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 X be a proper scheme and Z a prestack over X equipped with a flat connection. We give a local-to-global description of D-modules on the prestack S(Z) of flat sections of Z. Examples of S(Z) include the moduli stacks of principal G-bundles and de Rham local systems on X. We show that the category of D-modules is equivalent to the category of ind-coherent sheaves which are equivariant with respect to infinitesimal Hecke groupoids parametrized by finite subsets of X. We describe a number of applications to geometric representation theory and conformal field theory, including a derived enhance…
We show that the moduli spaces of stable sheaves on projective schemes admit certain non-commutative structures, which we call quasi NC structures, generalizing Kapranov's NC structures. The completion of our quasi NC structure at a closed point of the moduli space gives a pro-representable hull of the non-commutative deformation functor of the corresponding sheaf developed by Laudal, Eriksen, Segal and Efimov-Lunts-Orlov. We also show that the framed stable moduli spaces of sheaves have canonical NC structures.
We study marginal and relevant supersymmetric deformations of the N=4 super-Yang-Mills theory in four dimensions. Our primary innovation is the interpretation of the moduli spaces of vacua of these theories as non-commutative spaces. The construction of these spaces relies on the representation theory of the related quantum algebras, which are obtained from F-term constraints. These field theories are dual to superstring theories propagating on deformations of the AdS_5xS^5 geometry. We study D-branes propagating in these vacua and introduce the appropriate notion of algebraic geometry for non…
We propose an explicit relation between the cohomology of compactified and noncompactified moduli spaces of algebraic curves with punctures. This relationship generalizes one between commutative algebras and Lie algebras proposed by Lazard, Kontsevich and Ginzburg and Kapranov. More explicitly, we show that the cohomology of one moduli space is the graph complex decorated with the cohomology of the other, which in its turn generalizes Getzler's result for the genus zero moduli. We also show that a certain class of algebras over the chain operad of compactified moduli spaces has a natural struc…
We construct, using geometric invariant theory, a quasi-projective Deligne-Mumford stack of stable graded algebras. We also construct a derived enhancement, which classifies twisted bundles of stable graded A-infinity-algebras. The tangent complex of the derived scheme is given by graded Hochschild cohomology, which we relate to ordinary Hochschild cohomology. We obtain a version of Hilbert stability for non-commutative projective schemes.
The Hilbert scheme of point modules was introduced by Artin-Tate-Van den Bergh to study non-commutative graded algebras. The key tool is the construction of a map from the algebra to a twisted ring on this Hilbert scheme. In this paper, we study moduli stacks of more general "fat" point modules, and show that there is a similar map to a twisted ring associated to the stack. This is used to provide a sufficient criterion for a non-commutative projective surface to be birationally PI. It is hoped that such a criterion will be useful in understanding Mike Artin's conjecture on the birational clas…
We survey the basic theory of non-commutative K3 surfaces, with a particular emphasis to the ones arising from cubic fourfolds. We focus on the problem of constructing Bridgeland stability conditions on these categories and we then investigate the geometry of the corresponding moduli spaces of stable objects. We discuss a number of consequences related to cubic fourfolds including new proofs of the Torelli theorem and of the integral Hodge conjecture, the extension of a result of Addington and Thomas and various applications to hyperkähler manifolds. These notes originated from the lecture ser…
We attempt to develop a general algebro-geometric study of the moduli stack of commutative, 1-parameter formal Lie groups. We emphasize the pro-algebraic structure of this stack: it is the inverse limit, over varying n, of moduli stacks of n-buds, and these latter stacks are algebraic. Our main results pertain to aspects of the height stratification relative to fixed prime p on the stacks of buds and formal Lie groups. We conclude with a largely expository account of some foundational material on limits in bicategories.
