Pluri-canonical divisors on Kähler manifolds (Q1104575)

In this paper the author studies the behaviour of the pluri-genera under smooth deformation and shows that under certain conditions, the pluri- genera remain constant under deformation. The following is the main result of the paper. Let \(p: X\to M\) be a smooth proper map of complex manifolds, with connected fibers of dimension k. Suppose that \(p^{- 1}(0)=X_ 0\) is in the class \({\mathcal C}\). Let m be a positive integer and suppose that the divisor of some nonzero section in H \(0(X_ 0,(\Omega\) \(k_{X_ 0})^{\otimes m})\) is smooth. Then there is an analytic neighbourhood U of 0 in M such that for \(\ell =1,...,m\) the \(\ell\)-genus \(P_{\ell}(p^{-1}(z))\) is constant over U.