A Theorem on Bimeromorphic Maps of Kähler Manifolds and Its Applications

Abstract

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