r/) _> 0 for all semi-positive (n - p, n - p) forms r /on M.

(b) A real (io, p)-current T on M is strictly positive if there is a strictly positive (1, 1)-form on M such that T - wv is semi-positive; T is said to be strictly positive at a point x E M

if there is a neighborhood U of x such that TIu is a strictly positive current on U. Note that T is strictly positive on M if and only if T is strictly positive at each point of M. By the above, a smooth form is strictly positive [semi-positive] as a form if and only if it is strictly positive [semi-positive] as a current. If a (p,p)-current T is strictly positive [semi-positive], we write T > 0 [T > 0]. We also write S > T [S > T] if S - T > 0 IS - T > 0], for (p,p)-currents S , T .

The image under a holomorphic map of a semi-positive current is semi-positive. Furthermore, if

N

rc : M--* M

is a modification and T is a strictly positive (p,p)-current on M, then 7r.T is a strictly positive current on M. To verify this fact, let x E M be arbitrary, and let w be a positive (I, 1)-form

on M. Let U C C M be a neighborhood of x, and let ~ be a positive (1, 1)-form on M such that T _> ~v. We can choose a constant c > 0 such that ~ > cTr*w on 7r- l (U) . Then 7r.T >_ 7r.Sv >_ cVTr.(Tr*czv) = cVwV on U as desired.

R e m a r k . Elsewhere in the literature, a semi-positive current is called simply a "positive current." We have modified the usual definitions in order to have consistent terminology for currents and smooth forms. [ ]

If T is a closed, semi-positive (p,p)-current on M, we let n(T, x) denote the Lelong number of T at a point x E M. We also write

Ec(T) = {x e M : n (T , x ) >_ c},

for c > 0. By a theorem of Siu [12], Ec(T) is a subvariety of M of codimension at least p.

Definition 2.3. (a) A (1, 1)-current w on M is said to be a Kiihler current (cf. [7]) if it is d-closed and strictly positive on M. A (1, l)-current or a (1~ 1)-form is said to be integral if its de Rham cohomology class is in the image of the map

j . : H2(M,Z) , H2(M,R)

induced from the inclusion j : Z ~ R.


(b) Let L --~ M be a holomorphic line bundle over M. A singular hermitian metric h on L is a map h : L ~ [ - ~ , +cx~] that is given in any local trivialization (~-, 0) : T-l(U) ~ U • C by

hff) = I0(~)1 e-~(~(e)), for ~ ~ T-I(u)

where Cu C L~oc(U ). The curvature current of (L, h) is the d-closed (1, 1)-current c,(L, h) given by

c~(L,h) = x/'-S-f O0 Cu ,ff

on U, which is indeed independent of the choice of local trivialization. (Note that the constant is chosen so that cl (L, h) is a representative of the Chem class Cl (L) E H2(M, R) and thus is

integral.)

(c) A compact complex manifold M is called Moishezon if the transcendence degree of its meromorphic funcfi.on field is equal j o the dimension of M or, equivalently, if there exists a modification 7r : M ~ M where M is projective-algebraic. If M is compact K~tler, then M is projective-algebraic if and only if M is Moishezon [10].

Let X be an n-dimensional closed complex submanifold of C m , and let i : X ~ C m denote the inclusion map. Let T be a (p, q)-current on X. We shall give a method of approximating T by smooth forms. Recall that the (m - n + p, rn - n + q)-current i . T on C m is given by

(i.T, r = ( T , i ' r = (T , r

for any form r E 79n-P'n-q(cm). Recall that i . commutes with 0 and 0, and if T is a semi-

positive (p, p)-current on X then i . T is a semi-positive (m - n + p, m - n + p)-current on C m.

We shall let X, denote an approximate identity on C ~ of the form X,(Z) = e-2mx(z/e), where X E C ~ ( C m) is such that X > 0, suppx C {llzll < 1}, and f x -- 1. We have the smooth forms

X, * i .T E s for e > 0.

If T is smooth, the forms X~ * i .T can be written explicitly as follows: Let k = p + q, s = 2n, t = 2m, and consider T E C k (X) , where X is an oriented real s-dimensional submanifold of R t. We let x l , . . . ,x t denote the coordinates in ]~t. For a multiindex I = ( i l , . . . , i . )

[1 < il < "" < i . < t], we write IZl -- r and d z ' -- dx" A . . . A dz". We also let I • denote the multiindex with II11 = t - ,', Z U & = { X , . . . , t} . We then have the formula

( x , * i . T ) ( x ) = ~ a , { f ~ X , ( x - ~ ) T ( ~ ) A d ~ Z ~ } d x z EX I

(2.1)

(lII = t - s + k) for z c ~ ' , where ~ is the sign of the permutation { 1 . . . t} ~ /_/•


Rewriting (2.1) in complex coordinates with T E CP'q(X), we have

(x , * i . T ) ( z ) = E (r,z { / X~(Z - ~)T(~) A d~'= A d~J~- } d z ' A d2 J (2.2) I,J EX

( 1 1 1 --- - n + p, I J l = m - n + q, I/_LI = n - p , I J • - - n - q ) , where a z j = +1 .

