We compute the dimension of the tangent space to, and the Krull dimension of the pro-representable hull of two deformation functors. The first one is the ``algebraic'' deformation functor of an ordinary curve X over a field of positive charateristic with prescribed action of a finite group G, and the data are computed in terms of the ramification behaviour of X -> G\X. The second one is the ``analytic'' deformation functor of a fixed embedding of a finitely generated discrete group N in PGL(2,K) over a non-archimedean valued field K, and the data are computed in terms of the Bass-Serre representation of N via a graph of groups. Finally, if F is a free subgroup of N such that N is contained in the normalizer of F in PGL(2,K), then the Mumford curve associated to F becomes equipped with an action of N/F, and we show that the algebraic functor deforming the latter action coincides with the analytic functor deforming the embedding of N.
We prove the following results for toric Deligne-Mumford stacks, under minimal compactness hypotheses: the Localization Theorem in equivariant K-theory; the equivariant Hirzebruch-Riemann-Roch theorem; the Fourier--Mukai transformation associated to a crepant toric wall-crossing gives an equivariant derived equivalence.
For any finite abelian group G, the equivariant Gromov-Witten invariants of C^r/G can be viewed as a certain kind of abelian Hurwitz-Hodge integrals. In this note, we use Tseng's orbifold quantum Riemann-Roch theorem to express this kind of abelian Hurwitz-Hodge integrals as a sum over Feynman graphs, where the weight of each graph is expressed in terms of descendant integrals over moduli spaces of stable curves and representations of G. This expression will play a crucial role in the proof of the remodeling conjecture (arXiv:0709.1453, arXiv:0807.0597) for affine toric Calabi-Yau 3-orbifolds …
We give a simple proof of the Riemann-Roch theorem for Deligne-Mumford stacks using the equivariant Riemann-Roch theorem and the localization theorem in equivariant K-theory together with some basic commutative algebra of Artin rings.
Formulae for the number of branch points of one-dimensional orbifolds defined over a non-archimedean local field and uniformisable by discrete projective linear groups are given. They depend only on the uniformising group. The method of equivariant pasting reveals the possible relative position of the branch points.
We use perfectoid spaces associated to abelian varieties and Siegel moduli spaces to study torsion points and ordinary CM points. We reprove the Manin-Mumford conjecture i.e. Raynaud's theorem. We also prove the Tate-Voloch conjecture for a product of Siegel moduli spaces namely ordinary CM points outside a closed subvariety can not be p-adically too close to it.
Mumford defines a certain type of Shimura curves of Hodge type, parameterizing polarized complex abelian fourfolds. In this paper, we study the good reduction of such a curve in positive characteristic and give a characterization in the generically ordinary case.
In this expository article we give a categorical definition of the integral cohomology ring of a stack. We show that for quotient stacks the categorical cohomology may be identified with equivariant cohomology. Via this identification we show that for Deligne-Mumford quotient stacks this cohomology is rationally isomorphic to the rational cohomology of the coarse moduli space. The theory is presented with a focus on the stacks of smooth and stable curves.
Based on the methods used by the author to prove the Riemann-Roch formula for algebraic stacks, this paper contains a description of the rationnal G-theory of Deligne-Mumford stacks over general bases. We will use these results to study equivariant K-theory, and also to define new filtrations on K-theory of algebraic stacks.
We study the equivariant cobordism theory of schemes for torus actions. We give the explicit relation between the equivariant and the ordinary cobordism of schemes with torus action. We deduce analogous results for action of arbitrary connected linear algebraic groups. We prove some structure theorems for the equivariant and ordinary cobordism of schemes with torus action and derive important consequences. We establish the localization theorems in this setting. These are used to describe the structure of the ordinary cobordism ring of certain smooth projective varieties.
In the present article we investigate ordinary and equivariant Rost motives. We provide an equivariant motivic decomposition of the variety X of full flags of a split semisimple algebraic group over a smooth base scheme, study torsion subgroup of the Chow group of twisted forms of X, define some equivariant Rost motives over a field and ordinary Rost motives over a general base scheme, and relate equivariant Rost motives with classifying spaces of some algebraic groups.
