Holomorphic cohomology groups on compact Kähler complex spaces

AbstractLet (X,OX) be a compact (reduced) complex space, bimeromorphic to a Kähler manifold. The singular cohomology groups Hq(X,C) carry a mixed Hodge structure. In particular they carry a weight filtration W−lHq(X,C) (l=0,…,q), and the graded quotients W−lHq(X,C)W−l−1Hq(X,C) have a direct sum decomposition into holomorphic invariants as ⊕r+s=q−l(W−lHq(X,C)W−l−1Hq(X,C))(r,s). Here we investigate the relationships between the above invariants for r=0 and the cohomology groups Hq(X,O˜X), where O˜X is the sheaf of weakly holomorphic functions on X. Moreover, according to the smooth case, we characterize the topological line bundles L on X such that the class of c1(L) in W0H2(X,C)W−1H2(X,C) has pure type (1,1)