In this survey paper we study the relationships between the coarse moduli space which parameterizes the finite dimensional linear representations of an associative alegebra, the non commutative hilbert scheme and the affine scheme which is the spectrum of the abelianization of algebra of the divided powers. In particular we will show a map which specialize to the Hlibert - Chow morphism when the associative algebra is commutative. The extension to the positive characteristic case of some of the results due to L. Le Bruyn on noncommutative desingularization is outlined. The possibility to use o…
Let $k$ be a commutative ring and let $R$ be a commutative $k-$algebra. The aim of this paper is to define and discuss some connection morphisms between schemes associated to the representation theory of a (non necessarily commutative) $R-$algebra $A. $ We focus on the scheme $\ran//\GL_n$ of the $n-$dimensional representations of $A, $ on the Hilbert scheme $\Hilb_A^n$ parameterizing the left ideals of codimension $n$ of $A$ and on the affine scheme Spec $Γ_R^n(A)^{ab} $ of the abelianization of the divided powers of order $n$ over $A. $ We give a generalization of the Grothendieck-Deligne no…
The moduli space of rank-n commutative algebras equipped with an ordered basis is an affine scheme B_n of finite type over Z, with geometrically connected fibers. It is smooth if and only if n <= 3. It is reducible if n >= 8 (and the converse holds, at least if we remove the fibers above 2 and 3). The relative dimension of B_n is (2/27) n^3 + O(n^{8/3}). The subscheme parameterizing etale algebras is isomorphic to GL_n/S_n, which is of dimension only n^2. For n >= 8, there exist algebras are not limits of etale algebras. The dimension calculations lead also to asymptotic formulas for the numbe…
We construct a functor that associates to any dg cooperad of dg commutative algebras (satisfying some conditions) an augmented commutative algebra. When applied to the cohomology operad of Francis Brown's moduli spaces it produces an algebra that formally models the algebra of multiple zeta values. We prove that there is an injection from the graded dual of the Grothendieck-Teichmueller Lie algebra into the indecomposables of the algebra associated to the Gerstenhaber cooperad, and that there is a morphism from the algebra associated to Brown's moduli spaces to the algebra associated to the Ge…
To any finite group G in SL_2(C), and each `t' in the center of the group algebra of G, we associate a category, Coh_t. It is defined as a suitable quotient of the category of graded modules over (a graded version of) the deformed preprojective algebra introduced by Crawley-Boevey and Holland. The category Coh_t should be thought of as the category of coherent sheaves on a `noncommutative projective 2-space', equipped with a framing at the line at infinity. Our first result establishes an isomorphism between the moduli space of torsion free objects of Coh_t and the Nakajima quiver variety aris…
Inspired by ideas from non-commutative geometry, unions of moduli spaces of linear control systems are identified as open subsets of infinite Grassmannians.
We suggest to compactify the universal covering of the moduli space of complex structures by non-commutative spaces. The latter are described by certain categories of sheaves with connections which are flat along foliations. In the case of abelian varieties this approach gives quantum tori as a non-commutative boundary of the moduli space. Relations to mirror symmetry, modular forms and deformation theory are discussed.
In this paper, we will introduce Quantum Toric Varieties which are (non-commutative) generalizations of ordinary toric varieties where all the tori of the classical theory are replaced by quantum tori. Quantum toric geometry is the non-commutative version of the classical theory; it generalizes non-trivially most of the theorems and properties of toric geometry. By considering quantum toric varieties as (non-algebraic) stacks, we define their category and show that it is equivalent to a category of quantum fans. We develop a Quantum Geometric Invariant Theory (QGIT) type construction of Quantu…
We study moduli spaces of maximal orders in a ramified division algebra over the function field of a smooth projective surface. As in the case of moduli of stable commutative surfaces, we show that there is a Kollár-type condition giving a better moduli problem with the same geometric points: the stack of blt Azumaya algebras. One virtue of this refined moduli problem is that it admits a compactification with a virtual fundamental class.
The moduli spaces of stable sheaves on projective schemes admit certain gluing data of Kapranov's NC structures, which we call quasi NC structures. The formal completion of the quasi NC structure at a closed point coincides with the pro-representable hull of the non-commutative deformation functor of the corresponding sheaf. In this paper, we show the existence of smooth non-commutative dg-resolutions of the above quasi NC structures, and call them quasi NCDG structures. When there are no higher obstruction spaces, the quasi NCDG structures define the notion of NC virtual structure sheaves, th…
We prove that any non-commutative smooth projective variety with a Bridgeland stability condition of dimension less than $\frac{6}{5}$ must be a smooth projective curve. As a consequence, we deduce the non-existence of such categories with dimension in the interval $(1,\frac{6}{5})$. Moreover, we prove a sharp reconstruction result for smooth projective curves of higher genus using the moduli space of stable objects in a category of dimension $1$, and deduce a structural result for their semi-orthogonal decompositions.
