A Theorem on Bimeromorphic Maps of Kähler Manifolds and Its Applications
A Theorem on Bimeromorphic Maps of Kähler Manifolds and Its Applications
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凯勒流形双亚纯映射定理及其应用
DOI：
10.2977/prims/1195185272
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发表时间：
1981
影响因子：
1.2
通讯作者：
A. Fujiki
中科院分区：
文献类型：
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作者：
A. Fujiki
对于紧致复流形Z，记为P(Z)(Rep.SP(ZJ)H(Z，R)中的凸锥，由正的类组成。半肯定的)在Kodaira意义上。因此，我们有P(Z)SSP(Z)Gh&Gt；(Z)sh(Z，R)。特别地，Z是Kahler当且仅当P(Z)^0，在这种情况下SP(Z)是P(Z)在H(Z，R)中的闭包。设/：X-？是紧致复流形的双纯映射。设f诱导余维S和Gt；2的解析子集的补的同构，则作为本文的一个主要定理，我们将证明f是双全纯的，或者f*(P(Xj)ft SP(X)=0在H(Y，B)中，其中/*：H(X，R)-*H(Y，R)是由f诱导的同态。特别地，如果X是具有充分因子D的射影的，并且如果线性系统f#D在Y上是无基点的，则f一定是双全纯的，这一事实可以直接用自然同构r(X，Ox(D})^r(Y，0v(f*D))来证明。然而，在文[4，(1.13)]中，我们在某种特殊情况下给出了这一点的另一种证明，该证明实际上也适用于下面的引理3.1和众所周知的因子引理2.4中的变换公式(8)的一般情况。后一种证明的优点在于，它可以进一步推广到给出上述主要定理。为此，由于Kahler类，或者更广泛地说，半正类一般不能用因子表示，作为替代，我们考虑Lelong意义下的(1，1)型正流[10]。它们一方面包括作为特例的半正定形式，另一方面包括有效的
For a compact complex manifold Z we denote by P(Z) (resp. SP(ZJ) the convex cone in H(Z, R) consisting of classes which are positive (resp. semipositive) in the sense of Kodaira. Thus we have P(Z)sSP(Z)gH > (Z)sH(Z, R). In particular Z is Kahler if and only if P(Z)^0, and in this case SP(Z) is the closure of P(Z) in H(Z, R). Now let/: X-»ybe a bimeromorphic map of compact complex manifolds. Suppose that / induces an isomorphism of complements of analytic subsets of codimension S>2. Then as a main theorem of this note we shall show that either f is biholomorphic or f*(P(XJ) ft SP(X) = 0 in H(Y, B), where /*: H(X, R)-*H(Y, R) is the homomorphism induced by f. (See Theorem 3.2 for a little more general statement.) In particular if X is projective with an ample divisor D and if the linear system \f#D\ is base point free on Y, then/ must be biholomorphic, the fact which can be verified directly using the natural isomorphism r(X, Ox(D})^r(Y, 0v(f*D)). However, in [4, (1.13)] we have given another proof for this in a certain special case, which in fact is applicable also to the general case in view of Lemma 3.1 below and of the transformation formula (8) in Lemma 2.4, well-known for divisors. The advantage of the latter proof lies in the fact that it can further be generalized to give the main theorem as above. For this purpose, since a Kahler class, or more generally, a semipositive class cannot in general be represented by divisors, as substitutes we consider positive currents of type (1,1) in the sense of Lelong [10]. They include as special cases semipositive forms on the one hand, and effective