A derived category approach to generic vanishing

By Christopher D. Hacon. 2003

134 citations recorded by the source corpus.

We prove a Generic Vanishing Theorem for coherent sheaves on an abelian variety over an algebraically closed field k. When k = C this implies a conjecture of Green and Lazarsfeld.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/841f67c16d9a90f30dc7ffa55217bef531f408c0

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In Algebra and Number Theory

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Citations in the corpus

Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Paun (2016, methodology): “f twists by positive line bundles. An important addition was provided in work of Simpson [Sim93]. The results of Green-Lazarsfeld were extended to include higher direct images of canonical bundles in [Hac04]. From the point of view of this paper, the main statements to keep in mind are summarized in”
Kodaira dimension of algebraic fiber spaces over abelian varieties (2015, unknown): “[6], [7], [14], [15], [16], [18], [19], [21], [24], [25], [26], [30], [32], [34], [35] among many others.”
GENERIC VANISHING THEORY VIA MIXED HODGE MODULES (2011, methodology): “Why mixed Hodge modules? To motivate the introduction of mixed Hodge modules into the problem, let us briefly recall the very elegant proof of the generic vanishing theorem for ωX discovered by Hacon [14].”
On direct images of pluricanonical bundles (2014, background): “F ⊗f∗Pα) 6= 0 } ≥ k for every k ≥ 0. If f is generically ﬁnite, then by a special case of the generic vanishing theorem of Green and Lazarsfeld [GL87], ωX is a GV-sheaf. This was generalized by Hacon [Hac04] to the eﬀect that for an arbitrary f the higher direct images Rif∗ωX are GV-sheaves on A for”
Holonomic D-modules on abelian varieties (2013, unknown): “e L∈ Pic0(X). Hacon pointed out that the codimension bounds can be interpreted as properties of certain coherent sheaves on abelian varieties, and then reproved them using the Fourier-Mukai transform [Hac04]. His method applies particularly well to those 2010 Mathematics Subject Classiﬁcation. Prima”