The Abuaf-Ueda flop is a 7-dimensional flop related to $G_2$ homogeneous spaces. The derived equivalence for this flop was first proved by Ueda using mutations of semi-orthogonal decompositions. In this article, we give an alternative proof for the derived equivalence using tilting bundles. Our proof also shows the existence of a non-commutative crepant resolution of the singularity appearing in the flopping contraction. We also give some results on moduli spaces of finite-length modules over this non-commutative crepant resolution.
In this note we introduce a new family of non-commutative spaces that we call non-commutative toric varieties and we describe some of their main properties. The main technical tool in this investigation is a natural extension of LVM-theory for the irrational case. In order to introduce the moduli space of (non-commutative) toric varieties we use variations on the notion of diffeology as models for non-commutative spaces.
We study a toric degeneration of the Cox ring of the moduli of principal $SL_m(\mathbb{C})$ bundles on the projective line, with quasi parabolic data given by the the stabilizer of the highest weight vector in $\mathbb{C}^m$ and its dual $\bigwedge^{m-1}(\mathbb{C}^m)$. The affine semigroup algebra resulting from this degeneration is described using the $K-$Pieri rule from Kac-Moody representation theory. Along the way we give a proof of the $K-$Pieri rule which utilizes the classical Pieri rule and elements of commutative algebra, and we describe a relationship between the Cox ring and a clas…
We develop homotopical algebraic geometry (see math.AG/0207028) in the special context where the base symmetric monoidal model category is the category S of spectra, i.e. what might be called, after Waldhausen, ``brave new algebraic geometry''. We discuss various model topologies on the model category of commutative algebras in S, the associated theories of geometric S-stacks (a geometric S-stack being an analog of Artin notion of algebraic stack in Algebraic Geometry), and finally show how to define global moduli spaces of associative ring spectra structures and a moduli space related to topo…
In the first part of this paper we construct a model structure for the category of filtered cochain complexes of modules over some commutative ring $R$ and explain how the classical Rees construction relates this to the usual projective model structure over cochain complexes. The second part of the paper is devoted to the study of derived moduli of sheaves: we give a new proof of the representability of the derived stack of perfect complexes over a proper scheme and then use the new model structure for filtered complexes to tackle moduli of filtered derived modules. As an application, we const…
Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL_n. We calculate the equivariant cohomology rings of the Laumon moduli spaces in terms of Gelfand-Tsetlin subalgebra of U(gl_n), and formulate a conjectural answer for the small quantum cohomology rings in terms of certain commutative shift of argument subalgebras of U(gl_n).
We introduce a generalization of variations of Hodge structures living over moduli spaces of non-commutative deformations of complex manifolds. Hodge structure associated with a point of such moduli space is an element of Sato type grassmanian of semi-infinite subspaces in H^*(X,C)[[h^{-1},h]]. Periods associated with such semi-infinite Hodge structures serve in order to extend mirror symmetry relations in dimensions greater then three.
