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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].

14F17 Vanishing theorems in algebraic geometry
32L20 Vanishing theorems
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14C22 Picard groups