In general there is no canonical way to push Xe * i , T back onto X . However, the following well-known lemma lets us define a push-forward that approximates the current T .

L e m m a 2.4. [5, p. 257] Let X be as above. Then there is an open subset ~ C C '~ containing X and a holomorphic submersion p : Q ) X such that

p o i = idx

where i : X ~ C m is the inclusion map and idx is the identity map on X .

Thus, if T has compact support, we can define

L , ( T ) := P.(X, * i . T ) (2.3)

for e sufficiently small. The LE (T) are smooth approximations of T , as the following lemma

states precisely:

L e m m a 2.5. Let X be a Stein manifold and let K be a compact subset of X . Then there exists real C-linear mappings

,

for 0 < e < ~o, such that, writing L~ -- (~L p'q, we have for any T E c~'q:

(i) L , ( T ) ~ T weakly in C'P'q(X), as e ~ O; furthermore, if U C C X -

sing supp T, then L, (T) , T uniformly on U as e ---* O.

(ii) If p = q and T is a semi-positive current, then L , ( T ) is a semi-positive form.

(iii) L , ( 3 T ) = O(L , (T) ) and L , (OT) = O(L , (T) ) .

(iv) For each open U D suppT, there exists 5 > 0 such that s u p p L , ( T ) C U for O < e < 5 .

P roo f . Let K , X be as in the statement of the theorem and let T E C~ 'q. Let / : X r C "~ be an embedding of X as a closed submanifold of C m and let f2 C C m , p : f~ ---* X be as


in Lemma 2.4. Let e0 = dist (K , Of 2). Then for 0 < e < e0, we can use (2.3) to define L, (T ) E g'P'q ( X ) . Next we show that

L , ( T ) e DP'q (X) . (2.4)

To verify (2.4), we can choose local holomorphic coordinates in C TM so that X becomes a linear subspace and p is the projection onto X ; in particular, we replace C n by m s and C m by R s+t so

that p : R s+t ---* ]R s is the coordinate projection. It then suffices to show that p . r is smooth for

'0 E Dk(Rs+t) . We can write

r = Z r A dv 1 A " " A dv t + r (2.5)

where the coordinates in R s+t are u l , . . . , u S , v l , . . . , v t and r does not contain any terms containing dv 1 A �9 �9 �9 A dv t. We easily see that

P ' r = Z Y x d u ' (2.6)

where

f , ( u ) = f , r 1 A . . . A dv t, E~ t

and therefore p . r E 79k+t(l~s+t). Conclusions (ii) and (iii) of Lemma 2.5 follow immediately from the fact that p and i are holomorphic; (iv) follows immediately from (2.3). Conclusion (i) is a consequence of the following lemma and the localization property (iv).

Lemma 2.6. T E 79P'q(U), then

uniformly on U as e ~ O.

For each a E K , there is a neighborhood U o f a in X such that if

L , ( T ) , T

Proof. As above, we let X be an oriented real s-dimensional submanifold of R s+t. Let

us choose the neighborhood U of a as follows. Consider a coordinate neighborhood f2 of a in IR '+t with coordinates ( u : , . . . , u s, v l , . . . , v t) such that X f') f2 = {v: = . . . . v t = 0} and

u j op = u j on f2 for 1 < j < s, where p is given as in Lemma 2.4. We write u = ( u X , . . . , uS), v = ( v l , . . . , v t) and we let U = X f') f2 = v - l ( 0 ) . We thus have a commutative diagram

f21 H f2

U' h U (2.7)


where U', l')' are domains in R s, R ~+t respectively, h -~ = u, H -~ = (u, v), and 7r(ba, . . . , b ~+t) = ( b l , . . . , b~). We can further assume that i'2' = U' • V', where V' C R t.

Now suppose T E Z)~(U), and write T -- Z TjduJ" We have by (2.1) J

I,J EU' (2.8)

where Tj = Tj o h and

ff~ I J O(h~,..., h~-~) O(~m, . . . ,~u,-k )

where (#x , . . . , / z s -k ) ----- ( J , s + 1 , . . . , s + Q• ( U l , . . . , u s - k ) = I•

Hem we use the notation

O ( f l , . . . , f ' )

O(w',... ,w ")

(o:o) = det \ Ow ~ ]

By (2.7),

h ' ( L , T ) = h*p . (x , * i . T ) = 7r.H*(x, * i . T ) (2.9)

and hence

(2.10)

(L = ( / 1 , . . . , / k ) , l <_ Ii < "'" < lk _< s), where

K,JL (u, ~) = ~-'~ ~ H (~) f~ x ~ ( H ( u , v ) - H ( ( , O ) ) r ~ x L ( U , v ) d v l A . . . A d v t (2.11) I EV~

where

O ( H i , , . . . , Hi,+~)

~IL = O(Ul, . . . ,uZ~ vl . . . ,v~) "


Choose U0 CC U such that supp T C (.70. For b E h- l (U0) consider the linear approximation/s : ]Rs+t ~ iRs+t given by

s+ t bj O H , f ib(w) = H(b) + ~-~(w i - )-O-~(b). (2.12 / j = l

We modify the K [L to define the forms S~ E Dk(U ') by

S~= ~ [f~ K,bJL(u,~)7"s(~)d~' A ... A d~ s] du L J,L EU'

(2.131

w h e r e K b J L is given as in (2.11) except that H is replaced by /f/b; also @Ia and (Tgzz are

replaced by the resulting ~tbzj and ~ L , which are constant.

