arXiv subjects
Publications and source records attributed to Kai Behrend.
Recently, John Pardon proved the MNOP conjecture (on the GW-DT correspondence for CY3s) by introducing a new mathematical gadget, which we call the Pardon homology algebra of 1-cycles in 3-folds. We work out an analogous construction for 0-cycles in d-folds. This gives a new point of view on enumerative problems involving point-counting, such as, for example, the degree zero MNOP conjecture on the Hilbert scheme of points in projective 3-folds.
We prove that dg manifolds of finite positive amplitude, i.e. bundles of positively graded curved $L_\infty[1]$-algebras, form a category of fibrant objects. As a main step in the proof, we obtain a factorization theorem using path spaces. First we construct an infinite-dimensional factorization of a diagonal morphism using actual path spaces motivated by the AKSZ construction. Then we cut down to finite dimensions using the Fiorenza-Manetti method. The main ingredient in our method is the homotopy transfer theorem for curved $L_\infty[1]$-algebras. As an application, we study the derived intersections of manifolds.
We prove that an \'etale fibration between $L_\infty$-bundles admits local sections composed of several elementary morphisms of particularly simple and accessible type. As applications, we establish an inverse function theorem for $L_\infty$-bundles and provide an elementary proof that every weak equivalence of $L_\infty$-bundles induces a quasi-isomorphism of the differential graded algebras of global functions. Furthermore, we apply this inverse function theorem to show that the homotopy category of $L_\infty$-bundles admits a simple description in terms of homotopy classes of morphisms, when $L_\infty$-bundles are restricted to their germs around their classical loci.
We investigate the quantization problem of $(-1)$-shifted derived Poisson manifolds in terms of $\BV_\infty$-operators on the space of Berezinian half-densities. We prove that quantizing such a $(-1)$-shifted derived Poisson manifold is equivalent to the lifting of a consecutive sequences of Maurer-Cartan elements of short exact sequences of differential graded Lie algebras, where the obstruction is a certain class in the second Poisson cohomology. Consequently, a $(-1)$-shifted derived Poisson manifold is quantizable if the second Poisson cohomology group vanishes. We also prove that for any $\L$-algebroid $\Cc{\aV}$, its corresponding linear $(-1)$-shifted derived Poisson manifold $\Cc{\aV}^\vee[-1]$ admits a canonical quantization. Finally, given a Lie algebroid $A$ and a one-cocycle $s\in \sections{A^\vee}$, the $(-1)$-shifted derived Poisson manifold corresponding to the derived intersection of coisotropic submanifolds determined by the graph of $s$ and the zero section of the Lie Poisson $A^\vee$ is shown to admit a canonical quantization in terms of Evens-Lu-Weinstein module.
We develop the theory of derived differential geometry in terms of bundles of curved $L_\infty[1]$-algebras, i.e. dg manifolds of positive amplitudes. We prove the category of derived manifolds is a category of fibrant objects. Therefore, we can make sense of "homotopy fibered product" and "derived intersection" of submaifolds in a smooth manifold in the homotopy category of derived manifolds. We construct a factorization of the diagonal using path spaces. First we construct an infinite-dimensional factorization using actual path spaces motivated by the AKSZ construction, then we cut down to finite dimensions using the Fiorenza-Manetti method. The main ingredient is the homotopy transfer theorem for curved $L_\infty[1]$-algebras. We also prove the inverse function theorem for derived manifolds, and investigate the relationship between weak equivalence and quasi-isomorphism for derived manifolds.
We view the inertia construction of algebraic stacks as an operator on the Grothendieck groups of various categories of algebraic stacks. We show that the inertia operator is locally finite and diagonalizable. This is proved for the Grothendieck group of Deligne-Mumford stacks over a base scheme and the category of quasi-split Artin stacks defined over a field of characteristic zero. Motivated by the quasi-splitness condition, in [2] we consider the inertia operator of the Hall algebra of algebroids, and applications of it in generalized Donaldson-Thomas theory.