We seek an appropriate definition for a Shimura curve of Hodge type in positive characteristics via characterizing curves in positive characteristics which are reduction of Shimura curve over $\mathbb{C}$. In this paper, we study the liftablity of a curve in the moduli space of principally polarized abelian varieties over $k, \text{char} k=p$. We show that in the generic ordinary case, some tensor decomposition of the isocrystal associated to the family imply that this curve can be lifted to a Shimura curve.
We define the equivariant Cox ring of a normal variety with algebraic group action. We study algebraic and geometric aspects of this object and show how it is related to the ordinary Cox ring. Then, we specialize to the case of normal rational varieties of complexity one under the action of a connected reductive group G. We show that the G-equivariant Cox ring is then a finitely generated integral normal G-algebra. Under a mild additional condition, we give a presentation by generators and relations of its subalgebra of U-invariants, where U is the unipotent part of a Borel subgroup of G. The …
For a smooth quasi-projective scheme $X$ over a field $k$ with an action of a reductive group, we establish a spectral sequence connecting the equivariant and the ordinary higher Chow groups of $X$. For $X$ smooth and projective, we show that this spectral sequence degenerates, leading to an explicit relation between the equivariant and the ordinary higher Chow groups. We obtain several applications to algebraic $K$-theory. We show that for a reductive group $G$ acting on a smooth projective scheme $X$, the forgetful map $K^G_i(X) \to K_i(X)$ induces an isomorphism $K^G_i(X)/{I_G K^G_i(X)} \co…
In this article we describe the $\tG\times \tG$-equivariant $K$-ring of $X$, where $\tG$ is a {\it factorial} cover of a connected complex reductive algebraic group $G$, and $X$ is a regular compactification of $G$. Furthermore, using the description of $K_{\tG\times \tG}(X)$, we describe the ordinary $K$-ring $K(X)$ as a free module of rank the cardinality of the Weyl group, over the $K$-ring of a toric bundle over $G/B$, with fibre the toric variety $\bar{T}^{+}$, associated to a smooth subdivision of the positive Weyl chamber. This generalizes our previous work on the wonderful compactifica…
Springer fibers are subvarieties of the flag variety that play an important role in combinatorics and geometric representation theory. In this paper, we analyze the equivariant cohomology of Springer fibers for $GL_n(\mathbb{C})$ using results of Kumar and Procesi that describe this equivariant cohomology as a quotient ring. We define a basis for the equivariant cohomology of a Springer fiber, generalizing a monomial basis of the ordinary cohomology defined by De Concini and Procesi and studied by Garsia and Procesi. Our construction yields a combinatorial framework with which to study the equ…
In the paper we answer the following question: for a morphism of varieties (or, more generally, stacks), when the derived category of the base can be recovered from the derived category of the covering variety by means of descent theory? As a corollary, we show that for an action of a reductive group on a scheme, the derived category of equivariant sheaves is equivalent to the category of objects, equipped with an action of the group, in the ordinary derived category.
Let $\mathbb{F}_q$ be the finite field of order $q=p^h$ with $p>2$ prime and $h>1$, and let $\mathbb{F}_{\bar{q}}$ be a subfield of $\mathbb{F}_q$. From any two $\bar{q}$-linearized polynomials $L_1,L_2 \in \overline{\mathbb{F}}_q[T]$ of degree $q$, we construct an ordinary curve $\mathcal{X}_{(L_1,L_2)}$ of genus $(q-1)^2$ which is a generalized Artin-Schreier cover of the projective line $\mathbb{P}^1$. The automorphism group of $\mathcal{X}_{(L_1,L_2)}$ over the algebraic closure $\overline{\mathbb{F}}_q$ of $\mathbb{F}_q$ contains a semidirect product $Σ\rtimes Γ$ of an elementary abelian …
We give a positive equivariant Littlewood-Richardson rule also discovered independently by Molev. Our proof generalizes a proof by Stembridge of the ordinary Littlewood-Richardson rule. We describe a weight-preserving bijection between our indexing tableaux and the Knutson-Tao puzzles.