Let 6 > 0 be arbitrary. First we shall find a positive el (independent of b E h-l(Uo)) such that

sup IIS, b - h'TII < 6, for 0 < e < el- (2.141

OH ~ i By making a linear coordinate change in ~ (depending on b), so that -~ - (b ) ---- 6j, we can assume that

^ ^ { 1 i f I = ( J , s + l , . . . , s + t ) (I)~s = @~s = 0 otherwise.

Then I(~ JL = 0 for J y~ L and

fi~bJ J z f v ~,u,~)= X ~ ( U - ~ , v ) d v l A . . . A d v t = x ~ ( u - ~ ) EV ~

where X~ is an approximate identity on R s of the form X~(u) = e-sxb(u/e) where X b depends continuously on b. Thus we have

s~(u) = Z [ f x~(,.,-r162162 A . . . A d~ s] d~ ~. j EU ~

(2.151

We now return to the original coordinates ( u l , . . . , u s, v l , . . . , C ) and note that (2.15) is still valid. The existence of el satisfying (2.141 follows from (2.151.


Since X,(~) = e-~-tX(~/e) and supp X~ C {11~11 < c}, we can find a positive e 2 < c 1 (also independent of b) such that

[_ff,bJZ (b, ~) - K:,L(b,~)Id~a A . . . A dE" < 6 (2.16) EU'

for 0 < e < e2, for all J, .L. Thus by (2.10) and (2.13)

IIS,b(b)- h*(L,T)(b)][ < C6 for0 < e < e2. (2.17)

where C = g supj,~ ITJ(~)I and N is the number of multiindices JL. Therefore by (2.14) and (2.17),

I I h * ( L , T - T)(b)ll < (C + 1)6 for0 < e < e2. [ ]

3. Global smoothing of (p,p)-currents

In this section, we give a method of smoothing certain (P, p)-currents on compact complex manifolds, which we then use to prove Theorem 1.4. We begin with some general facts on the existence of smooth currents.

Let Xo be an open set in a complex manifold X. Consider the sheaves ~P'q on X given by

Note that

.T'P'q(u) = {u e :D'P'q(u): sing supp u C Xo f') U}.

Yf'q= { D',,q, E~,q,

i f x EXo,

i f x ~Xo .

Thus we have the exact sequence

o - a , - 7 ,,~ L 7 p,1 2 . . . L ~-,,~ --, o.

By the de Rham theorem, we have a commutative diagram of isomorphisms:

H"(CP,'(X)) ~ H~(Y~' (X))

g q ( x , f2p) = Hq(X, f2v)

(3.1)

Remark. If Xo = X, then .)rp,q = D,p,q. []

48 Shanyu J i a n d Bernard Shi f fman

Lemma 3.1. If u E :TP,q(X) such that du E C p+q+1 (X) , then there exist ol E cP 'q(x ) and v E 5 r P - l ' q - l ( S ) with

u = a + OOv.

P r o o f . Consider the bicomplex (A, d ' , d") [A = @A p'q, d', d" : A ~ A have bidegrees

(1 ,0) and (0, 1) respectively, d'd" = d"d', d '2 = d ''2 = 0] given by

Ap,q = .~'P'q ( z ) / ~P'q ( s ) ,

d' = O, d 't = ( -1)qO. We have the long exact cohomology sequence

t q ~/-q-4-1 t q+l �9 .----, Hq, , (C(X)) Hq, , ( .T(X)) ~ H~,,(A) ---, --d,, ( C ( X ) ) ~ . - . .

By (3.1) L q is an isomorphism of all q. Therefore Hd,,(A) = 0. Since d ' f = d: d" f , we

also have Hd,(A) = 0. Lemma 3.1 then becomes a special case of the following fact:

Lemma 3.2. Let (A, d', d") be a bicomplex with A "'q = 0 for min(p, q) < 0, and Hd, (A) = Hd,, (A) = 0 for all p, q. I f a E A p'q such that d'a = d"a = O, then there exists b E A p-l'q-1 such that a = d'd"b.

P r o o f . Let d = d ' + ( - 1 ) q d ' ' . Let a be as in the lemma. We can assume that min(p, q) >_

1, since otherwise a = 0. Since Ha,,(A) = 0, we can find x = xl + " " + Xq E A with

x j E A p-l+j'q-j such that dx = a. Define length (x) to be the largest j such that x j # 0. Choose an x as above with minimum length l. We claim that l _< 1. For, suppose instead that x = X l + " - + x z , xt 5 ~ 0, I _> 2. Then d'xt = 0. Choose c E A p-2+t'q-t such that

dtc = xt, and let 5c = x - dc = xx + " " + ~ct-1. Then d~ = a, but lengthS: = 1 - 1,

which is a conu'adiction. Hence x = xx E A "'q-i, d%1 = a, d'xl = 0. Then we can choose b E A p- l 'q-1 such that xl = d'b and then a = d"d'b. []

We remark that the following generalization of a well-known fact (see [4]) follows from

Lemma 3.1:

C o r o l l a r y 3.3. Let

P,q H o ~ ( X ) = {a E C " q ( X ) : da=O}/OOCP-~ 'q - l (X) ,

~ p,q goo ( Z ) = {a E .TP 'q(X) : da = O } / O ~ . ~ P - l ' q - l ( Z ) .

p,q ~ p ,q Then the inclusion gP'q(X) ~ .TP'q(x) induces an isomorphism Ho~ ( x ) ~, Ho~ ( X ) .


