Click the icons on the right side to show/hide the abstracts.


Classics revisited: Éléments de géométrie algébrique
April 2018, pdf, submitted to Jahresbericht der DMV, 50 p.
About 50 years ago, Éléments de Géométrie Algébrique (EGA) by A. Grothendieck and J. Dieudonné appeared, an encyclopedic work on the foundations of Grothendieck's algebraic geometry. We sketch some of the most important concepts developed there, comparing it to the classical language, and mention a few results in algebraic and arithmetic geometry which have since been proved using the new framework.
Stratifications of affine Deligne-Lusztig varieties
January 2018, arxiv:1802.02225, 21 p.
Affine Deligne-Lusztig varieties are analogues of Deligne-Lusztig varieties in the context of affine flag varieties and affine Grassmannians. They are closely related to moduli spaces of $p$-divisible groups in positive characteristic, and thus to arithmetic properties of Shimura varieties. We compare stratifications of affine Deligne-Lusztig varieties attached to a basic element $b$. In particular, we show that the stratification defined by Chen and Viehmann using the relative position to elements of the group $\mathbb J_b$, the $\sigma$-centralizer of $b$, coincides with the Bruhat-Tits stratification in all cases of Coxeter type, as defined by X. He and the author.
Erratum: $P$-alcoves and nonemptiness of affine Deligne-Lusztig varieties
with X. He, S. Nie, March 2017, pdf, 3 p.
We correct two small mistakes in [G., He, Nie, $P$-alcoves and nonemptiness of affine Deligne-Lusztig varieties]. All the main results remain unaffected.
Fully Hodge-Newton decomposable Shimura varieties
with X. He, S. Nie, arXiv:1610.05381
The motivation for this paper is the study of arithmetic properties of Shimura varieties, in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure. This is closely related to the structure of Rapoport-Zink spaces and of affine Deligne-Lusztig varieties.
We prove a Hodge-Newton decomposition for affine Deligne-Lusztig varieties and for the special fibres of Rapoport-Zink spaces, relating these spaces to analogous ones defined in terms of Levi subgroups, under a certain condition (Hodge-Newton decomposability) which can be phrased in combinatorial terms.
Second, we study the Shimura varieties in which every non-basic σ-isogeny class is Hodge-Newton decomposable. We show that (assuming the axioms of \cite{HR}) this condition is equivalent to nice conditions on either the basic locus, or on all the non-basic Newton strata of the Shimura varieties. We also give a complete classification of Shimura varieties satisfying these conditions.
While previous results along these lines often have restrictions to hyperspecial (or at least maximal parahoric) level structure, and/or quasi-split underlying group, we handle the cases of arbitrary parahoric level structure, and of possibly non-quasi-split underlying groups.

