We determine the mod p cohomological invariants for several affine group schemes G in chararacteristic p. These are invariants of G-torsors with values in etale motivic cohomology, or equivalently in Kato's version of Galois cohomology based on differential forms.
In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod p etale motivic cohomology of fields, extending Vial's computations of the operations on the mod p Milnor K-theory of fields.
We explain how to exploit Rost's theory of Chow groups with coefficients to carry some computations of cohomological invariants (i.e. characteristic classes for G-torsors in Galois cohomology, for an algebraic group G). In particular, we use the idea of the ``stratification method'' introduced by Vezzosi. We recover a number of known results, with very different proofs. We obtain some new information on spin groups..
We give a formula for the cohomological invariants of a root stack, which we apply to compute the cohomological invariants and the Brauer group of the stack of admissible double coverings.
In this paper, we generalize Serre's splitting theorem for cohomological invariants of the symmetric group to finite Coxeter groups, provided that the ground field has characteristic zero. We then use this principle to determine all the cohomological invariants of Weyl groups of classical type with coefficients modulo 2.
In a previous paper, we showed that all the cohomological invariants of Weyl groups are completely determined by their restrictions to the abelian subgroups generated by reflections. Using this principle, we describe all the cohomological invariants of the Weyl groups of classical type modulo 2.
In a recent paper A. Merkurjev constructed an exact sequence which includes as one of the terms the group of degree 3 normalized cohomological invariants of a semisimple algebraic group G, greatly extending results of M. Rost for simply connected quasi-simple groups. Furthermore, in the aforementioned paper, Merkurjev uses his exact sequence to determine the groups of invariants for all semisimple adjoint groups of inner type. The goal of this paper is to use Merkurjev's sequence to compute the group of invariants for the remaining split cases, namely groups of types A and D that are neither s…
We compute the cohomological invariants with coefficients in $\mathbb{Z}/p\mathbb{Z}$ of the stack $\mathscr{H}_3$ of hyperelliptic curves of genus $3$ over an algebraically closed field.
Let $g$ be an even positive integer, and $p$ be a prime number. We compute the cohomological invariants with coefficients in $\mathbb{Z}/p\mathbb{Z}$ of the stacks of hyperelliptic curves $\mathscr{H}_g$ over an algebraically closed field $k_0$.
We derive a combinatorial formula for quantized Donaldson-Thomas invariants of the m-loop quiver. Our main tools are the combinatorics of noncommutative Hilbert schemes and a degenerate version of the Cohomological Hall algebra of this quiver.
In the present article we discuss different approaches to cohomological invariants of algebraic groups over a field. We focus on the Tits algebras and on the Rost invariant and relate them to the Morava K-theory. Furthermore, we discuss oriented cohomology theories of affine varieties and the Rost motives for different cohomology theories in the sense of Levine-Morel.
The purpose of this paper is to lay the foundations of a theory of invariants in étale cohomology for smooth Artin stacks. We compute the invariants in the case of the stack of elliptic curves, and we use the theory we developed to get some results regarding Brauer groups of algebraic spaces.
Let $\mathrm{k}_{0}$ be a field and $W$ a finite orthogonal reflection group over $\mathrm{k}_{0}$. We prove Serre's splitting principle for cohomological invariants of $W$ with values in Rost's cycle modules (over $\mathrm{k}_{0}$) if the characteristic of $\mathrm{k}_{0}$ is coprime to $|W|$. We then show that this principle for such groups holds also for Witt- and Milnor-Witt $K$-theory invariants.
When the genus $g$ is even, we extend the computation of mod 2 cohomological invariants of $\mathcal{H}_g$ to non algebraically closed fields, we give an explicit functorial description of the invariants and we completely describe their multiplicative structure. In the Appendix, we show that the cohomological invariants of the compactification $\overline{\mathcal{H}}_g$ are trivial, and use our methods to give a very short proof of a result by Cornalba on the Picard group of the compactification $\overline{\mathcal{H}}_g$ and extend it to positive characteristic
For a complex algebraic variety $X$, we show that triviality of the sheaf cohomology group $H^0(X,\mathcal{H}^3)$ occurring on the second page of the Bloch-Ogus spectral sequence follows from a condition on the integral Chow group $CH^2X$ and the integral cohomology group $H^3(X, Z)$. In the case that $X$ is an appropriate approximation to the classifying stack $BG$ of a finite $p$-group $G$, this result states that the group $G$ has no degree three cohomological invariants. As a corollary we show that the nonabelian groups of order $p^3$ for odd prime $p$ have no degree three cohomological in…
We compute the invariants so of Weyl groups in mod 2 Milnor K-theory and more general cycle modules, which are annihilated by 2. Over a base field of characteristic coprime to the group order, the invariants decompose as direct sums of the coefficient module. All basis elements are induced either by Stiefel-Whitney classes or specific invariants in the Witt ring. The proof is based on Serre's splitting principle that guarantees detection of invariants on elementary abelian 2-subgroups generated by reflections.