Proof. Surjectivity is an immediate consequence of Lemma 3.1. We now verify injectivity: Suppose o~ E gP'q(X), tx = OOu, u E 5rP- l ' q - l (X) . We must find "7 E g P - l ' q - l ( X ) such that ~ = 00% Let v = Ou E .TP'q-I(X) . Since dv = - a E gP+q(X), by Lemma 3.1 there exist w E .7:P-l 'q-2(X) and/3 E gp,q-i ( X ) such that v = /~ + OOw. Since/3 = O(u - Ow),

by (3.1) applied to/~, we can find 7 E g P - l ' q - l ( X ) with/3 = 0'7. Thus

[]

We now give a useful smoothing lemma on Stein manifolds:

L e m m a 3.4. Let X be a Stein manifold, and let w be a strictly positive (1, 1)-form on X . Let T be a real, d-closed (p,p)-current on X and suppose there is a real (p,p)-form 0 on X such that T >_ O. Let U, V , W be open subsets of X such that W C C V C X and sing supp T C U, and let E = ( V - W ) f-1 -U. Then there exist real (p, p)-currents T~ and real (p - 1, p - 1)-currents S , on X , for 0 < e < 1, such that

(i)

(ii)

(iii)

(iv)

(v) _< 5(a).

T = T, +

T, ---* T weakly as e ~ O,

supp S~ C V,

sing supp T~ C U - W , and

for each a > O, there exists 5(a) > 0 such that Te q- aw p >_ 0 on X - E for all

Remark . Note that T = T~ outside of V. [ ]

Proof. Choose open sets W1, W2, W3 in X such that W C C W1 C C W2 C C W3 C C V. Then choose an open set Uo such that sing supp T C [.To C U and Uo N W3 C C U. By Lemma 3.1, we can find a smooth real (p,p)-form P on X and a real (p - 1,p - 1)-current u on X such that sing supp u C [70 and

T = P + v/-A--fObu. (3.2)

Take a function r e C a ( X ) such that r ---- 1 and supp (r C W2. Let

P' = T - v/E-lOO(Cu) = P + ~/~100[(1 - +)u] (3.3)

and note that sing supp P' C Uo - W1. For 0 < e < 1, we can define

S, = Cu - L, ( r (3.4)

50

and

Shanyu Ji and Bernard Shiffman

T, = T - v/Z-fOOS, = P ' + v/L--'fOOL,(r = P ' + L , ( T - P ' ) (3.5)

where L~ is given by Lemma 2.5 with K = W3. By part (iv) of Lemma 2.5, we can choose 61 > 0 such that supp L, (r C W2 for 0 < e < 61. It then follows from Lemma 2.5 and the above that conclusions (i)-(iv) are satisfied.

It remains to verify (v). Take another cut-off function p E C ~ ( X ) such that p _> 0,

P[w2 = 1 and supp (p) C W3. On W2,

T, - 0 = pP ' - pO + L, [p(T - P')] = L, ( p T - pig) + [(pP' - pig) - L , (pP' - pO)]. (3.6)

Let a > 0 be arbitrary. By part (i) of Lemma 2.5,

L , ( p P ' - p O ) ~ p P ' - p O (3.7)

uniformly on (W2 - U) U W. By part (ii) of Lemma 2.5, L , ( p T - p/9) _> 0, and therefore by (3.6) and (3.7) we can choose ~(a) E (0, 61) such that

T , - O + a w p > O on ( W 2 - U ) U W (3.8)

for e _< 6(a). Since T~ = T on X - W2, conclusion (v) follows from (3.8). [ ]

As a special case of Lemrna 3.4, we obtain the following result on the global smoothing of currents with singularities inside a Stein domain:

L e m m a 3.5. Let M be a compact complex manifold, and let w be a strictly positive (1, 1)-form on M . Let V be an open subset of a Stein open set X in M , and let T be a real, d-closed (p,p)-current on M such that sing supp T C C V . Suppose there is a real (p,p)-form

O on M such that T >_ O. Then for all a > 0 there is a real (p,p)-form ~" on M anda real

(p -- 1, p -- 1)-current S on M such that supp S C V, T = T + v/ZTOOS, and

+ aw v > 0.

Proof. Assume without loss of generality that V CC X and choose open sets U, W such that

s i n g s u p p T C U C C W C C V .

