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Register a new user$ell$-adic Tautological Systems
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Tautological systems was introduced in Lian-Yau as the system of differential equations satisfied by period integrals of hyperplane sections of some complex projective homogenous varieties. We introduce the $ell$-adic tautological systems for the case where the ground field is of characteristic $p$.
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We lift the classical Hasse--Weil zeta function of varieties over a finite field to a map of spectra with domain the Grothendieck spectrum of varieties constructed by Campbell and Zakharevich. We use this map to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map $mathbb{S} to K(Var_k)$ induced by the inclusion of $0$-dimensional varieties is not surjective on $pi_1$ for a wide range of fields $k$. The methods used in this paper should generalize to lifting other motivic measures to maps of $K$-theory spectra.
Let $X$ be a smooth connected projective algebraic curve over an algebraically closed field, and let $S$ be a finite nonempty closed subset in $X$. We study deformations of $overline{mathbb F}_ell$-sheaves. The universal deformation space is a formal scheme. Its generic fiber has a rigid analytic space structure. By studying this rigid analytic space, we prove a conjecture of Katz which says that if a lisse $overline{mathbb Q}_ell$-sheaf $mathcal F$ on $X-S$ is irreducible and rigid, then we have $mathrm{dim}, H^1(X,j_astmathcal End(mathcal F))=2g$, where $j:X-Sto X$ is the open immersion, and $g$ is the genus of $X$.
Let C be a complex curve of genus g, let J(C) be its Jacobian and let R(C) be its tautological ring, that is, the group of algebraic cycles modulo algebraic equivalence. We study the algebraic structure of R(C). In particular, we give a detailed description of all the possibilities that may occur for g<9: we construct convenient basis and we determine the matrices representing the Fourier transform and both intersection and Pontryagin products explicitly. In particular, we estimate the dimension of R(C).
We take a new look at the curvilinear Hilbert scheme of points on a smooth projective variety $X$ as a projective completion of the non-reductive quotient of holomorphic map germs from the complex line into $X$ by polynomial reparametrisations. Using an algebraic model of this quotient coming from global singularity theory we develop an iterated residue formula for tautological integrals over curvilinear Hilbert schemes.
Relations among tautological classes on the moduli space of stable curves are obtained via the study of Wittens r-spin theory for higher r. In order to calculate the quantum product, a new formula relating the r-spin correlators in genus 0 to the representation theory of sl2 is proven. The Givental-Teleman classification of CohFTs is used at two special semisimple points of the associated Frobenius manifold. At the first semisimple point, the R-matrix is exactly solved in terms of hypergeometric series. As a result, an explicit formula for Wittens r-spin class is obtained (along with tautological relations in higher degrees). As an application, the r=4 relations are used to bound the Betti numbers of the tautological ring of the moduli of nonsingular curves. At the second semisimple point, the form of the R-matrix implies a polynomiality property in r of Wittens r-spin class. In the Appendix (with F. Janda), a conjecture relating the r=0 limit of Wittens r-spin class to the class of the moduli space of holomorphic differentials is presented.
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