We establish a geometric interpretation of orientifold Donaldson-Thomas invariants of $σ$-symmetric quivers with involution. More precisely, we prove that the cohomological orientifold Donaldson-Thomas invariant is isomorphic to the rational Chow group of the moduli space of $σ$-stable self-dual quiver representations. As an application we prove that the Chow Betti numbers of moduli spaces of stable $m$-tuples in classical Lie algebras can be computed numerically. We also prove a cohomological wall-crossing formula relating semistable Hall modules for different stabilities.
Let $G$ be a split semisimple linear algebraic group over a field and let $X$ be a generic twisted flag variety of $G$. Extending the Hilbert basis techniques to Laurent polynomials over integers we give an explicit presentation of the Grothendieck ring $K_0(X)$ in terms of generators and relations in the case $G=G^{sc}/μ_2$ is of Dynkin type ${\rm A}$ or ${\rm C}$ (here $G^{sc}$ is the simply-connected cover of $G$); we compute various groups of (indecomposable, semi-decomposable) cohomological invariants of degree 3, hence, generalizing and extending previous results in this direction.
In this paper we determine all possible cohomological invariants of Aut(J)-torsors in Galois cohomology with mod 2 coefficients (characteristic of the base field not 2), for J a split central simple Jordan algebra of odd degree n>=3. This has already been done for J of orthogonal and exceptional type, and we extend these results to unitary and symplectic type. We will use our results to compute the essential dimensions of some groups, for example we show that ed(PSp(2n))=n+1 for n odd.
In this note I give a description of Lyubeznik's local cohomology invariants for a certain natural class of local rings, namely the ones which have the same local cohomology vanishing as one expects from an isolated singularity. This strengthens the results of Bondu and myself in math.AG/0406265 while at the same time somewhat simplifying the proofs. Through examples I further point out the bad behavior of these invariants under reduction to positive characteristic.
We define a new type of Hall algebras associated e.g. with quivers with polynomial potentials. The main difference with the conventional definition is that we use cohomology of the stack of representations instead of constructible sheaves or functions. In order to take into account the potential we introduce a generalization of theory of mixed Hodge structures, related to exponential integrals. Generating series of our Cohomological Hall algebra is a generalization of the motivic Donaldson-Thomas invariants introduced in arXiv:0811.2435. Also we prove a new integrality property of motivic Dona…
Let G be a group of type E8 of compact type over the field of rational numbers, let K be a field of characteristic 0, and q the 5-fold Pfister form which is the sum of 32 squares. J-P. Serre posed in a letter to M. Rost written on June 23, 1999 the following problem: Is it true that G is split over K if and only if q is hyperbolic over K? In the present article we construct a cohomological invariant of degree 5 for groups of type E8 with trivial Rost invariant over any field k of characteristic 0, and putting the field of rational numbers for k answer positively this question of Serre. Aside f…
In 2009 T. Ekedahl introduced certain cohomological invariants for finite groups. In this work we introduce these invariants and we show some of their properties. We also give an equivalent definition for such invariants.
We prove that there are no unexpected universal integral linear relations and congruences between Hodge, Betti and Chern numbers of compact complex manifolds and determine the linear combinations of such numbers which are bimeromorphic or topological invariants. This extends results in the Kähler case by Kotschick and Schreieder. We then develop a framework to tackle the more general questions taking into account `all' cohomological invariants (e.g. the dimensions of the higher pages of the Frölicher spectral sequence, Bott-Chern and Aeppli cohomology). This allows us to reduce the general que…
We construct GLSM invariants for a general choice of stability in both the narrow and broad sector cases and prove they form a Cohomological Field Theory. This is obtained by forming the analogue of a virtual fundamental class which lives in the local cohomology of the twisted Hodge complex. This general construction comes from the use of two new ingredients. First, the use of the Thom-Sullivan and Godement resolutions applied to matrix factorizations are introduced to handle poorly behaved (non-separated) moduli spaces. Second, a localized Chern character map built from the Atiyah class of a …
We compute the Brauer group of the moduli stack of hyperelliptic curves $\mathcal{H}_g$ over any field of characteristic zero. In positive characteristic, we compute the part of the Brauer group whose order is prime to the characteristic of the base field.