Then ( W - W) fqU = 0. Let S, E 7)'P-x'v-a(X), T, E 7)'P'P(X) be given as in Lemma 3.4 for some e < 6(a), and let S E 7) 'v-I 'P-I(M) be the extension of S, to M with S = 0 on M - X. By (iv), TE is smooth on all of X. Since T = T~ on X - V, our desired (p,p)-form

:T e Dv,v(M) can be given by T = T, on X and :F = T on M - V. []


The following theorem of Chow and Kodaira [1] (for the case of smooth surfaces) is a consequence of Lemma 3.5:

Corol lary 3.6. Every 2-dimensional Moishezon manifold is projective-algebraic.

Proof. Let M be a 2-dimensional Moishezon manifold. There is a modification 7r : M M, where M is projective-algebraic. Let ~ be an integral K/ihler form on M, and let T = 7r,~. Then T is a strictly positive, closed (1, 1)-current, and the de Rham class [T] is integral. Note that sing supp T is a finite set. Thus by Lemma 3.5, [T] contains a smooth K/ihler form and thus M is projective-algebraic by the Kodaira embedding theorem. []

Remark . A similar type of proof of the Chow-Kodaira Theorem was given by Miya-

oka [9]. [ ]

Proof of Theorem 1.4. Let T1, T2 be given as in the theorem and write T = T1. Fix a strictly positive (1, 1)-form w on M. If T1 and T2 are strictly positive at some point x E M, choose 0 E 7)v'v(M) such that 0 is strictly positive at z and T > 0 > 0 on M. Otherwise set 0 = 0. Choose open sets U, V, W, X in M such that X is Stein, sing supp TI C U,

(sing supp T1) M (sing supp T2) C W CC V CC X, (3.9)

and

(U - W) fl sing supp T2 = 0. (3.10)

Let E, T~, 6(a) be given as in Lemma 3.4, and regard T~ as a current on M by setting T, = T on M - X. For k = 1 , 2 , 3 , . . . , write e(k) = min{1/k,6(1/k)} and let

1 T~ = T,(~ + gwv. (3.11)

Then

Tk>_0 on M - E , (3.12)

sing supp Tk C U - W, and Tk ~ T weakly as k --~ +00. By (3.10), we can choose ~b E C~176 such that 0 < q5 < 1, ~b -- 1 on a neighborhood of U - W, and q5 ___ 0 on a

neighborhood of sing supp T2. Note that qST2 and (1 - ~b)Tk are smooth.

52

Since

we have

(T, . T:)

and thus by (3.12)


sing supp T,(k) fq sing supp T2 = 0,

= (T~(k) " T2) = (Te(k) A T2, 1) 1

= ( T ~ , r 2 4 7 1 6 2 p)

1 (T~-T2) _> (Tk, r + (T2, ( 1 - r ~(T2, ~ p)

for k = 1, 2, 3 , . . . . Letting k ~ + o o in (3.13), we obtain

(T1. T:) > (T, r + (T2, (1 - r

(O, r + (T2, (1 - r = ( T 2 , 0 ) > _ O .

In the case where T1, T2 and 0 are strictly positive at a point x, we have (T2, 0) > 0. [ ]

(3.13)

4. Global smoothing of (1,1)-currents

Let M be a compact complex manifold and consider the fiber bundle P T M 2* .A/f, where TM denotes the holomorphic tangent bundle of M , and for each x E X , the fiber (PTM)x is the projective space of hyperplanes in TM,z. We have the standard tautological line bundle

--~ PTM, which has the property that (l~-'(~) is the hyperplane section bundle for each x E X (see [11, pp. 91-93]). We can choose a smooth hermitian metric h on ~ and a strictly positive (Is 1)-form w on M such that

Cl (s h) + zr*u > 0. (4.1)

(For example, by a partition of unity argument we can choose h such that cl (~, h)[~-l(x) > 0 for all x E M . If we let r /be an arbitrary positive (1, 1)-form on M , we can then take w = cr/ for c sufficiently large.)

We shall use the following global regularization theorem of Demailly:

L e m m a 4.1. (DemaiUy) Let M be a compact complex manifold. Let T be a d-closed (1, 1)-current with T > O, where 0 is a continuous real (1, 1)-form. Let w be a smooth positive (1, 1)-form satisfying (4.1). Then for any c > O, there is a sequence of closed (1, 1)-currents


Tc,k in the de Rham class of T and a decreasing sequence of continuous functions/z~,k on M such that

(1) T~,k = T + v/Z-lOOqJ~,k, where {gbc,k} is a decreasing sequence of locally integrable functions on M such that qJc,k ~ 0 pointwise as k ---+ -q-oc (and thus T~,k ---* T weakly in V'I:(M));

(2) T~,k is smooth on M - E ~ ( T ) f o r all c, k;

(3) l imk_+~ #c,k(x) = min{n(T, x) , c} for all x E M , and #~,k < c + 1/k;

(4) T~,k _> 0 - #~,kaJ.

R e m a r k . Demailly's smoothing theorem [4, Theorem 1.1] is stronger than the statement of Lemma 4.1. To obtain this lemma from Demailly's smoothing theorem, we let u = w in [4, Theorem 1.1], and after passing to a subsequence so that we may take ek = 1/k , we let r = ~ , k -- ~ and/-~c,k = min{~k, c} + ek. In fact, for our purposes we could use instead a more elementary result of Demailly [4, Theorem 2.14] to obtain the conclusion of Lemma 4.1 for w sufficiently large so that it satisfies an inequality that is stronger than (4.1). [ ]

We note the following consequence of Lemma 4.1:

T h e o r e m 4.2. Let M be a compact complex manifold of dimension n. Let T be a d-closed semi-positive (1, 1)-current on M . Suppose there is a point a E M such that the Lelong number n ( T , a) = O. Then the intersection number

(T. [C]) > 0

for all irreducible curves C containing a.

