We define and study coisotropic structures on morphisms of commutative dg algebras in the context of shifted Poisson geometry, i.e. $P_n$-algebras. Roughly speaking, a coisotropic morphism is given by a $P_{n+1}$-algebra acting on a $P_n$-algebra. One of our main results is an identification of the space of such coisotropic structures with the space of Maurer--Cartan elements in a certain dg Lie algebra of relative polyvector fields. To achieve this goal, we construct a cofibrant replacement of the operad controlling coisotropic morphisms by analogy with the Swiss-cheese operad which can be of independent interest. Finally, we show that morphisms of shifted Poisson algebras are identified with coisotropic structures on their graph.
We extend results about $n$-shifted coisotropic structures from part I of this work to the setting of derived Artin stacks. We show that an intersection of coisotropic morphisms carries a Poisson structure of shift one less. We also compare non-degenerate shifted coisotropic structures and shifted Lagrangian structures and show that there is a natural equivalence between the two spaces in agreement with the classical result. Finally, we define quantizations of $n$-shifted coisotropic structures and show that they exist for $n>1$.
We define (iterated) coisotropic correspondences between derived Poisson stacks, and construct symmetric monoidal higher categories of derived Poisson stacks where the $i$-morphisms are given by $i$-fold coisotropic correspondences. Assuming an expected equivalence of different models of higher Morita categories, we prove that all derived Poisson stacks are fully dualizable, and so determine framed extended TQFTs by the Cobordism Hypothesis. Along the way we also prove that the higher Morita category of $E_{n}$-algebras with respect to coproducts is equivalent to the higher category of iterate…
We give a definition of coisotropic morphisms of shifted Poisson (i.e. $P_n$) algebras which is a derived version of the classical notion of coisotropic submanifolds. Using this we prove that an intersection of coisotropic morphisms of shifted Poisson algebras carries a Poisson structure of shift one less. Using an interpretation of Hamiltonian spaces as coisotropic morphisms we show that the classical BRST complex computing derived Poisson reduction coincides with the complex computing coisotropic intersection. Moreover, this picture admits a quantum version using brace algebras and their mod…
We relate the brace construction introduced by Calaque and Willwacher to an additivity functor. That is, we construct a functor from brace algebras associated to an operad $O$ to associative algebras in the category of homotopy $O$-algebras. As an example, we identify the category of $P_{n+1}$-algebras with the category of associative algebras in $P_n$-algebras. We also show that under this identification there is an equivalence of two definitions of derived coisotropic structures in the literature.
We give an explicit finite-dimensional model for the derived moduli stack of flat connections on $\mathbb{C}^k$ with logarithmic singularities along a weighted homogeneous Saito free divisor. We investigate in detail the case of plane curves of the form $x^p = y^q$ and relate the moduli spaces to the Grothendieck-Springer resolution. We also discuss the shifted Poisson geometry of these moduli spaces. Namely, we conjecture that the map restricting a logarithmic connection to the complement of the divisor admits a shifted coisotropic structure and we construct a shifted Poisson structure on the…
Let $G$ be an affine group scheme over a noetherian commutative ring $R$. We show that every $G$-equivariant vector bundle on an affine toric scheme over $R$ with $G$-action is extended from $\Spec(R)$ for several cases of $R$ and $G$. We show that given two affine schemes with group scheme actions, an equivalence of the equivariant derived categories implies isomorphism of the equivariant $K$-theories as well as equivariant $K'$-theories.
Let Spec(A) be an affine derived stack. We give two proofs of the existence of a canonical map from the moduli space of shifted Poisson structures (in the sense of Pantev-Toën-Vaquié-Vezzosi, see http://arxiv.org/abs/1111.3209 ) on Spec(A) to the moduli space of homotopy (shifted) Poisson algebra structures on A. The first makes use of a more general description of the Poisson operad and of its cofibrant models, while the second in more computational and involves an explicit resolution of the Poisson operad.
Let $R$ be a commutative Noetherian ring such that $X=Spec R$ is connected. We prove that the category $D^b(coh X)$ contains no proper full triangulated subcategories which are regular. We also bound from below the dimension of a regular category $T$, if there exists a triangulated functor $T \to D^b(coh X)$ with certain properties. Applications are given to cohomological annihilator of $R$ and to point-like objects in $T$.
Moduli spaces of stable objects in the derived category of a $K3$ surface provide a large class of holomorphic symplectic varieties. In this paper, we study the interplay between Chern classes of stable objects and zero-cycles on holomorphic symplectic varieties which arise as moduli spaces. First, we show that the second Chern class of any object in the derived category lies in a suitable piece of O'Grady's filtration on the $\mathrm{CH}_0$-group of the $K3$ surface. This solves a conjecture of O'Grady and improves on previous results of Huybrechts, O'Grady, and Voisin. Then we propose a cand…
We describe the derived Picard group of an Azumaya algebra A on an affine scheme X in terms of global sections of the constant sheaf of integers on X, the Picard group of X, and the stabilizer of the Brauer class of A under the action of Aut(X). In particular, we find that the derived Picard group of an Azumaya algebra is generally not isomorphic to that of the underlying scheme. In the case of the trivial Azumaya algebra, our result refines Yekutieli's description of the derived Picard group of a commutative algebra. We also get, as a corollary, an alternate proof of a result of Antieau which…
Relying on recent advances in the theory of motives we develope a general formalism for derived categories of motives with Q-coefficients on perfect (ind-)schemes. As an application we give a motivic refinement of Zhu's geometric Satake equivalence for Witt vector affine Grassmannians in this set-up.
