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We characterize primary operations in differential cohomology via stacks, and illustrate by differentially refining Steenrod squares and Steenrod powers explicitly. This requires a delicate interplay between integral, rational, and mod p cohomology, as well as cohomology with U(1) coefficients and differential forms. Along the way we develop computational techniques in differential cohomology, including a K\"unneth decomposition, that should also be useful in their own right, and point to applications to higher geometry and mathematical physics., Comment: 36 pages, minor corrections

The cohomology of the degree-$n$ general linear group over a finite field of characteristic $p$, with coefficients also in characteristic $p$, remains poorly understood. For example, the lowest degree previously known to contain nontrivial elements is exponential in $n$. In this paper, we introduce a new system of characteristic classes for representations over finite fields, and use it to construct a wealth of explicit nontrivial elements in these cohomology groups. In particular we obtain nontrivial elements in degrees linear in $n$. We also construct nontrivial elements in the mod $p$ homology and cohomology of the automorphism groups of free groups, and the general linear groups over the integers. These elements reside in the unstable range where the homology and cohomology remain poorly understood., Accepted to the Advances in Mathematics

We introduce the log etale cohomology in the second (full log etale) sense after K. Kato and prove several fundamental results including base change theorems. To prove them, we use some corresponding results in the log etale cohomology in the first (kummer log etale) sense. Conversely, they are used to show some new results in the kummer log etale cohomology.

We introduce and study the homotopy theory of motivic spaces and spectra parametrized by quotient stacks [X/G], where G is a linearly reductive linear algebraic group. We extend to this equivariant setting the main foundational results of motivic homotopy theory: the (unstable) purity and gluing theorems of Morel and Voevodsky and the (stable) ambidexterity theorem of Ayoub. Our proof of the latter is different than Ayoub's and is of interest even when G is trivial. Using these results, we construct a formalism of six operations for equivariant motivic spectra, and we deduce that any cohomology theory for G-schemes that is represented by an absolute motivic spectrum satisfies descent for the cdh topology., Comment: v4: added subsection 2.5 to fix a mistake

Associated to any manifold equipped with a closed form of degree >1 is an `L-infinity algebra of observables' which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a map is a L-infinity morphism from the Lie algebra of the group into the observables which lifts the infinitesimal action. We establish the relationship between homotopy moment maps and equivariant de Rham cohomology, and analyze the obstruction theory for the existence of such maps. This allows us to easily and explicitly construct a large number of examples. These include results concerning group actions on loop spaces and moduli spaces of flat connections. Relationships are also established with previous work by others in classical field theory, algebroid theory, and dg geometry. Furthermore, we use our theory to geometrically construct various L-infinity algebras as higher central extensions of Lie algebras, in analogy with Kostant's quantization theory. In particular, the so-called `string Lie 2-algebra' arises this way., Final version will appear in Advances in Mathematics. Results concerning equivariant cohomology strengthened. In particular, we exhibit the explicit relationship between equivariant de Rham cocycles of arbitrary degree and homotopy moment maps. 62 pages. Comments are welcome. arXiv admin note: text overlap with arXiv:1402.0144 by other authors

Let a torus T act in a Hamiltonian fashion on a compact symplectic manifold ( M , ω ) . The assignment ring A T ( M ) is an extension of the equivariant cohomology ring H T ( M ) ; it is modeled on the GKM description of the equivariant cohomology of a GKM space. We show that A T ( M ) is a finitely generated S ( t ⁎ ) -module, and give a criterion guaranteeing that a given set of assignments generates (alternatively, is a basis for) this module. We define two new types of assignments, delta classes and bridge classes, and show that if the torus T is 2-dimensional, then all assignments of sufficiently high degree are generated by cohomological, delta, and bridge classes. In particular, if M is 6-dimensional, then we can find a basis of such classes.

Let $G$ be a connected reductive algebraic group and $B$ be a Borel subgroup defined over an algebraically closed field of characteristic $p>0$. In this paper, the authors study the existence of generic $G$-cohomology and its stability with rational $G$-cohomology groups via the use of methods from the authors' earlier work. New results on the vanishing of $G$ and $B$-cohomology groups are presented. Furthermore, vanishing ranges for the associated finite group cohomology of $G({\mathbb F}_{q})$ are established which generalizes earlier work of Hiller, in addition to stability ranges for generic cohomology which improves on seminal work of Cline, Parshall, Scott and van der Kallen., Corrected proof of Theorem 5.1.1, added Remark 5.2.3

In this paper we study the notion of representation up to homotopy of a Lie groupoid and the resulting derived category, and show that the adjoint representation is well defined as a representation up to homotopy. As an application, we extend Bottʼs spectral sequence converging to the cohomology of classifying spaces of Lie groups to the case of Lie groupoids. We explain the relation of this construction with the models of Cartan and Getzler for equivariant cohomology. Our work is closely related to and inspired by Behrendʼs [3] , Bottʼs [4] , and Getzlerʼs [9] .