Proof . Let M, T, a be as in Theorem 4.2 and let C be an irreducible curve containing a. Choose a positive (1, 1)-form ~o satisfying (4.1). Let c = 1 + max~eM n ( T , x ) , which is finite by the upper semi-continuity of the Lelong number. Let To,k,/zc,k be given as in Lemma 4.1, with O = 0. Since Ec = @ by the choice of c, Tc,k 6 E I : ( M ) , for k = 1 , 2 , . . . . Thus

(T [Cl) = (Tc k [Cl)= ([C1,Tck) : fc >_ - fc r~g mg

(4.2)

By the Lebesgue dominated convergence theorem

C,,g #c'kCa

54

and hence


(T. [c]) >_ - / ~ . ~(T, z)~(~). (4.3)

For each positive integer k, Es/a N C is a subvariety of C by Siu's theorem [12]. Since C is irreducible and a ~_ E1/~, it follows that/~l/k f3 C is finite. Thus rz(T, z) = 0 for all z E C outside a countable set. The conclusion follows from (4.3). [ ]

Theorem 4.2 for the case where M is projective-algebraic was proved by H. Tsuji in an unpublished manuscript. A related consequence of Lemma 4.1 is the following restatement of a result of Demailly [4, Corollary 6.4]:

Theorem 4.3. (Demailly) Let M be a compact complex manifoM of dimension n. Let T1 and T2 be d-closed, semi-positive currents of bidegrees (I, 1) and (n - 1, n - I), respectively, on M such that r~(T1, a) = O for all a outside a countable subset of M . Then the intersection number

(T, . T:) >_o.

Furthermore, if in addition there is a point Xo E M such that T1 and T2 are strictly positive at Xo, then

(T~ �9 T:) >0.

Proof. Let a > 0 be arbitrary. Let/9 = 0, or if TI and T2 are strictly positive at z0, let E s (M) such that 8(z0) > 0 and T1 .> 8 > 0. Let To,k, #r be given as in Lemma 4.1,

with T = 7"1 and ~v satisfying (4.1). then

Tc,k > 0 -- # ~ > 0 -- (c + 1 / k ) ~ . (4.4)

Choose c > 0 small enough and k large enough so that c + 1/k < a/2. Thus Tc,k > O - (a /2)w. Since Tc,k is smooth outside the finite set Ec(T) , by Lemma 3.5 we can find a smooth (1, 1)-form

such that IT] = ITs,k] = [T] and T + aw > 0 on M. Thus

(T, . T2) = (T2 ,~) >_ (T2,e) - a(T=,~).

Since a > 0, the conclusion follows. []


A consequence of Theorem 4.3 (or of Theorem 4.2) is the following sufficient condition for numerical effectiveness originally given by H. Tsuji (unpublished):

Corol la ry 4.4. (Tsuji) Let L be a holomorphic line bundle over a projective-algebraic manifold M . If L has a singular hermitian metric h such that cl (L, h) is a semi-positive current and n(c l (L , h), a) = O for all a outside a countable subset of M, then L is nef.

Proof of Theorem 1.1.

(1) ~ (2): If M is a Moishezon manifold, there is a modification 7r : M , M such

that the manifold M is projective-algebraic. Let ~ be an integral K/ilaler form on M; then the push-forward current 7r,~ is an integral K/ihler current on M (see Definitions 2.2b and 2.3a).

(2) ::~ (1): Choose a positive (1, 1)-form 0 such that T _> 0, and let w be a positive (1, 1)-form satisfying (4.1). Choose a > 0 such that aw < 0. Let To,k, #c,k be given as in Lemma 4.1, and choose c, k such that c + 1/k < a/2. Then

10 Tc ,k>_O--#kw>_O-- c + co> 2 (4.5)

Hence Tc,k is an integral Kahler current that is smooth outside the analytic subset Ec(T). Then by [8, Theorem 1.1], it follows that M is a Moishezon manifold.

The equivalence of (2) and (3) is a consequence of the following lemma:

Lemma 4.5. (cf. [11, Lemma 2.36] or [8, Lemma 2.1]) Let M be a complex manifold and let r l be a closed integral (1, 1)-current of order 0 on M. Then there exists a holomorphic line bundle L on M with a singular hermitian metric h such that

r] = cI(L, h).

Let us recall the asymptotic behavior of the dimension of the space H ~ L k) as k --* r Suppose that E is a holomorphic line bundle with a nontrivial holomorphic section on a compact complex manifold. Associated to E we have the meromorphic map

~E : M - - - - , P ( H ~

(where PV denotes the space of hyperplanes in a finite-dimensional, complex vector space V) given by

�9 E(z ) = {s E H ~ s ( x ) = 0 } .


