Bounds for the stalks of perverse sheaves in characteristic p and a conjecture of Shende and Tsimerman
- PDF / 411,543 Bytes
- 32 Pages / 439.37 x 666.142 pts Page_size
- 68 Downloads / 192 Views
Bounds for the stalks of perverse sheaves in characteristic p and a conjecture of Shende and Tsimerman Will Sawin1
Received: 9 October 2019 / Accepted: 9 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove a characteristic p analogue of a result of Massey which bounds the dimensions of the stalks of a perverse sheaf in terms of certain intersection multiplicities of the characteristic cycle of that sheaf. This uses the construction of the characteristic cycle of a perverse sheaf in characteristic p by Saito. We apply this to prove a conjecture of Shende and Tsimerman on the Betti numbers of the intersections of two translates of theta loci in a hyperelliptic Jacobian. This implies a function field analogue of the Michel– Venkatesh mixing conjecture about the equidistribution of CM points on a product of two modular curves. 1 Introduction Massey used the polar multiplicities of a Lagrangian cycle in the cotangent bundle of a smooth complex manifold to bound the Betti numbers of the stalk of a perverse sheaf at a point [9, Corollary 5.5]. In this paper, we prove an analogous result in characteristic p. We use the characteristic cycles for constructible sheaves on varieties of characteristic p defined by Saito [12, Definition 5.10], building heavily on work of Beilinson [1]. Before stating our main theorem, let us define the polar multiplicities.
With an appendix by Jacob Tsimerman.
B Will Sawin
[email protected]
1
Department of Mathematics, Columbia University, New York, NY 10027, USA
123
W. Sawin
Definition 1.1 We say a closed subset, or algebraic cycle, on a vector bundle is conical if it is invariant under the Gm action by dilation of vectors. Definition 1.2 For a vector bundle V on a variety X , let P(V ) = Proj(Sym∗ (V ∨ )) be its projectivization, whose dimension dim X +rank V −1, which is equivalent to the quotient of the affine bundle V , minus its zero section, by Gm . For a conical cycle C on V , let P(C) the quotient of C, minus its intersection with the zero section, by Gm . Definition 1.3 Let X be a smooth variety of dimension n. Let C be a conical cycle on the cotangent bundle T ∗ X of X of dimension n and let x be a point on X . For 0 ≤ i < dim X , let V be a sub-bundle of T ∗ X defined over a neighborhood of x, with rank i + 1. such that the fiber Vx is a general point of the Grassmanian of i + 1-dimensional subspaces of (T ∗ X )x . Then we define the ith polar multiplicity of C at x, γCi (x), as the multiplicity of the pushforward π∗ (P(C) ∩ P(V )) at x, where π : P(T ∗ X ) → X is the projection. We define the nth polar multiplicity of C at x to be the multiplicity of the zero-section in C. Here π∗ (P(C) ∩ P(V )) is interpreted as an algebraic cycle, and the multiplicity of an algebraic cycle at a point is the appropriate linear combination of the multiplicities of its irreducible components. We will check that this multiplicity is independent of the choice of V with Vx sufficiently general in Sect. 3 below. Our result is as follows: Theorem
Data Loading...