We introduce two families of symmetric functions generalizing the factorial Schur P - and Q -functions due to Ivanov. We call them K -theoretic analogues of factorial Schur P - and Q -functions. We prove various combinatorial expressions for these functions, e.g. as a ratio of Pfaffians, a sum over set-valued shifted tableaux, and a sum over excited Young diagrams. As a geometric application, we show that these functions represent the Schubert classes in the K -theory of torus equivariant coherent sheaves on the maximal isotropic Grassmannians of symplectic and orthogonal types. This generalizes a corresponding result for the equivariant cohomology given by the authors. We also discuss a remarkable property enjoyed by these functions, which we call the K -theoretic Q -cancellation property. We prove that the K -theoretic P -functions form a (formal) basis of the ring of functions with the K -theoretic Q -cancellation property.

We give an interpretation of the Brauer group of a purely inseparable extension of exponent 1 , in terms of restricted Lie–Rinehart cohomology. In particular, we define and study the category p - LR ( A ) of restricted Lie–Rinehart algebras over a commutative algebra A . We define cotriple cohomology groups H p - L R ( L , M ) for L ∈ p - LR ( A ) and M a Beck L -module. We classify restricted Lie–Rinehart extensions. Thus, we obtain a classification theorem for regular extensions considered by Hochschild.

Let X be a subanalytic compact pseudomanifold. We show a de Rham theorem for L ∞ forms on the nonsingular part of X . We prove that their cohomology is isomorphic to the intersection cohomology of X in the maximal perversity.

There are two main approaches to the problem of realizing a Π-algebra (a graded group Λ equipped with an action of the primary homotopy operations) as the homotopy groups of a space X. Both involve trying to realize an algebraic free simplicial resolution G• of Λ by a simplicial space W•, and proceed by induction on the simplicial dimension. The first provides a sequence of Andre–Quillen cohomology classes in Hn+2(Λ;ΩnΛ) (n⩾1) as obstructions to the existence of successive Postnikov sections for W• (cf. Dwyer et al. (1995) [27]). The second gives a sequence of geometrically defined higher homotopy operations as the obstructions (cf. Blanc (1995) [8]); these were identified in Blanc et al. (2010) [16] with the obstruction theory of Dwyer et al. (1989) [25]. There are also (algebraic and geometric) obstructions for distinguishing between different realizations of Λ. In this paper we (a) provide an explicit construction of the cocycles representing the cohomology obstructions; (b) provide a similar explicit construction of certain minimal values of the higher homotopy operations (which reduce to “long Toda brackets”); and (c) show that these two constructions correspond under an evident map.

In this work we describe the Chen-Ruan cohomology of the moduli stacks of smooth and stable genus 2 pointed curves, and its algebraic counterpart: the stringy Chow ring. In the first half of the paper we compute the additive structure of the Chen-Ruan cohomology ring for the moduli stack of stable pointed genus 2 curves, describing it as a rationally graded vector space. In the second part we give generators for the even Chen--Ruan cohomology ring as an algebra on the ordinary cohomology., 48 pages, 3 Figures. Final version to appear in Advances in Mathematics

We refine the cyclic cohomological apparatus for computing the Hopf cyclic cohomology of the Hopf algebras associated to infinite primitive Cartan-Lie pseudogroups, and for the transfer of their characteristic classes to foliations. The main novel feature is the precise identification as a Hopf cyclic complex of the image of the canonical homomorphism from the Gelfand-Fuks complex to the Bott complex for equivariant cohomology. This provides a convenient new model for the Hopf cyclic cohomology of the geometric Hopf algebras, which allows for an efficient transport of the Hopf cyclic classes via characteristic homomorphisms. We illustrate the latter aspect by indicating how to realize the universal Hopf cyclic Chern classes in terms of explicit cocycles in the cyclic cohomology of foliation groupoids., Comment: Final version, to appear in Advances in Mathematics

We show that Dirac cohomology of the Jacquet module of a Harish-Chandra module is a Harish-Chandra module for the corresponding Levi subgroup. We obtain an explicit formula of Dirac cohomology of the Jacquet module for most of the principal series, based on our determination of Dirac cohomology of irreducible generalized Verma modules with regular infinitesimal characters.