We show that the existence of locally finite stability conditions on the bounded derived category $\mathbf{D}^{b}(X)$ of coherent sheaves on an affine Noetherian scheme $X$ is equivalent to $\dim X=0$. We also study the spaces of stability conditions on the category of morphisms $\mathbf M_{X}$ in the derived category of the scheme $X$ and show that the spaces of stability conditions on $\mathbf{D}^{b}(X)$ and $\mathbf{M}_{X}$ are homotopy equivalent to each other.
We introduce a new algebra called the shifted $q=0$ affine algebra, which arises naturally from the study of coherent sheaves on Grassmannians and n-step partial flag varieties via a natural correspondence. It has similar presentation as the shifted quantum affine algebra defined by Finkelberg-Tsymbaliuk arXiv:1708.01795v6. We then give a definition of its categorical action and prove that there is a categorical action on the bounded derived categories of coherent sheaves on n-step partial flag varieties. Finally, as an application, we use it to construct a categorical action of the $q=0$ affi…
We construct 2-representations of quantum affine algebras from 2-representations of quantum Heisenberg algebras. The main tool in this construction are categorical vertex operators, which are certain complexes in a Heisenberg 2-representation that recover vertex operators after passing to the Grothendieck group. As an application we categorify the Frenkel-Kac-Segal homogeneous realization of the basic representation of (simply laced) quantum affine algebras. This gives rise to categorical actions of quantum affine (and toroidal) algebras on derived categories of coherent sheaves on Hilbert sch…
The geometric Satake isomorphism is an equivalence between the categories of spherical perverse sheaves on affine Grassmanian and the category of representations of the Langlands dual group. We provide a similar description for derived categories of l-adic sheaves on an affine flag variety which are geometric counterparts of a maximal commutative subalgebra in the Iwahori Hecke algebra; of the anti-spherical module over this algebra; and of the space of Iwahori-invariant Whitakker functions.
We give a universal construction of a derived affine group scheme and its representation category from a symmetric monoidal infinity-category, which we shall call the tannnakization of a symmetric monoidal infinity-category. This can be viewed as infinity-categorical generalization of the work of Joyal-Street and Nori. We then apply it to the stable infinity-category of mixed motives equipped with the realization functor of a mixed Weil cohomology and obtain a derived motivic Galois group whose representation category has a universality, and which represents the automorphism group of the reali…
We study the derived critical locus of a function $f:[X/G]\to \mathbb{A}_{\mathbb{K}}^1$ on the quotient stack of a smooth affine scheme $X$ by the action of a smooth affine group scheme $G$. It is shown that $\mathrm{dCrit}(f) \simeq [Z/G]$ is a derived quotient stack for a derived affine scheme $Z$, whose dg-algebra of functions is described explicitly. Our results generalize the classical BV formalism in finite dimensions from Lie algebra to group actions.
Let $U$ be a maximal unipotent subgroup of a connected semisimple group $G$ and $U'$ the derived group of $U$. We study actions of $U'$ on affine $G$-varieties. First, we consider the algebra of $U'$ invariants on $G/U$. We prove that $k[G/U]^{U'}$ is a polynomial algebra of Krull dimension $2r$, where $r=rk(G)$. A related result is that, for any simple finite-dimensional $G$-module $V$, $V^{U'}$ is a cyclic $U/U'$-module. Second, we study "symmetries" of Poincare series for $U'$-invariants on affine conical $G$-varieties. Third, we obtain a classification of simple $G$-modules $V$ with polyno…
The paper is concerned with cohomology of the small quantum group at a root of unity, and of its upper triangular subalgebra, with coefficients in a tilting module. We relate it to a certain t-structure on the derived category of equivariant coherent sheaves on the Springer resolution, and to equivariant coherent IC sheaves on the nil-cone. The support of these cohomology is described in terms of cells in affine Weyl groups, and the basis in the Grothendieck group they provide is related to the Kazhdan-Lusztig basis, as predicted by J. Humphreys and V. Ostrik. The proof is based on earlier res…
In this article, we show that the categorical action of the shifted $q=0$ affine algebra can be used to construct (or induce) t-structures on the weight categories. The main idea is to interpret the exceptional collection constructed by Kapranov as convolution of Fourier-Mukai kernels in the categorical action via using the Borel-Weil-Bott theorem. In particular, when the categories are the bounded derived category of coherent sheaves on Grassmannians. The t-structure we obtain is precisely the exotic t-structure defined by Bezrukavnikov of the exceptional collections given by Kapranov. As an …
Let G^\vee be a complex simple algebraic group. We describe certain morphisms of G^\vee(\calO)-equivariant complexes of sheaves on the affine Grassmannian \Gr of G^\vee in terms of certain morphisms of G-equivariant coherent sheaves on \frakg, where G is the Langlands dual group of G^\vee and \frakg is its Lie algebra. This can be regarded as an example of the derived Satake correspondence.
In this paper we present an approach to quadratic structures in derived algebraic geometry. We define derived n-shifted quadratic complexes, over derived affine stacks and over general derived stacks, and give several examples of those. We define the associated notion of derived Clifford algebra, in all these contexts, and compare it with its classical version, when they both apply. Finally, we prove three main existence results for derived shifted quadratic forms over derived stacks, define a derived version of the Grothendieck-Witt group of a derived stack, and compare it to the classical on…