Let L be a holomorphic line bundle on M. Then the litaka dimension of L, written tz(L), is given by

a (L) = max dim ffL ~ (M) . (4.6) k>0

( I f / - /~ L k) = 0 for all k > 0, then one writes g(L) = -oe . ) Note that

a (L) _< c~(M) < dim M, (4.7)

where a ( M ) is the algebraic dimension of M, i.e., the transcendence degree of the meromorphic function field of M. A line bundle L is said to be big if a (L) = dim M. Thus a compact complex manifold is Moishezon if and only if it carries a big line bundle.

The following theorem, which was proven by Demailly [3, Proposition 4.2b] for the case where M is assumed to be Kfihler, is a consequence of the proof of Theorem 1.1:

Theorem 4.6. Let L be a holomorphic line bundle on a compact complex manifold M. Then L is big if and only if L has a singular hermitian metric h such that the curvature current cl ( L, h) is strictly positive.

Proof. If L is big, then M is Moishezon, so we can find a modification 7r : M ~ M , where M is projective-algebraic. Let

L = 7r'(L),

which is also big [ 16, Theorem 5.13]. Then by [3, Proposition 4.2b], I, carries a singular hermitian metric h with strictly positive curvature current cl (L, h). Then h descends to a singular hermitian metric h on L, as follows: Let U be an open set in M on which L has a trivialization, and let

= 7r-l(U). Let gb~ E L~oc(U) define h as in Definition 2.3b. We let q~u = 7r,(~ E 7)'~ Since q~u E L~o~(U), we obtain a conection {4t~} that defines a singular metric h on L. Since

v) = =

it follows from Definitions 2.2b and 2.3b that

el(L, h) = .cl(L, h) > O.

Now suppose that Cl (L, h) is strictly positive. By the proof of Theorem 1.1, we may assume that h is smooth on M - ,_q, where S is a proper analytic subset of M. By a minor modification


of the proof of Theorem 1.1 in [8] (as in the proof of Proposition 4.2(b) in [3]), we can show that for any holomorphic line bundle E on M , there is a positive integer k such that

dinl (~ L k |174 M -~- dimM.

Letting E = K ~ 1, we have t~(L) = dim M. []

Theorem 4.6 is the singular analogue of the form of the Kodaira embedding theorem that says that a holomorphic line bundle is ample if and only if it is positive. Next, we shall prove the following sufficient condition for a line bundle to be big, which together with Lemma 4.5 yields Theorem i.2:

Theorem 4.7. Let M be a compact complex manifold of dimension n. Let L be a holomorphic line bundle over M with a singular metric h such that cl ( L, h) is semi-positive on M and is strictly positive at some point Xo E M , and suppose that sing supp ci (L, h) is contained in a Stein open subset of M . Then L is big.

In order to prove Theorem 4.7, we need the following three lemmas:

Lemma 4.8. [i6, Theorem 8.1] Let L be a holomorphic line bundle over a compact complex manifold M such that t~(L) >_ O. Then there are positive constants C, C ~ and a positive integer ko such that

C k '~(L) < dim H ~ L k) <_ C ' k ~(L) for k _> ko.

A consequence of Lemma 4.8 is that a holomorphic line bundle L on a compact complex manifold M of dimension n is big if and only if there is a positive constant C such that

d i m H ~ k) >_ C k n,

for k sufficiently large.

L e m m a 4.9. [15, p. 176] (see also [14, (5.2)]) Let M be an n-dimensional compact complex manifold, and let w be a strictly positive (1, 1)-form on M. Then there exists a positive constant C ( M , w ), which depends only on M and w, with the following property: Suppose G is an open subset of M , a and b are positive numbers, and L is a holomorphic line bundle on M admitting a smooth hermitian metric h such that ci ( L, h) >_ aw on G and cl ( L, h) > - b w on

M - G . I f

C ( M , w ) [ 1 + log+(b/a)]n(b2/a)" f M - G w" < ( c , ( L ) " , M ) ,

58

then

for k sufficiently large.


k72 dimH~ k) > 2-~.) (cx(L)n,M)

Lemma 4.10. Let M be an n-dimensional compact complex manifold. Let T be as in Theorem 1.2. Then

([T] n, M) ~ 1.

where IT] denotes the de Rham class o fT in H2(M, •).

Proof. Choose 8 6 C 1'1 (M) such that 8(xo) > 0 and T >/9 > 0. Choose open sets V, X in M such that sing supp T C V C C X and X is Stein. Let u be a strictly plurisubharmonic C ~ function on X. Let p G C ~ ( M ) such that p > 0, supp p C X and ply = 1. Let

r / = X/r-~a0(/Tu). (4.8)

m

Then dr] = 0, r/is strictly positive on V and supp r/ C X. Let C > 0 be arbitrary. Then we can choose a > 0 such that

and

/M_v(T + arl) ~ >_ /M_V T" - C >_ fM_V 8~ -- C (4.9)

Since T is integral, the conclusion of the lemma follows. []

Since C > 0 is arbitrary,

([TIn, M ) > fM On > O.

( [ T + a~]n ,M) ~ ([T]~,M) + C. (4.10)

Choose w 6 E 1'1 ( M ) such that w > 0 on M and w = r/on V , and let T be as in the conclusion of Lemma 3.5. Then

([T]" ,M) >_ fM(T-t-a~) n - C

=

> fMOn--2C.


