A geometry of Kähler cones (Q1103075)

The purpose of the present paper is to introduce the notion of pseudo- positivity of holomorphic line bundles over a compact complex manifold in the class C and to investigate properties of such line bundles. Let X be an n-dimensional compact complex manifold. When there exists a Kähler metric on X, we define the Kähler cone KC(X) of X as the image of the set of all Kähler forms on X by the natural projection \(Z_{{\mathbb{R}}}^{1,1}(X)\to H_{{\mathbb{R}}}^{1,1}(X).\) Let X be an n- dimensional compact complex manifold in the class C and let \(\pi\) : \(Z\to X\) be a surjective bimeromorphic morphism such that Z is an n-dimensional compact Kähler manifold. A holomorphic line bundle L over X is defined to be pseudo-positive when \(c_ 1(\pi \quad *L)\in \overline{KC(Z)},\) where \(\overline{KC(Z)}\) is the closure of KC(Z) in \(H_{{\mathbb{R}}}^{1,1}(Z)\). Then our main theorem is as follows. Theorem. Let L be a pseudo-positive holomorphic line bundle over an n- dimensional compact complex manifold X in the class C, and moreover assume that L satisfies \(c_ 1(L)\) \(n[X]>0\). Then H \(q(X,K_ X\otimes L)=0\) for \(1\leq q\leq n.\) As an application of this result, we also show the following theorem. Theorem. Let \(p: {\mathcal M}\to D\) be a smooth and proper map of a complex manifold \({\mathcal M}\) to a disk D such that the fibres are complex manifolds of dimension n. Let \({\mathcal L}\) be a holomorphic line bundle over \({\mathcal M}\). For any t in D, we set \(M|_ t:=p^{-1}(t)\) and \(L|_ t:={\mathcal L}| (M)|_ t)\). For a positive integer a, we define a function \(P_{L,a}: D\to {\mathbb{Z}}\) by \(P_{L,a}(t):=h\) \(0(M|_ t,L\) \(a|_ t)\). Assume that \(M|_ 0\) is in the class C and that \((L\quad a|_ 0)\otimes K^{-1}_{M|_ 0}\) is pseudo-positive and \(c_ 1((L\quad a|_ 0)\otimes K^{- 1}_{M|_ 0})\quad n[M|_ 0]>0.\) Then \(P_{L,a}\) takes a constant on a neighborhood of origin.