Refereed publications

Basic loci in Shimura varieties of Coxeter type
with X. He, Cambridge J. of Math. 3 (2015), no. 3, 323-353, arXiv:1311.6263
This paper is a contribution to the general problem of giving an explicit description of the basic locus in the reduction modulo p of Shimura varieties. Motivated by work of Vollaard-Wedhorn and Rapoport-Terstiege-Wilson, we classify the cases where the basic locus is (in a natural way) the union of classical Deligne-Lusztig sets associated to Coxeter elements. We show that if this is satisfied, then the Newton strata and Ekedahl-Oort strata have many nice properties.
Erratum to: Basic loci in Shimura varieties of Coxeter type
with X. He, Cambridge J. of Math. 6 (2018), no. 1, 89-92, pdf
We correct a few mistakes in [G., He, Basic loci in Shimura varieties of Coxeter type]. The main results remain unaffected.
$P$-alcoves and nonemptiness of affine Deligne-Lusztig varieties
with X. He, S. Nie, Ann. sci. ENS 48 (2015), Fasc. 3, 647-665, arXiv:1211.3784
We study affine Deligne-Lusztig varieties in the affine flag manifold of an algebraic group, and in particular the question, which affine Deligne-Lusztig varieties are non-empty. Under mild assumptions on the group, we provide a complete answer to this question in terms of the underlying affine root system. In particular, this proves the corresponding conjecture for split groups stated in Görtz et al. (2010). The question of non-emptiness of affine Deligne-Lusztig varieties is closely related to the relationship between certain natural stratifications of moduli spaces of abelian varieties in positive characteristic.
The supersingular locus in Siegel modular varieties with Iwahori level structure
with C.-F. Yu, Math. Annalen 353 no. 2(2012), 465-498, SpringerLink, arXiv:math/0807.1229v2
We study moduli spaces of abelian varieties in positive characteristic, more specifically the moduli space of principally polarized abelian varieties on the one hand, and the analogous space with Iwahori type level structure, on the other hand. We investigate the Ekedahl-Oort stratification on the former, the Kottwitz-Rapoport stratification on the latter, and their relationship. In this way, we obtain structural results about the supersingular locus in the case of Iwahori level structure, for instance a formula for its dimension in case g is even.
Ekedahl-Oort strata and Kottwitz-Rapoport strata
with M. Hoeve, J. Algebra 351 (2012), 160-174, doi: 10.1016/j.jalgebra.2011.10.039 , arXiv:math/0808.2537
We study the moduli space A_g of g-dimensional principally polarized abelian varieties in positive characteristic, and its variant A_I with Iwahori level structure. Both supersingular Ekedahl-Oort strata and supersingular Kottwitz-Rapoport strata are isomorphic to disjoint unions of Deligne-Lusztig varieties (see [Hoeve 2008] and [Goertz, Yu 2008], resp.). Here we compare these isomorphisms. We also give an explicit description of Kottwitz-Rapoport strata contained in the supersingular locus in the general parahoric case. Finally, we show that every Ekedahl-Oort stratum is isomorphic to a parahoric Kottwitz-Rapoport stratum.
Dimensions of affine Deligne-Lusztig varieties in affine flag varieties
Affine Deligne-Lusztig varieties are analogs of Deligne-Lusztig varieties in the context of an affine root system. We prove a conjecture stated in the paper arXiv:0805.0045v4 by Haines, Kottwitz, Reuman, and the first named author, about the question which affine Deligne-Lusztig varieties (for a split group and a basic $\sigma$-conjugacy class) in the Iwahori case are non-empty. If the underlying algebraic group is a classical group and the chosen basic $\sigma$-conjugacy class is the class of $b=1$, we also prove the dimension formula predicted in op. cit. in almost all cases.
Affine Deligne-Lusztig varieties in affine flag varieties
with T. Haines, R. Kottwitz, D. Reuman, Compositio Math. 146 (2010), 1339-1382, arXiv:math/0805.0045v2
This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, extends previous conjectures concerning their dimensions, and generalizes the superset method.
Supersingular Kottwitz-Rapoport strata and Deligne-Lusztig varieties
We investigate Siegel modular varieties in positive characteristic with Iwahori level structure. On these spaces, we have the Newton stratification, and the Kottwitz-Rapoport stratification; one would like to understand how these stratifications are related to each other. We give a simple description of (conjecturally all) KR strata which are entirely contained in the supersingular locus as disjoint unions of Deligne-Lusztig varieties. We also give an explicit numerical description of the KR stratification in terms of abelian varieties.
On the connectedness of Deligne-Lusztig varieties
We give a criterion which determines when a union of one-dimensional Deligne-Lusztig varieties has a connected closure. We also obtain a new, short proof of the connectedness criterion for Deligne-Lusztig varieties due to Lusztig.
Alcove walks and nearby cycles on affine flag manifolds
Using Ram's theory of alcove walks, we give a proof of the Bernstein presentation of the affine Hecke algebra. The method works also in the case of unequal parameters. We also discuss how these results help in studying sheaves of nearby cycles on affine flag manifolds.
The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties.
We study the Jordan-Hoelder series for nearby cycles on certain Shimura varieties and Rapoport-Zink local models, and on finite-dimensional pieces of Beilinson's deformation of the affine Grassmannian to the affine flag variety (and their p-adic analogues). We give a formula for the multiplicities of irreducible constituents in terms of certain cohomology groups, and we also provide an algorithm to compute multiplicities, in terms of the affine Hecke algebra.
Arithmetic intersection numbers
We define the arithmetic intersection number of three modular divisors and interpret it from the point of view of algebraic stacks. A criterion is given when the intersection of three modular divisors is finite. Furthermore, the final result about the arithmetic intersection numbers, as given by Gross and Keating, is stated and the strategy of its proof, carried out in the subsequent chapters, is explained.
A sum of representation numbers
This article contains the proof of a formula stated in the paper by Gross and Keating on intersections of modular correspondences, for a certain sum of representation densities.
Dimensions of some affine Deligne-Lusztig varieties
This paper concerns the dimensions of certain affine Deligne-Lusztig varieties, both in the affine Grassmannian and in the affine flag manifold. Rapoport conjectured a formula for the dimensions of the varieties X_mu(b) in the affine Grassmannian. We prove his conjecture for b in the split torus; we find that these varieties are equidimensional; and we reduce the general conjecture to the case of superbasic b. In the affine flag manifold, we prove a formula that reduces the dimension question for X_x(b) with b in the split torus to computations of dimensions of intersections of Iwahori orbits with orbits of the unipotent radical. Calculations using this formula allow us to verify a conjecture of Reuman in many new cases, and to make progress toward a generalization of his conjecture.
Bounds on weights of nearby cycles and Wakimoto sheaves on affine flag manifolds.
We study certain nearby cycles sheaves on an affine flag manifold which arise naturally in the Beilinson-Gaitsgory deformation of the affine flag manifold to the affine Grassmannian. We study the multiplicity functions we introduced in an earlier paper, which encode the data of the Jordan-Hoelder series. We prove the multiplicity functions are polynomials in q, and we give a sharp bound for their degrees. Our results apply as well to the nearby cycles in the p-adic deformation of Laumon-Haines-Ngo, and also to Wakimoto sheaves.
Topological flatness of local models in the ramified case.
Local models are schemes defined in terms of linear algebra which can be used to study the local structure of integral models of certain Shimura varieties, with parahoric level structure. We investigate the local models for groups of the form ResF/QpGLn and ResF/QpGSp2n, where F/Qp is a totally ramified extension, as defined by Pappas and Rapoport, and show that they are topologically flat. In the linear case, flatness can be deduced from this.
Computing the alternating trace of Frobenius on the sheaves of nearby cycles on local models for GL_4 and GL_5.
Consider a Shimura variety defined over some number field, and assume we have a model over the ring of integers at some prime of bad reduction. It is then interesting to know the alternating trace of Frobenius on the invariants under the inertia group of the sheaf of nearby cycles, since these traces are related to the local factor of the Hasse-Weil zeta function. In this article we compute the semi-simple alternating trace of Frobenius for Shimura varieties associated to the groups GU(2,2) and GU(3,2), with level structure of Iwahori type, by investigating the equations of the local model defined by Rapoport and Zink and by performing explicit blowing ups.
On the flatness of local models for the symplectic group
We investigate the bad reduction of certain Shimura varieties (associated to the symplectic group). More precisely, we look at a model of the Shimura variety at a prime p, with parahoric level structure at p. We show that this model is flat, as conjectured by Rapoport and Zink, and that its special fibre is reduced.
On the flatness of models of certain Shimura varieties of PEL type
Consider a PEL-Shimura variety associated to a unitary group that splits over an unramified extension of Q_p. Rapoport and Zink have defined a model of the Shimura variety over the ring of integers of the completion of the reflex field at a place lying over p, with parahoric level structures at p. We show that this model is flat, as conjectured by Rapoport and Zink, and that its special fibre is reduced.
Coherent modules and their descent on relative rigid spaces.
We prove that faithfully flat descent holds for coherent modules on (classical) rigid spaces, also over non-noetherian bases.