It is a fundamental problem in geometry to decide which moduli spaces of polarized algebraic varieties are embedded by their period maps as Zariski open subsets of locally Hermitian symmetric domains. In the present work we prove that the moduli space of Calabi-Yau threefolds coming from eight planes in $\mathbb{P}^3$ does {\em not} have this property. We show furthermore that the monodromy group of a good family is Zariski dense in the corresponding symplectic group. Moreover, we study a natural sublocus which we call hyperelliptic locus, over which the variation of Hodge structures is natura…
We discuss aspects of the algebraic geometry of compact non-commutative Calabi-Yau manifolds. In this setting, it is appropriate to consider local holomorphic algebras which can be glued together into a compact Calabi-Yau algebra. We consider two examples: a toroidal orbifold T^6/Z_2 x Z_2, and an orbifold of the quintic in CP_4, each with discrete torsion. The non-commutative geometry tools are enough to describe various properties of the orbifolds. First, one describes correctly the fractionation of branes at singularities. Secondly, for the first example we show that one can recover explici…
Gaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno-Drinfeld Lie algebra t_n. We show that Gaudin subalgebras form a variety isomorphic to the moduli space of stable curves of genus zero with n+1 marked points. In particular, this gives an embedding of the moduli space in a Grassmannian of (n-1)-planes in an n(n-1)/2-dimensional space. We show that the sheaf of Gaudin subalgebras over the moduli space is isomorphic to a sheaf of twisted first order differential operators. For each representation of the Kohno--Drinfeld Lie algebra with fixed ce…
We discuss the Hodge theory of algebraic non-commutative spaces and analyze how this theory interacts with the Calabi-Yau condition and with mirror symmetry. We develop an abstract theory of non-commutative Hodge structures, investigate existence and variations, and propose explicit construction and classification techniques. We study the main examples of non-commutative Hodge structures coming from a symplectic or a complex geometry possibly twisted by a potential. We study the interactions of the new Hodge theoretic invariants with mirror symmetry and derive non-commutative analogues of some…
In this paper, we study non-commutative projective schemes whose associated non-commutative graded algebras are finite over their centers. We study their moduli spaces of stable sheaves, and construct a symmetric obstruction theory in the Calabi-Yau-3 case. This allows us to define Donaldson-Thomas type invariants. We also discuss the simplest examples, called quantum Fermat quintic threefolds.
It is well known that the Fano scheme of lines on a cubic 4-fold is a symplectic variety. We generalize this fact by constructing a closed p-form with p=2n-4 on the Fano scheme of lines on a (2n-2)-dimensional hypersurface Y of degree n. We provide several definitions of this form - via the Abel-Jacobi map, via Hochschild homology, and via the linkage class, and compute it explicitly for n = 4. In the special case of a Pfaffian hypersurface Y we show that the Fano scheme is birational to a certain moduli space of sheaves on a p-dimensional Calabi--Yau variety X arising naturally in the context…
This is a noncommutative-geometric study of the semiclassical dynamics of finite topological D-brane systems. Starting from the formulation in terms of A -infinity categories, I show that such systems can be described by the noncommutative symplectic supergeometry of Z2-graded quivers, and give a synthetic formulation of the boundary part of the generalized WDVV equations. In particular, a faithful generating function for integrated correlators on the disk can be constructed as a linear combination of quiver necklaces, i.e. a function on the noncommutative symplectic superspace defined by the …
We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P^2. The generic noncommutative plane corresponds to the Sklyanin algebra S constructed from an automorphism sigma of infinite order on an elliptic curve E < P^2. In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic (1-n) provides a sy…
Hausel introduced a commutative algebra -- the multiplicity algebra -- associated to a fixed point of the C^*-action on the Higgs bundle moduli space. Here we describe this algebra for a fixed point consisting of a very stable rank 2 vector bundle and zero Higgs field for a curve of low genus. Geometrically, the relations in the algebra are described by a family of quadrics and we focus on the discriminant of this family, providing a new viewpoint on the moduli space of stable bundles. The discriminant in our examples demonstrates that as the bundle varies, we obtain a continuous variation in …
These are expanded notes from some talks given during the fall 2002, about ``homotopical algebraic geometry'' (HAG) with special emphasis on its applications to ``derived algebraic geometry'' (DAG) and ``derived deformation theory''. We use the general framework developed in Toen, Vezzosi, ``Homotopical Algebraic Geometry I: Topos theory'', and in particular the notions of model topology, model sites and stacks over them, in order to define various derived moduli functors and study their geometric properties. We start by defining the model category of D-stacks, with respect to an extension of …
