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Duality between Cartier crystals and perverse $\mathbb{F}_p$-sheaves, and application to generic vanishing
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abstract
We show that on any Noetherian $F$-finite $\mathbb{F}_p$-scheme, there is an anti-equivalence of categories between Cartier crystals and \'etale perverse $\mathbb{F}_p$-sheaves, commuting with derived proper pushforwards. We use this duality to construct an upper shriek functor for Cartier crystals, and give new proofs of Kashiwara's equivalence and the finite length of Cartier crystals. Finally, we deduce a generic vanishing statement for perverse $\overline{\mathbb{F}}_p$-sheaves on abelian varieties of characteristic $p > 0$, reminiscent of the characteristic zero and $l$-adic statements.
Forward citations
Cited by 2 Pith papers
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On the Euler characteristic of weakly ordinary varieties of maximal Albanese dimension
Smooth proper weakly ordinary varieties of maximal Albanese dimension satisfy chi(X, omega_X) >= 0, with chi = 0 for non-general-type examples and the Albanese image then fibered by ordinary abelian varieties.
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Generic vanishing theory in positive characteristic
The paper proves an equivalence between Cartier crystals and V-crystals on dual abelian varieties and derives H^0(X,ω_X)≠0, with S^0(X,ω_X)≠0 in the ordinary case, for normal proper varieties of maximal Albanese dimension.
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