Algebraic geometry I. Schemes, with Examples and Exercises
with T. Wedhorn, Vieweg (2010), 615 p.

Non-refereed publications/contributions to proceedings volumes

Book review: James Arthur, The Endoscopic Classification of Representations. Orthogonal and Symplectic Groups. AMS Colloquium Publ. 61 (2013)
in: Jahresbericht der DMV 116, Nr. 2 (2014) 123-128.
Abel-Preis für Pierre Deligne
Mitteilungen der DMV 21.3 (2013), 151-155
Eine Fields-Medaille für Bao Châu Ngô
Mitteilungen der DMV 19.4 (2011), 198-203
Affine Springer fibers and affine Deligne-Lusztig varieties
in: A. Schmitt (ed.), Proceedings of Affine Flag Manifolds and Principal Bundles (Berlin 2008), Trends in Mathematics, Birkhäuser (2010), pdf
Reduction of Shimura varieties and Deligne-Lusztig varieties
in Math. Forschungsinst. Oberwolfach Report 2010, no. 19, 1121-1124, pdf
Matrixgleichungen und Familien abelscher Varietäten in positiver Charakteristik
Mitteilungen der DMV 17.1 (2009) pdf
The supersingular locus in Siegel modular varieties, and Deligne-Lusztig varieties
in Math. Forschungsinst. Oberwolfach Report 2008, no. 5, 283-286, pdf.
Affine Deligne-Lusztig varieties
in Math. Forschungsinst. Oberwolfach Report 2008, no. 3, 136-139 pdf
Affine Deligne-Lusztig varieties
in: Math. Forschungsinst. Oberwolfach, Report 2007, no. 30, 1785-1787
Lokale Modelle von Shimura-Varietäten und ihre Garben der verschwindenden Zykel
in: Y. Tschinkel, Seminars 2003/04, Math. Inst. der Univ. Göttingen.

Reviews of these papers in MathSciNet.