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Register a new userHodge numbers are not derived invariants in positive characteristic
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We study a pair of Calabi-Yau threefolds X and M, fibered in non-principally polarized Abelian surfaces and their duals, and an equivalence D^b(X) = D^b(M), building on work of Gross, Popescu, Bak, and Schnell. Over the complex numbers, X is simply connected while pi_1(M) = (Z/3)^2. In characteristic 3, we find that X and M have different Hodge numbers, which would be impossible in characteristic 0. In an appendix, we give a streamlined proof of Abuafs result that the ring H^*(O) is a derived invariant of complex threefolds and fourfolds. A second appendix by Alexander Petrov gives a family of higher-dimensional examples to show that h^{0,3} is not a derived invariant in any positive characteristic.
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In this paper, we compute the number of covers of curves with given branch behavior in characteristic p for one class of examples with four branch points and degree p. Our techniques involve related computations in the case of three branch points, and allow us to conclude in many cases that for a particular choice of degeneration, all the covers we consider degenerate to separable (admissible) covers. Starting from a good understanding of the complex case, the proof is centered on the theory of stable reduction of Galois covers.
We study isogenies between K3 surfaces in positive characteristic. Our main result is a characterization of K3 surfaces isogenous to a given K3 surface $X$ in terms of certain integral sublattices of the second rational $ell$-adic and crystalline cohomology groups of $X$. This is a positive characteristic analog of a result of Huybrechts, and extends results of the second author. We give applications to the reduction types of K3 surfaces and to the surjectivity of the period morphism. To prove these results we describe a theory of B-fields and Mukai lattices in positive characteristic, which may be of independent interest. We also prove some results on lifting twisted Fourier--Mukai equivalences to characteristic 0, generalizing results of Lieblich and Olsson.
We start with a curve over an algebraically closed ground field of positive characteristic $p>0$. By using specialization techniques, under suitable natural coprimality conditions, we prove a cohomological Simpson Correspondence between the moduli space of Higgs bundles and the one of connections on the curve. We also prove a new $p$-multiplicative periodicity concerning the cohomology rings of Dolbeault moduli spaces of degrees differing by a factor of $p$. By coupling this $p$-periodicity in characteristic $p$ with lifting/specialization techniques in mixed characteristic, we find, in arbitrary characteristic, cohomology ring isomorphisms between the cohomology rings of Dolbeault moduli spaces for different degrees coprime to the rank. It is interesting that this last result is proved as follows: we prove a weaker version in positive characteristic; we lift and strengthen the weaker version to the result in characteristic zero; finally, we specialize the result to positive characteristic. The moduli spaces we work with admit certain natural morphisms (Hitchin, de Rham-Hitchin, Hodge-Hitchin), and all the cohomology ring isomorphisms we find are filtered isomorphisms for the resulting perverse Leray filtrations.
We study the existence of Fuchsian differential equations in positive characteristic with nilpotent p-curvature, and given local invariants. In the case of differential equations with logarithmic local mononodromy, we determine the minimal possible degree of a polynomial solution.
We discuss two elementary constructions for covers with fixed ramification in positive characteristic. As an application, we compute the number of certain classes of covers between projective lines branched at 4 points and obtain information on the structure of the Hurwitz curve parametrizing these covers.
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