Vanishing Theorems on Compact Hyper-kähler Manifolds

We prove that if B is a k-positive holomorphic line bundle on a compact hyper-kähler manifold M, then HpM,Ωq⊗B=0 for P>n+[k/2] with q a nonnegative integer. In a special case, k=0 and q=0, we recover a vanishing theorem of Verbitsky’s with a little stronger assumption.

EXAMPLES OF A NON-VANISHING CONJECTURE OF KAWAMATA

Let M be a compact complex manifold with a positive holomorphic line bundle L, and K be its canonical line bundle. We give some sufficient conditions for the non-vanishing of H0(M, K + L). We also show that the criterion can be applied to interesting classes of examples including all compact locally hermitian symmetric spaces of non-compact type, Mostow–Siu [10] surfaces, Kähler threefolds given by Deraux [3] and examples of Zheng [17].

Convexité holomorphe intermédiaire des revetements d'un domaine pseudoconvexe

Let M be a complex manifold and L → M be a positive holomorphic line bundle over M equipped with a Hermitian metric h of class C2. If D ⊂⊂ M is a pseudoconvex domain which is relatively compact in M then there exists an integer r0 such that for every r ≥ r0 and for every connected holomorphic covering π: the covering is holomorphically convex with respect to holomorphic sections of .