This review gives an introduction to cohomological Donaldson-Thomas theory: the study of a cohomology theory on moduli spaces of sheaves on Calabi-Yau threefolds, and of complexes in 3-Calabi-Yau categories, categorifying their numerical DT invariant. Local and global aspects of the theory are both covered, including representations of quivers with potential. We will discuss the construction of the DT sheaf, a nontrivial topological coefficient system on such a moduli space, along with some cohomology computations. The Cohomological Hall Algebra, an algebra structure on cohomological DT spaces…
We discuss semicanonical bases from the point of view of Cohomological Hall algebras via the "dimensional reduction" from 3-dimensional Calabi-Yau categories to 2-dimensional ones. Also, we discuss the notion of motivic Donaldson-Thomas invariants (as defined by M. Kontsevich and Y. Soibelman) in the framework of 2-dimensional Calabi-Yau categories. In particular we propose a conjecture which allows one to define Kac polynomials for a 2-dimensional Calabi-Yau category (this is a theorem of S. Mozgovoy in the case of preprojective algebras).
Given any smooth toric surface S, we prove a SYM-HILB correspondence which relates the 3-point, degree zero, extended Gromov-Witten invariants of the n-fold symmetric product stack [Sym^n(S)] of S to the 3-point extremal Gromov-Witten invariants of the Hilbert scheme Hilb^n(S) of n points on S. As we do not specialize the values of the quantum parameters involved, this result proves a strengthening of Ruan's Cohomological Crepant Resolution Conjecture for the Hilbert-Chow morphism from Hilb^n(S) to Sym^n(S) and yields a method of reconstructing the cup product for Hilb^n(S) from the orbifold i…
We start with a curve over an algebraically closed ground field of positive characteristic $p>0$. By using specialization techniques, under suitable natural coprimality conditions, we prove a cohomological Simpson Correspondence between the moduli space of Higgs bundles and the one of connections on the curve. We also prove a new $p$-multiplicative periodicity concerning the cohomology rings of Dolbeault moduli spaces of degrees differing by a factor of $p$. By coupling this $p$-periodicity in characteristic $p$ with lifting/specialization techniques in mixed characteristic, we find, in arbitr…
Donaldson-Thomas theory on a Calabi-Yau can be described in terms of a certain six-dimensional cohomological gauge theory. We introduce a certain class of defects in this gauge theory which generalize surface defects in four dimensions. These defects are associated with divisors and are defined by prescribing certain boundary conditions for the gauge fields. We discuss generalized instanton moduli spaces when the theory is defined with a defect and propose a generalization of Donaldson-Thomas invariants. These invariants arise by studying torsion free coherent sheaves on Calabi-Yau varieties w…
We introduce a new class of representations of the cohomological Hall algebras of Kontsevich and Soibelman, which we call cohomological Hall modules, or CoHM for short. These representations are constructed from self-dual representations of a quiver with contravariant involution $σ$ and provide a mathematical model for the space of BPS states in orientifold string theory. We use the CoHM to define a generalization of the cohomological Donaldson-Thomas theory of quivers which allows the quiver representations to have orthogonal and symplectic structure groups. The associated invariants are call…
In the present paper Mori extremal rays of a smooth projective manifold X are divided into two classes: L-supported and L-negligible (where ``L'' stands for ``Lefschetz'' since the division comes from Hard Lefschetz Theorem). Roughly speaking: L-supported rays are strongly distinguishable in topology while L-negligible rays have very mild geometry. Each L-supported ray R defines hyperplane in H^2(X,R) on which Lefschetz duality degenerates so it is a cohomology ring invariant. The hyperplane carries a multiplicity (cohomology ring invariant) which is related to the geometry of the ray R. The n…
The number of the relations of a Kahler group is bounded below by the number of the generators and some geometric invariants of the corresponding compact Kahler manifold, like the irregularity, the Albanese dimension and the Albanese genera. Appropriate functions of the aforementioned geometric invariants are shown to be lower bounds for the Betti numbers of the Kahler group within the range, determined by the Albanese dimension.
