Related papers: l-adic Monodromy and Shimura curves in positive ch…

For a connected reductive group $G$ over a finite field, we define partial Hasse invariants on the stack of $G$-zip flags. We obtain similar sections on the flag space of Shimura varieties of Hodge-type. They are mod $p$ automorphic forms…

Let (P,X) be Shimura data, M=M(P,X,K) the Shimura variety of level K. To an algebraic representation of P, one can associate a mixed sheaf (variation of Hodge structure, l-adic sheaf) on M. In the paper, we study the degeneration of such…

We prove results that imply, under various hypotheses, that every elliptic curve over a number field $k$ corresponding to a point on a modular curve has bad reduction at a certain prime $p$ of $\mathcal{O}_k$. For example, every elliptic…

The purpose of this paper is to prove a local p-adic monodromy theorem for ordinary abelian surfaces and K3 surfaces with bad reduction in characteristic p. As an application, we get a finiteness result for the reduction of their Hecke…

We study curves of negative self-intersection on algebraic surfaces. We obtain results for smooth complex projective surfaces X on the number of reduced, irreducible curves C of negative self-intersection C^2. The only known examples of…

Let f:X-->Y be a semi-stable family of complex abelian varieties over a curve Y of genus q, and smooth over the complement of s points. If F(1,0) denotes the non-flat (1,0) part of the corresponding variation of Hodge structures, the…

For $4 \nmid L$ and $g$ large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level $L$ structures. In particular, we determine the divisibility properties of the…

We compute the image of the $p$-adic period map for polarized K3 surfaces with supersingular reduction. This gives rise to a Rapoport-Zink type uniformization of their moduli space by an explicit open rigid analytic subvariety of a local…

Let S be a K3 surface and Aut D(S) the group of auto-equivalences of the derived category of S. We construct a natural representation of Aut D(S) on the cohomology of all moduli spaces of stable sheaves (with primitive Mukai vectors) on S.…

We introduce a new approach of computing the automorphism group and the field of moduli of points $\p=[C]$ in the moduli space of hyperelliptic curves $\H_g$. Further, we show that for every moduli point $\p \in \H_g(L)$ such that the…

In this article, we define the l-adic homology for a morphism of schemes satisfying certain finiteness conditions. This homology has these functors similar to the Chow groups: proper push-forward, flat pull-back, base change, cap-product,…

Let $X$ be the special fiber of a unitary Shimura variety of hyperspecial level at a prime $p$ inert in the totally real field $F$. Let $Y\to X$ be the associated flag space. For every $L$-dominant weight $\lambda$, let…

Given an automorphism of a smooth complex algebraic curve, there is an induced action on the moduli space of semi-stable rank 2 holomorphic bundles with fixed determinant. We give a complete description of the fixed variety in terms of…

This survey article discusses some results on the structure of families f:V-->U of n-dimensional manifolds over quasi-projective curves U, with semistable reduction over a compactification Y of U. We improve the Arakelov inequality for the…

In this paper we study the local geometry of the stack of pointed $A_r$-stable curves. In particular, we analyze the deformation theory of $A_r$-stable curves and their automorphism groups in order to study the combinatorics of families of…

We show that the moduli spaces of Scholze's $p$-adic shtukas with framing satisfy a $p$-adic rigid analytic version of Borel's extension theorem. In particular, this holds for local Shimura varieties, for all local Shimura data…

We prove several related results on the low-degree Hodge numbers of proper smooth rigid analytic varieties over non-archimedean fields. Our arguments rely on known structure theorems for the relevant Picard varieties, together with recent…

We extend fundamental inequalities related to the canonical map of surfaces of general type to positive characteristic. Next, we classify surfaces on the Noether lines, i.e., even and odd Horikawa surfaces, in positive characteristic. We…

We introduce and study the category of modular (i.e. with coefficient of positive characteristic) monodromic perverse sheaves on complex stratified $T$-varieties, with $T$ a complex algebraic torus. In particular, we show that under…

We construct integral models over $p=2$ for some Shimura varieties of abelian type with parahoric level structure, extending the previous work of Kim-Madapusi, Kisin, Pappas, and Zhou. For Shimura varieties of Hodge type, we show that our…