A theory of dg schemes is developed so that it becomes a homotopy site, and the corresponding infinity category of stacks is equivalent to the infinity category of stacks, as constructed by Toen and Vezzosi, on the site of dg algebras whose cohomologies have finitely many generators in each degree. Stacks represented by dg schemes are shown to be derived schemes under this correspondence.
For a derived stack obtained as a quotient of a derived affine scheme by a reductive group, we show that shifted symplectic structures can be characterized by the Cartan-de Rham complex. For non-reductive groups, we also show the analogous statement for Getzler's extension of the Cartan-de Rham complex. Dually, we construct a Cartan model for polyvector fields on global quotients by reductive groups, and show that shifted Poisson structures can be characterized by it.
We show the existence of a full exceptional collection in the graded stable derived category of a Gorenstein isolated quotient singularity using a result of Orlov (arXiv:math/0503632). We also show that the equivariant graded stable derived category of a Gorenstein Veronese subring of a polynomial ring with respect to an action of a finite group has a full strong exceptional collection, even if the corresponding quotient singularity is neither isolated nor Gorenstein.
Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier's notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory. More than a half of the text is devoted to an overview of "well known" definitions and results. As a result, the e-print is essentially self-contained.
We provided two explicit formulas for the intersection cohomology (as a graded vector space with pairing) of the symplectic quotient by a circle in terms of the $S^1$ equivariant cohomology of the original symplectic manifold and the fixed point data. The key idea is the construction of a small resolution of the symplectic quotient.
We define a derived enhancement of the classical quot functor of quotients associated to a coherent sheaf on a nonsingular quasiprojective variety. We prove its representability and show that it has the expected tangent complex. The derived quot scheme of points can be covered by affine charts obtained as spectra of commutative graded differential algebras (cdgas) and we compute an example. As a demonstration of the usefulness of this presentation, we write down an explicit shifted form on this example and conjecture that it agrees with the shifted symplectic structure on the derived stack of …
We construct toroidal partial compactifications of the moduli spaces of mixed Hodge structures with polarized graded quotients. They are moduli spaces of log mixed Hodge structures with polarized graded quotients. We construct them as the spaces of nilpotent orbits.
We announce the construction of toroidal partial compactifications of the moduli spaces of mixed Hodge structures with polarized graded quotients. They are moduli spaces of log mixed Hodge structures with polarized graded quotients. We include an application to the analyticity of zero loci of some functions.
We compute the Hilbert series, and the graded vector space structure, of Ext-algebras of quotients of Koszul algebras with almost linear resolution. The example of the generic determinantal varieties is treated in detail.
We give a formulation for derived analytic geometry built from commutative differential graded algebras equipped with entire functional calculus on their degree 0 part, a theory well-suited to developing shifted Poisson structures and quantisations. In the complex setting, we show that this formulation recovers equivalent derived analytic spaces and stacks to those coming from Lurie's structured topoi. In non-Archimedean settings, there is a similar comparison, but for derived dagger analytic spaces and stacks, based on overconvergent functions.
It is shown that there are globally defined Lagrangian distributions on the stable loci of derived Quot-stacks of coherent sheaves on Calabi--Yau four-folds. Dividing by these distributions produces perfectly obstructed smooth stacks with globally defined $-1$-shifted potentials, whose derived critical loci give back the stable loci of smooth stacks of sheaves in global Darboux form.
We study the symplectic reduction of the phase space describing $k$ particles in $\mathbb{R}^n$ with total angular momentum zero. This corresponds to the singular symplectic quotient associated to the diagonal action of $\operatorname{O}_n$ on $k$ copies of $\mathbb{C}^n$ at the zero value of the homogeneous quadratic moment map. We give a description of the ideal of relations of the ring of regular functions of the symplectic quotient. Using this description, we demonstrate $\mathbb{Z}^+$-graded regular symplectomorphisms among the $\operatorname{O}_n$- and $\operatorname{SO}_n$-symplectic qu…
We study the shifted Poisson structure on the cochain complex C*(g) of a graded Lie algebra arising from shifted Lie bialgebra structure on g. We apply this to construct a 1-shifted Poisson structures on an infinitesimal quotient of the derived variety of complexes RCom(V) by a subgroup of the automorphisms of V, and a non-shifted Poisson structure on an appropriately defined derived variety of 1-periodic complexes, extending the standard Kirillov-Kostant Poisson structure on gl*_n. We also show that in the case of RCom(V) the 1-shifted structure is up to homotopy a Batalin-Vilkovisky algebra …
We construct a period mapping for deformations of a differential graded algebra, that generalizes Griffiths' period mapping. It is constructed as a morphism between differential graded Lie algebras which has a moduli-theoretic interpretation, where the domain of the morphism is the shifted Hochschild cocohain complex. We then use this period mapping to give some applications such as unobstructedness of deformations.