A simple, combinatorial construction of the sl ˆ ( n ) k -WZNW fusion ring, also known as Verlinde algebra, is given. As a byproduct of the construction one obtains an isomorphism between the fusion ring and a particular quotient of the small quantum cohomology ring of the Grassmannian Gr k , k + n . We explain how our approach naturally fits into known combinatorial descriptions of the quantum cohomology ring, by establishing what one could call a ‘Boson–Fermion-correspondence’ between the two rings. We also present new recursion formulae for the structure constants of both rings, the fusion coefficients and the Gromov–Witten invariants.

We develop a framework for derived deformation theory, valid in all characteristics. This gives a model category reconciling local and global approaches to derived moduli theory. In characteristic 0, we use this to show that the homotopy categories of DGLAs and SHLAs (L infinity algebras) considered by Kontsevich, Hinich and Manetti are equivalent, and are compatible with the derived stacks of Toen--Vezzosi and Lurie. Another application is that the cohomology groups associated to any classical deformation problem (in any characteristic) admit the same operations as Andre--Quillen cohomology., Comment: 55 pages; v4 split in two - other half is now arXiv:0908.1963; v5 final version, to appear in Adv. Math; v6 incorporates contents of a subsequent corrigendum concerning geometric weak equivalences; v7 corrects an error in Lemma 4.44 found by Andrey Lazarev

We give a complete description of the bigraded Bredon cohomology ring of smooth projective real quadrics, with coefficients in the constant Mackey functor Z̲. These invariants are closely related to the integral motivic cohomology ring, which is not known for these varieties. Some of the results and techniques introduced can be applied to other geometrically cellular real varieties.

We study the geometry and cohomology of the (generic fibres) of formal deformation schemes of one-dimensional formal modules of finite height. By the work of Boyer (in mixed characterististic) and Harris and Taylor, the l-adic etale cohomology of these spaces realizes simultaneously the local Langlands and Jacquet-Langlands correspondences. The proofs given so far use Drinfeld modular varieties or Shimura varieties to derive this local result. In this paper we show without the use of global moduli spaces that the Jacquet-Langlands correspondence for supercuspidal representations is realized by the Euler-Poincare characteristic of the cohomology. Under a certain finiteness assumption on the cohomology groups, it is shown that the correspondence is realized in only one degree. One main ingredient of the proof consists in analyzing the boundary of the deformation spaces and in studying larger spaces which can be considered as compactifications of the deformation spaces., Comment: 72 pages, 1 figure, submitted

We give new weighted decompositions for simple polytopes, generalizing previous results of Lawrence–Varchenko and Brianchon–Gram. We start with Witten's non-abelian localization principle in equivariant cohomology for the norm-square of the moment map in the context of toric varieties to obtain a decomposition for Delzant polytopes. Then, by a purely combinatorial argument, we show that this formula holds for any simple polytope. As an application, we study Euler–Maclaurin formulas.

Suppose that an algebraic torus $G$ acts algebraically on a projective manifold $X$ with generically trivial stabilizers. Then the Zariski closure of the set of pairs $\{(x,y)\in X\times X\mid y=gx \text{for some}g\in G\}$ defines a nonzero equivariant cohomology class $[\Delta_G]\in H^*_{G\times G}(X\times X)$. We give an analogue of this construction in the case where $X$ is a compact symplectic manifold endowed with a hamiltonian action of a torus, whose complexification plays the role of $G$. We also prove that the Kirwan map sends the class $[\Delta_G]$ to the class of the diagonal in each symplectic quotient. This allows to define a canonical right inverse of the Kirwan map., Comment: A substatially revised version of the paper "A right inverse to the Kirwan map". Improved exposition. Singular quotients are also considered in the new version. 23 pages. Accepted for publication in Adv. in Math

Certain low-dimensional symmetric cohomology groups of G-modules, for any given group G, are computed as the cohomology of an explicit cochain complex. This result is used to establish natural one-to-one correspondences between elements of the 3rd symmetric cohomology groups of G-modules, G-equivariant pointed 2-connected homotopy 4-types, and equivalence classes of G-graded Picard categories. The simplicial nerve of a G-graded Picard category is also constructed and studied.

