On generic vanishing for pluricanonical bundles

Abstract

We study cohomology support loci and higher direct images of (log) pluricanonical bundles of smooth projective varieties or log canonical pairs. We prove that the 0-th cohomology support loci of log pluricanonical bundles are finite unions of torsion translates of subtori, and we give a generalization of the generic vanishing theorem to log canonical pairs. We also construct an example of morphism from a smooth projective variety to an abelian variety such that a higher direct image of a pluricanonical bundle to the abelian variety is not a GV-sheaf.

On the global generation of higher direct images of pluricanonical bundles

“…Note that R i f * (K m X ) are not necessarily GV -sheaves. In fact, [Shi16] constructed such a counterexample; see Example 4.5 there. After that T. Shibata [Shi16] (Proposition 4.8) proved that R i f * (K m X ) are GV -sheaves with the assumptions that dim X = 2 and κ(X) 0.…”

“…See Question in [PS14] after Corollary 5.4. Note that T. Shibata [Shi16] provided an example to show that…”

Pushforwards of klt pairs under morphisms to abelian varieties

“…We start with a corollary of Theorem 1.2. Note that [PS14] and [Shi16] have some results related to statements (i) and (ii) of Corollary 4.1 under different assumptions.…”

On the characterization of abelian varieties for log pairs in zero and positive characteristic

“…The author would like to thank Christopher for his constant support, encouragement and many enlightening discussions. He is also indebted to Karl Schwede who provided lots of help on F -singularities, and Sofia Tirabassi who pointed out the very important reference [Shi15]. Moreover he thanks Omprokash Das, Tommaso de Fernex, Yoshinori Gongyo, Zsolt Patakfalvi, Mihnea Popa and Shunsuke Takagi for helpful conversations.…”