We propose a generalization of the topological vertex, which we call the "non-commutative topological vertex". This gives open BPS invariants for a toric Calabi-Yau manifold without compact 4-cycles, where we have D0/D2/D6-branes wrapping holomorphic 0/2/6-cycles, as well as D2-branes wrapping disks whose boundaries are on D4-branes wrapping non-compact Lagrangian 3-cycles. The vertex is defined combinatorially using the crystal melting model proposed recently, and depends on the value of closed string moduli at infinity. The vertex in one special chamber gives the same answer as that computed…
Let R be a non-commutative field. We prove that generic triples of flags in an m-dimensional R-vector space are described by flat R-line bundles on the honeycomb graph with (m-1)(m-2)/2 holes. Generalising this, we prove that non-commutative stacks X(m,S) of framed rank m flat R-vector bundles of on decorated surfaces S are birationally identified with the moduli spaces of flat line bundles on a spectral surface assigned to certain bipartite graphs on S. We introduce non-commutative cluster Poisson varieties related to bipartite ribbon graphs. They carry canonical non-commutative Poisson struc…
In this article, we construct a non-commutative crepant resolution (=NCCR) of a minimal nilpotent orbit closure $\overline{B(1)}$ of type A, and study relations between an NCCR and crepant resolutions $Y$ and $Y^+$ of $\overline{B(1)}$. More precisely, we show that the NCCR is isomorphic to the path algebra of the double Beilinson quiver with certain relations and we reconstruct the crepant resolutions $Y$ and $Y^+$ of $\overline{B(1)}$ as moduli spaces of representations of the quiver. We also study the Kawamata-Namikawa's derived equivalence between crepant resolutions $Y$ and $Y^+$ of $\ove…
For each simply connected, simple complex group $G$ we show that the direct sum of all vector bundles of conformal blocks on the moduli stack $\bar{\mathcal{M}}_{g, n}$ of stable marked curves carries the structure of a flat sheaf of commutative algebras. The fiber of this sheaf over a smooth marked curve $(C, \vec{p})$ agrees with the Cox ring of the moduli of quasi-parabolic principal $G-$bundles on $(C, \vec{p})$. We use the factorization rules on conformal blocks to produce flat degenerations of these algebras. These degenerations are toric in the case $G = SL_2(\mathbb{C}),$ and the resul…
The derived category of a general complete intersection of four quadrics in P^{2n-1} has a semi-orthogonal decomposition < O(-2n+9), ..., O(-1), O, D >, where D is the derived category of twisted sheaves on a certain non-algebraic complex 3-fold coming from a moduli problem. In particular, when n=4 we obtain a (twisted) derived equivalence of Calabi-Yau 3-folds predicted by Gross. This differs from Kuznetsov's result in that our construction is geometric and avoids non-commutative varieties.
In 2008, Rogalski and Zhang showed that if R is a strongly noetherian connected graded algebra over an algebraically closed field, then R has a canonical birationally commutative factor. This factor is, up to finite dimension, a twisted homogeneous coordinate ring B(X, L, s); here X is the projective parameter scheme for point modules over R, as well as tails of points in qgr-R. (As usual, s is an automorphism of X, and L is an s-ample invertible sheaf on X.) We extend this result to a large class of noetherian (but not strongly noetherian) algebras. Specifically, let R be a noetherian connect…
Log Calabi-Yau mirror symmetry and non-archimedean disks
Sean Keel, Department of Mathematics, 1 University Station C1200, Austin, TX 78712-0257, USA；Logan White, Department of Mathematics, 1 University Station C1200, Austin, TX 78712-0257, USA；Tony Yue YU, Department of Mathematics, M/C 253-37, Caltech, 1200 E.\ California Blvd., Pasadena, CA 91125, USA
It is shown that every algebra over the chain operad of the little disks operad gives naturally rise to a Hertling-Manin's F-manifold, that is a smooth manifold equipped with an integrable graded commutative associative product on the tangent sheaf. In particular, moduli spaces of extended deformations of complex/symplectic structures are shown to have a canonical structure of F-manifold. With the help of the $G_\infty$-operad a strong homotopy version of the notion of F-manifold is constructed. Among natural examples of $F_\infty$-manifolds one finds formal manifolds associated with the Hochs…
Let $L\subset X$ be a not necessarily orientable relatively $Pin$ Lagrangian submanifold in a symplectic manifold $X$. We construct a family of cyclic unital curved $A_\infty$ structures on differential forms on $L$ with values in the local system of graded non-commutative rings given by the tensor algebra of the orientation local system of $L$. The family of $A_\infty$ structures is parameterized by the cohomology of $X$ relative to $L$ and satisfies properties analogous to the axioms of Gromov-Witten theory. On account of the non-orientability of $L,$ the evaluation maps of moduli spaces of …