For several important classes of manifolds acted on by the torus, the information about the action can be encoded combinatorially by a regular n-valent graph with vector labels on its edges, which we refer to as the torus graph. By analogy with the GKM-graphs, we introduce the notion of equivariant cohomology of a torus graph, and show that it is isomorphic to the face ring of the associated simplicial poset. This extends a series of previous results on the equivariant cohomology of torus manifolds. As a primary combinatorial application, we show that a simplicial poset is Cohen-Macaulay if its face ring is Cohen-Macaulay. This completes the algebraic characterisation of Cohen-Macaulay posets initiated by Stanley. We also study blow-ups of torus graphs and manifolds from both the algebraic and the topological points of view., 26 pages, LaTeX2e; examples added, some proofs expanded

We construct a new equivariant cohomology theory for a certain class of differential vertex algebras, which we call the chiral equivariant cohomology. A principal example of a differential vertex algebra in this class is the chiral de Rham complex of Malikov-Schechtman-Vaintrob of a manifold with a group action. The main idea in this paper is to synthesize the algebraic approach to classical equivariant cohomology due to H. Cartan, with the theory of differential vertex algebras, by using an appropriate notion of invariant theory. We also construct the vertex algebra analogues of the Mathai-Quillen isomorphism, the Weil and the Cartan models for equivariant cohomology, and the Chern-Weil map. We give interesting cohomology classes in the new theory that have no classical analogues., 76 pages, final version, to appear in Advances in Mathematics

Let G be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p . Let B be a Borel subgroup of G and U be its unipotent radical. In this paper, we determine the second cohomology groups of B and its Frobenius kernels for all simple B -modules. We also consider the standard induced modules obtained by inducing a simple B -module to G and compute all second cohomology groups of the Frobenius kernels of G for these induced modules. Also included is a calculation of the second ordinary Lie algebra cohomology group of Lie( U ) with coefficients in k .

One of the main obstacles for proving Riemann–Roch for algebraic stacks is the lack of cohomology and homology theories that are closer to the K-theory and G-theory of algebraic stacks than the traditional cohomology and homology theories for algebraic stacks. In this paper we study in detail a family of cohomology and homology theories which we call Bredon-style theories that are of this type and in the spirit of the classical Bredon cohomology and homology theories defined for the actions of compact topological groups on topological spaces. We establish Riemann–Roch theorems in this setting: it is shown elsewhere that such Riemann–Roch theorems provide a powerful tool for deriving formulae involving virtual fundamental classes associated to dg-stacks, for example, moduli stacks of stable curves provided with a virtual structure sheaf associated to a perfect obstruction theory. We conclude the present paper with a brief application of this nature.

We study local cohomology of rings of global sections of sheafs on the Alexandrov space of a partially ordered set. We give a criterion for a splitting of the local cohomology groups into summands determined by the cohomology of the poset and the local cohomology of the stalks. The face ring of a rational pointed fan can be considered as the ring of global sections of a flasque sheaf on the face poset of the fan. Thus we obtain a decomposition of the local cohomology of such face rings. Since the Stanley-Reisner ring of a simplicial complex is the face ring of a rational pointed fan, our main result can be interpreted as a generalization of Hochster's decomposition of local cohomology of Stanley-Reisner rings., 24 pages; To appear in Adv. Math. Revised version of the paper

We use (non-)additive sheaves to introduce an (absolute) notion of Hochschild cohomology for exact categories as Ext's in a suitable bisheaf category. We compare our approach to various definitions present in the literature., 27 pages

We study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface. We show that, under certain Hodge theoretic conditions, the cohomology ring of the complement of the hypersurface functorially determines that of the boundary. When the hypersurface defines a hyperplane arrangement, the cohomology of the boundary is completely determined by the combinatorics of the underlying arrangement and the ambient dimension. We also study the LS category and topological complexity of the boundary manifold, as well as the resonance varieties of its cohomology ring., 31 pages; accepted for publication in Advances in Mathematics

We study the T-equivariant quantum cohomology of the Grassmannian. We prove the vanishing of a certain class of equivariant quantum Littlewood-Richardson coefficients, which implies an equivariant quantum Pieri rule. As in the equivariant case, this implies an algorithm to compute the equivariant quantum Littlewood-Richardson coefficients., 24 pages, LaTeX

In 1998, Goresky, Kottwitz, and MacPherson showed that for certain projective varieties X equipped with an algebraic action of a complex torus T, the equivariant cohomology ring HT*(X) can be described by combinatorial data obtained from its orbit decomposition. In this paper, we generalize their theorem in three different ways. First, our group G need not be a torus. Second, our space X is an equivariant stratified space, along with some additional hypotheses on the attaching maps. Third, and most important, we allow for generalized equivariant cohomology theories EG* instead of HT*. For these spaces, we give a combinatorial description of EG*(X) as a subring of ∏EG*(Fi), where the Fi are certain invariant subspaces of X. Our main examples are the flag varieties G/P of Kac–Moody groups G, with the action of the torus of G. In this context, the Fi are the T-fixed points and EG* is a T-equivariant complex oriented cohomology theory, such as HT*, KT* or MUT*. We detail several explicit examples.