An action of a finite group on a smooth projective curve over an algebraically closed field of positive characteristic is called restrained, if all second ramification groups are trivial (e.g., every group action on an ordinary curve is restrained). When the ramification indices satisfy certain numerical criteria, we construct a degenerating equivariant quasi-projective family to which the given curve belongs, and which in a sense is the unique building block for all such restrained equivariant families that ramify above a fixed set of points. The result is used to inductively study automorphi…
In this paper we study ideals of points lying on rational normal curves defined in projective plane and projective $ 3 $-space. We give an explicit formula for the value of Castelnuovo-Mumford regularity for their ordinary powers. Moreover, we compare the $ m $-th symbolic and ordinary powers for such ideals in order to show whenever the $ m $-th symbolic defect is non-zero.
In this short note, we prove a comparision theorem between Levine-Serpé's equivariant higher Chow groups of an algebraic variety equipped with an action of a finite group and ordinary higher Chow groups of its fixed points. As a consequence, we show that the equivariant motivic spectral sequence degenerates rationally. This yields a Riemann-Roch Theorem for equivariant algebraic $K$-theory.
Supersymmetric curves are the analogue of Riemann surfaces in super geometry. We establish some foundational results about complex Deligne-Mumford superstacks, and we then prove that the moduli superstack of supersymmetric curves is a smooth complex Deligne-Mumford superstack. We then show that the superstack of supersymmetric curves admits a coarse complex superspace, which, in this case, is just an ordinary complex space. In the second part of this paper we discuss the period map. We remark that the period domain is the moduli space of ordinary abelian varieties endowed with a symmetric thet…
We compute section class relative equivariant Gromov-Witten invariants of the total space of P^2-bundles of the form P(O+L1+L2)-->C where C is a genus g curve, O is the trivial bundle, and L1 (resp. L2) is an arbitrary line bundle of degree k1 (resp. k2) over C. We prove a gluing formula for the partition functions of these invariants. Using this gluing formula together with localization techniques, we construct three explicit 3x3 matrices G, U1 and U2 with entries in Q((u))(t0,t1,t2), where u is the genus parameter, and t0,t1,t2 are the equivariant parameters. Then we prove that the partition…
By using vector field techniques, we compute the ordinary and equivariant cohomology rings of Hilbert scheme of points in the projective plane in relation with that of a Grassmann variety.
We generalize our puzzle formula for ordinary Schubert calculus on Grassmannians, to a formula for the T-equivariant Schubert calculus. The structure constants to be calculated are polynomials in {y_{i+1} - y_i}; they were shown (abstractly) to have positive coefficients in [Graham] math.AG/9908172. Our formula is the first to be manifestly positive in this sense. In particular this gives a new and self-contained proof of the ordinary puzzle formula, by an induction backwards from the "most equivariant" case. The proof of the formula is mostly combinatorial, but requires no prior combinatorics…
Let ${mathcal M}_g^n$ be the moduli space of n-pointed Riemann surfaces of genus g. Denote by ${\bar {\mathcal M}}_g^n$ the Deligne-Mumford compactification of ${mathcal M}_g^n$. In the present paper, we calculate the orbifold and the ordinary Euler characteristic of ${\bar {\mathcal M}}_g^n$ for any g and n such that n>2-2g.
In this article we describe the $G\times G$-equivariant $K$-ring of $X$, where $X$ is a regular compactification of a connected complex reductive algebraic group $G$. Furthermore, in the case when $G$ is a semisimple group of adjoint type, and $X$ its wonderful compactification, we describe its ordinary $K$-ring $K(X)$. More precisely, we prove that $K(X)$ is a free module over $K(G/B)$ of rank the cardinality of the Weyl group. We further give an explicit basis of $K(X)$ over $K(G/B)$, and also determine the structure constants with respect to this basis.