Given a quiver with potential associated to a toric Calabi-Yau threefold, the numerical Donaldson-Thomas invariants for the moduli space of framed representations can be computed by using toric localization, which reduces the problem to the enumeration of molten crystals. We provide a refinement of this localization procedure, which allows to compute motivic Donaldson-Thomas invariants. Using this approach, we prove a universal formula which gives the BPS invariants of any toric quiver, up to undetermined contributions which are invariant under Poincaré duality. When the toric Calabi-Yau three…
We determine the group of reductive cohomological degree $3$ invariants of all split semisimple groups of types $B$, $C$, and $D$. We also present a complete description of the cohomological invariants. As an application, we show that the group of degree $3$ unramified cohomology of the classifying space $BG$ is trivial for all split semisimple groups $G$ of types $B$, $C$, and $D$.
Let $R=\oplus_{i\in \N_0}R_n$ be a standard graded ring, $R_+ :=\oplus_{i\in \N}R_n$ be the irrelevant ideal of $R$ and $\fa_0$ be an ideal of $R_0$. In this paper, as a generalization of the concept of Castelnouvo-Mumford regularity $\reg(M)$ of a finitely generated graded $R$-module $M$, we define the regularity of $M$ with respect to $\fa_0+ R_+$, say $\reg_{\fa_0+ R_+}(M)$. And we study some relations of this new invariant with the classic one. To this end, we need to consider the cohomological dimension of some finitely generated $R_0$-modules. Also, we will express $\reg_{\fa_0+ R_+}(M)$…
Let $G$ be a finite group, $k$ be a field and $G\to GL(V_{\rm reg})$ be the regular representation of $G$ over $k$. Then $G$ acts naturally on the rational function field $k(V_{\rm reg})$ by $k$-automorphisms. Define $k(G)$ to be the fixed field $k(V_{\rm reg})^G$. Noether's problem asks whether $k(G)$ is rational (resp. stably rational) over $k$. When $k=\bQ$ and $G$ contains a normal subgroup $N$ with $G/H\simeq C_8$ (the cyclic group of order $8$), Jack Sonn proves that $\bQ(G)$ is not stably rational over $\bQ$, which is a non-abelian extension of a theorem of Endo-Miyata, Voskresenskii, L…
Let U be a connected scheme of finite cohomological dimension in which every finite set of points is contained in an affine open subscheme. Suppose that alpha is a class in H^2(U_et,Gm)_{tors}. For each positive integer m, the K-theory of alpha-twisted sheaves is used to identify obstructions to alpha being representable by an Azumaya algebra of rank m^2. The etale index of alpha, denoted eti(alpha), is the least positive integer such that all the obstructions vanish. Let per(alpha) be the order of alpha in H^2(U_{et},Gm)_{tors}. Methods from stable homotopy theory give an upper bound on the e…
We prove that the group of normalized cohomological invariants of degree 3 modulo the subgroup of semidecomposable invariants of a semisimple split linear algebraic group G is isomorphic to the torsion part of the Chow group of codimension 2 cycles of the respective versal G-flag. In particular, if G is simple, we show that this factor group is isomorphic to the group of indecomposable invariants of G. As an application, we construct nontrivial cohomological classes for indecomposable central simple algebras.
For any finite group G we define the moduli space of pointed admissible G-covers and the concept of a G-equivariant cohomological field theory (G-CohFT), which, when G is the trivial group, reduce to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. We prove that by taking the "quotient" by G, a G-CohFT reduces to a CohFT. We also prove that a G-CohFT contains a G-Frobenius algebra, a G-equivariant generalization of a Frobenius algebra, and that the "quotient" by G agrees with the obvious Frobenius algebra structure on the space of G-invariants, after re…
In this paper we give an upper bound, in characteristic 0, for the cohomological dimension of a graded ideal in a polynomial ring such that the quotient has depth at least 3. In positive characteristic the same bound holds true by a well-known theorem of Peskine and Szpiro. As a corollary, we give new examples of prime ideals that are not set-theoretically Cohen-Macaulay.