Let $K$ be a compact Lie group with complexification $G$, and let $V$ be a unitary $K$-module. We consider the real symplectic quotient $M_0$ at level $0$ of the homogeneous quadratic moment map as well as the complex symplectic quotient, defined here as the complexification of $M_0$. We show that if $(V, G)$ is $3$-large, a condition that holds generically, then the complex symplectic quotient has symplectic singularities and is graded Gorenstein. This in particular implies that the real symplectic quotient is graded Gorenstein. In the case that $K$ is a torus or $\operatorname{SU}_2$, we sho…
Realizing a part of the Derived Deformation Theory program, we construct a "derived" analog of the Grothendieck's Quot scheme parametrizing subsheaves in a given coherent sheaf F on a smooth projective variety X. This analog is a differential graded manifold RQuot_h(F) (so it is always smooth in an appropriate sense) whose tangent space at a point represented by a subsheaf K in F, is a cochain complex quasiisomorphic to RHom(K, F/K).
Recent results in geometric invariant theory (GIT) for non-reductive linear algebraic group actions allow us to stratify quotient stacks of the form [X/H], where X is a projective scheme and H is a linear algebraic group with internally graded unipotent radical acting linearly on X, in such a way that each stratum [S/H] has a geometric quotient S/H. This leads to stratifications of moduli stacks (for example, sheaves over a projective scheme) such that each stratum has a coarse moduli space.
In this paper we develop a graded tilting theory for gauged Landau-Ginzburg models of regular sections in vector bundles over projective varieties. Our main theoretical result describes - under certain conditions - the bounded derived category of the zero locus $Z(s)$ of such a section $s$ as a graded singularity category of a non-commutative quotient algebra $Λ/\langle s\rangle$: $D^b(\mathrm{coh} Z(s))\simeq D^{\mathrm{gr}}_{\mathrm{sg}}(Λ/\langle s\rangle)$. Our geometric applications all come from homogeneous gauged linear sigma models. In this case $Λ$ is a non-commutative resolution of t…
We give an intrinsic characterization of Cox sheaves on Krull schemes in terms of their valuative algebraic properties. We also provide a geometric characterization of their graded relative spectra in terms of good quotients of graded schemes, extending the work of Arzhantsev, Derenthal, Hausen and Laface on relative spectra of Cox sheaves on normal varieties. Moreover, we obtain an irredundant characterization of Cox rings which in turn produces a normality criterion for certain graded rings.
We introduce a formalism for derived moduli functors on differential graded associative algebras, which leads to non-commutative enhancements of derived moduli stacks and naturally gives rise to structures such as Hall algebras. Descent arguments are not available in the non-commutative context, so we establish new methods for constructing various kinds of atlases. The formalism permits the development of the theory of shifted bi-symplectic and shifted double Poisson structures in the companion paper.
Let A denote the ring of differential operators on the affine line with its two usual generators t and d/dt given degrees +1 and -1 respectively. Let X be the stack having coarse moduli space the affine line Spec k[z] and isotropy groups Z/2 at each integer point. Then the category of graded A-modules is equivalent to the category of quasi-coherent sheaves on X. Version 2: corrected typos and deleted appendix at referee's suggestion.
We establish a system of formal noncommutative calculus for differential forms and polyvector fields, which forms the foundations for the study of pre-Calabi-Yau categories. Using an explicit trace map, we show that any $n$-Calabi-Yau structure on a non-positively graded dg algebra $A$ induces a $(2-n)$-shifted symplectic structure on its derived moduli stack of representations; while any $n$-pre-Calabi-Yau structure on $A$ induces a $(2-n)$-shifted Poisson structure on this derived moduli stack.