We study the symplectic nature of the moduli stack classifying dormant curves over a field $K$ of positive characteristic, i.e., proper hyperbolic curves over $K$ equipped with a dormant indigenous bundle. The central objects of the present paper are the following two Deligne-Mumford stacks. One is the cotangent bundle ${^\circledcirc T^{\vee ^\mathrm{Zzz...}}_{g,K}}$ of the moduli stack ${^\circledcirc \mathfrak{M}^{^\mathrm{Zzz...}}_{g,K}}$ classifying ordinary dormant curves over $K$ of genus $g$. The other is the moduli stack ${^\circledcirc \mathfrak{S}^{^\mathrm{Zzz...}}_{g,K}}$ classify…
In this paper we develop an equivariant intersection theory for actions of algebraic groups on algebraic schemes. The theory is based on our construction of equivariant Chow groups. They are algebraic analogues of equivariant cohomology groups which satsify the formal properties of ordinary Chow groups. In addition, they enjoy many of the properties of equivariant cohomology. The principal results are: (1) We prove the existence of canonical intersection products on the Chow groups of geometric quotients of smooth varieties- even when the stabilizers of geometric points are non-reduced. (2) We…
This paper defines and studies permutation representations on the equivariant cohomology of Schubert varieties, as representations both over C and over C[t_1, t_2,...,t_n]. We show these group actions are the same as an action of simple transpositions studied geometrically by M. Brion, and give topological meaning to the divided difference operators studied by Berstein-Gelfand-Gelfand, Demazure, Kostant-Kumar, and others. We analyze these representations using the combinatorial approach to equivariant cohomology introduced by Goresky-Kottwitz-MacPherson. We find that each permutation represent…
Peterson varieties are a special class of Hessenberg varieties that have been extensively studied e.g. by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley-Monk formula, for the ordinary and Borel-equivariant cohomology of the Peterson variety $Y$ in type $A_{n-1}$, with respect to a natural $S^1$-action arising from the standard action of the maximal torus on flag varieties. As far as we know, this is the first example of positive Schubert calculu…
Let $\mathrm{R}$ be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semi-algebraic subsets of $\mathrm{R}^k$ in terms of the number and degrees of the defining polynomials has been an important problem in real algebraic geometry with the first results due to Ole{\uı}nik and Petrovski{\uı}, Thom and Milnor. These bounds are all exponential in the number of variables $k$. Motivated by several applications in real algebraic geometry, as well as in theoretical computer science, where such bounds have found applications, we consider in this paper the problem of bo…
We compute the spectrum of the category of derived Mackey functors (in the sense of Kaledin) for all finite groups. We find that this space captures precisely the top and bottom layers (i.e. the height infinity and height zero parts) of the spectrum of the equivariant stable homotopy category. Due to this truncation of the chromatic information, we are able to obtain a complete description of the spectrum for all finite groups, despite our incomplete knowledge of the topology of the spectrum of the equivariant stable homotopy category. From a different point of view, we show that the spectrum …
The equivariant and ordinary cohomology rings of Hilbert schemes of points on the minimal resolution C^2//G for cyclic G are studied using vertex operator technique, and connections between these rings and the class algebras of wreath products are explicitly established. We further show that certain generating functions of equivariant intersection numbers on the Hilbert schemes and related moduli spaces of sheaves on C^2//G are tau functions of 2-Toda hierarchies.
Warwick Mathematics Institute（华威数学研究所）；Univ. Artois, UR 2462, Laboratoire de Mathématiques de Lens（阿图瓦大学，UR 2462，伦斯数学实验室）；Department of Mathematics, University of Georgia（佐治亚大学数学系）
Let $G:=\widehat{SL_2}$ denote the affine Kac-Moody group associated to $SL_2$ and $\bar{\mathcal{X}}$ the associated affine Grassmannian. We determine an inductive formula for the Schubert basis structure constants in the torus-equivariant Grothendieck group of $\bar{\mathcal{X}}$. In the case of ordinary (non-equivariant) $K$-theory we find an explicit closed form for the structure constants. We also determine an inductive formula for the structure constants in the torus-equivariant cohomology ring, and use this formula to find closed forms for some of the structure constants.
Let $\mathrm{R}$ be a real closed field. We prove that for any fixed $d$, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $\mathrm{R}^k$ defined by polynomials of degrees bounded by $d$ vanishes in dimensions $d$ and larger. This vanishing result is tight. Using a new geometric approach we also prove an upper bound of $d^{O(d)} s^d k^{\lfloor d/2 \rfloor-1} $ on the equivariant Betti numbers of closed symmetric semi-algebraic subsets of $\mathrm{R}^k$ defined by quantifier-free formulas involving $s$ symmetric polynomials of degrees bounded by $d$, wher…