Proof of Theorem 4.7. Let T = cl (L, h), where L, h are as in the statement of the theorem. Choose 0 as in the proof of Lemma 4.10, and let co be a strictly positive ( 1, 1)-form on M. Let G be an open neighborhood of x 0 and let a > 0 such that

T > 0 > aw on G. (4.11)

Then choose a real number b such that 0 < b < a and

C(M,w) (1 + l o g + b b)n ( ba-~-~)n fM ,,, a/~ <__ l, a - - -G

(4.12)

where C(M, w) is as in Lemma 4.9. By Lemma 3.5, we can find a distribution S e ~D'(M) and a smooth (1, 1)-form T such that T = T - v/-Z'TOOS and T + bw >_ O. Hence T >__ (a - b)w on G and T > - b w on M .

Note that S is locally the difference of plurisubha[monic functions, and thus S E L~o~(M ). Therefore we can define the singular hermitian metric h = he ~s on L with curvature current

C 1 ( L , h ) = c 1 (L, h) - v/-~--lOOS = T.

By Lemma 4.9, it follows that

dimH~ k) >_ ~ ( [ T ] n , M )

for k sufficiently large. Then by Lemmas 4.8 and 4.10 it follows that L is big. [ ]

The following intersection inequality of Demailly tells us that if M is compact K~hler, then the conclusion of Lemma 4.10 is valid if T satisfies the weaker hypothesis of Theorem 1.3:

L e m m a 4.11. (Demailly [4, Corollary 7.6]) Let T be a d-closed semi-positive (1, 1)- current on a compact Kiihler manifold M. Suppose that the Lelong number n(T, x) -~ O for all x E M except for a countable subset. Then

([T]~' M ) >- ~ n(T'x)'~ + /M T:~, xEM

where Tabc denotes the absolutely continuous part of the Lebesgue decomposition of T.


Theorem 1.3 is likewise a consequence of the following result:

Theorem 4.12. Let M be a compact complex surface or a compact Kithler manifold. Let L be a holomorphic line bundle over M with a singular metric h such that ci (L~ h) is semipositive on M and is strictly positive at some point :Co 6 M . I f cl (L, h) has zero Lelong numbers outside of a countable subset of M , then L is big.

Proof. Let T = cl (L, h) and choose 0 as in the proof of Lemma 4.10. We first show that

(IT]n, M ) > 0. (4.13)

If d i m M = 2, then (4.13) follows from Theorem 4.3. On the other hand, if M is K~ler, we can apply Lemma 4.11 to conclude that

(IT1 o, M) > { _> { oo > o. JM JM

Let w satisfy (4.1), and choose G, a, b, as in the proof of Theorem 4.7 so that the inequalities (4.11) and (4.12) are satisfied. Let To,k, #c,k, r be given as in Lemma 4.1; then

T~,k >_ 0 -- I.Zkw >_ 0 - (c + 1/k)w. (4.14)

Choose c > 0 small enough and k large enough so that c + 1/k < b/2. Thus Tc,k > 0 - (b/2)w. Since Tc,k is smooth outside the finite set E~(T) , by Lemma 3.5 we can find a distribution

S 6 :D'(M) and a smooth (1, 1)-form T such that

= T~,k + v~OOS = T + x/=100(S-+- Co,k)

and T -I- bw _> 0 on M. Then we can use the same argument as in the proof of Theorem 4.7 to conclude that L is big. [ ]

To obtain Theorem 1.3 from Theorem 4.12, we use Lemma 4.8 as before to conclude that M is Moishezon. If dim M = 2, then M must be projective-algebraic by the Chow-Kodaira Theorem (see Corollary 3.6). If M is ~h le r , the conclusion follows by a result of Moishezon [I0] (cf. Definition 2.3c).

Added note. J.-P. Demailly informed the authors that he has obtained a proof that a holomorphic line bundle L on a compact complex surface M is big if (and only if) it has a singular hermitian metric h such that cl (L, h) is semi-positive over M and is strictly positive at one point. As a consequence, a compact complex surface is projective-algebraic if and only if it carries a d-closed, integral, semi-positive (1, 1)-current that is stricfl,y positive at one point.


References

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[3] Demailly, J.-P. Singular Hermitian metrics on positive line bundles. In: Complex Algebraic Varieties, Lecture Notes in Mathematics 1507, pp. 87-104. New York: Springer-Verlag 1992..

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[11] Shiffman, B., and Sommese, A. J. Vanishing Theorems on Complex Manifolds. Boston: Birkh~iuser 1985. [12] Siu, Y.-T. Analyficity of sets associated to Lelong numbers and the extension of closed positive currents. Invent.

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(1984). [15] Siu, Y.-T. Some recent results in complex manifold theory related to vanishing theorems for the semipositive case.

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439. New York: Springer-Verlag 1975.

Received June 2, 1992

Department of Mathematics, University of Houston, Houston, TX 77204 USA Department of Mathematics, The Johns Hopkins University, Baltimore, MD 21218 USA

Properties of compact complex manifolds carrying closed ...shiffman/publications/34 closed positive...

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