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Holomorphic line bundles with partially vanishing cohomology. (English) Zbl 0859.14005
Teicher, Mina (ed.), Proceedings of the Hirzebruch 65 conference on algebraic geometry, Bar-Ilan University, Ramat Gan, Israel, May 2-7, 1993. Ramat-Gan: Bar-Ilan University, Isr. Math. Conf. Proc. 9, 165-198 (1996).
Let $$X$$ denote a complex manifold of dimension $$n$$. The authors study holomorphic line bundles $$L$$ on $$X$$ with partially vanishing cohomology (or having metrics with positive eigenvalues of curvature). They define $$\sigma_+(L)$$ to be the smallest integer $$q$$ with the following property: There exists an ample divisor $$D$$ on $$X$$ and a constant $$c>0$$ such that $$H^j(X, mL-pD)=0$$ for all $$j>q$$ and $$mp\geq 0$$, $$m\geq c(p+1)$$. Note that $$\sigma_+(L)=0$$ if and only if $$L$$ is ample while $$\sigma_+(L)=n$$ if and only if $$c_1(L^*)$$ is in the closure of the cone of effective divisors. An ample $$q$$-flag is defined as a sequence $$Y_q\subset Y_{q+1}\subset \cdots\subset Y_n=X$$ of subvarieties $$Y_k$$ of $$X$$ such that $$\dim Y_k=k$$ and $$Y_k$$ is the image of an ample Cartier divisor in the normalization of $$Y_{k+1}$$. Then a line bundle $$L$$ is called $$q$$-flag positive if for some ample $$q$$-flag, $$L\mid Y_q$$ is positive.
Vanishing theorem: If $$L\in\text{Pic}X$$ is $$q$$-flag positive then $$\sigma_+(L)\leq n-q$$.
The converse of this theorem is not true in general. A counter example and a positive result (of converse) for $$\mathbb{P}_{n-1}$$ bundles over a curve are given. The structure of projective 3-folds with $$\sigma_+(-K_X)=1$$, $$K_X=$$ canonical bundle, is investigated. One has $$\sigma_+(-K_X)=0$$ if and only if $$X$$ is Fano and $$\sigma_+(-K_X)\leq 2$$ if and only if $$\kappa(X)= -\infty$$. The authors also study various cones in $$NX(X) \otimes\mathbb{R}$$, $$NX(X)$$ being Néron-Severi group, i.e. the group of divisors modulo numerical equivalence. All these cones coincide for surfaces.
For the entire collection see [Zbl 0828.00035].

##### MSC:
 14F17 Vanishing theorems in algebraic geometry 32L20 Vanishing theorems 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14C22 Picard groups
##### Keywords:
flag; holomorphic line bundles; vanishing cohomology