We study the action of the inertia operator on the motivic Hall algebra, and prove that it is diagonalizable. This leads to a filtration of the Hall algebra, whose associated graded algebra is commutative. In particular, the degree 1 subspace forms a Lie algebra, which we call the Lie algebra of virtually indecomposable elements, following Joyce. We prove that the integral of virtually indecomposable elements admits an Euler characteristic specialization. In order to take advantage of the fact that our inertia groups are unit groups in algebras, we introduce the notion of algebroid.
We characterize which quadratic regular algebras of global dimension 3 are stable in the sense of Behrend-Noohi. (This notion of stability is a non-commutative analogue of Hilbert stability.) We describe the quasi-projective stack of stable quadratic regular algebras of dimension 3. This provides the first non-trivial, non-commutative examples of stability a la Behrend-Noohi.
In an enriched setting, we show that higher groupoids and higher categories form categories of fibrant objects. The nerve of a differential graded algebra is a higher category in the category of algebraic varieties, where covers are defined to be smooth epimorphisms.
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.
We establish the general machinery of string topology for differentiable stacks. This machinery allows us to treat on an equal footing free loops in stacks and hidden loops. In particular, we give a good notion of a free loop stack, and of a mapping stack $\map(Y,\XX)$, where $Y$ is a compact space and $\XX$ a topological stack, which is functorial both in $\XX$ and $Y$ and behaves well enough with respect to pushouts. We also construct a bivariant (in the sense of Fulton and MacPherson) theory for topological stacks: it gives us a flexible theory of Gysin maps which are automatically compatible with pullback, pushforward and products. Further we prove an excess formula in this context. We introduce oriented stacks, generalizing oriented manifolds, which are stacks on which we can do string topology. We prove that the homology of the free loop stack of an oriented stack and the homology of hidden loops (sometimes called ghost loops) are a Frobenius algebra which are related by a natural morphism of Frobenius algebras. We also prove that the homology of free loop stack has a natural structure of a BV-algebra, which together with the Frobenius structure fits into an homological conformal field theories with closed positive boundaries. Using our general machinery, we construct an intersection pairing for (non necessarily compact) almost complex orbifolds which is in the same relation to the intersection pairing for manifolds as Chen-Ruan orbifold cup-product is to ordinary cup-product of manifolds. We show that the hidden loop product of almost complex is isomorphic to the orbifold intersection pairing twisted by a canonical class. Finally we gave some examples including the case of the classifying stacks $[*/G]$ of a compact Lie group.
Given a smooth complex threefold X, we define the virtual motive of the Hilbert scheme of n points on X. In the case when X is Calabi-Yau, this gives a motivic refinement of the n-point degree zero Donaldson-Thomas invariant of X. The key example is affine three-space, where the Hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety, and its virtual motive is defined in terms of the Denef-Loeser motivic nearby fiber. A crucial technical result asserts that if a function is equivariant with respect to a suitable torus action, its motivic nearby fiber is simply given by the motivic class of a general fiber. This allows us to compute the generating function of the virtual motives of the Hilbert schemes of affine three-space via a direct computation involving the motivic class of the commuting variety. We then give a formula for the generating function for arbitrary X as a motivic exponential, generalizing known results in lower dimensions. The weight polynomial specialization leads to a product formula in terms of deformed MacMahon functions, analogous to Gottsche's formula for the Poincare polynomials of the Hilbert schemes of points on surfaces.
We introduce differentiable stacks and explain the relationship with Lie groupoids. Then we study $S^1$-bundles and $S^1$-gerbes over differentiable stacks. In particular, we establish the relationship between $S^1$-gerbes and groupoid $S^1$-central extensions. We define connections and curvings for groupoid $S^1$-central extensions extending the corresponding notions of Brylinski, Hitchin and Murray for $S^1$-gerbes over manifolds. We develop a Chern-Weil theory of characteristic classes in this general setting by presenting a construction of Chern classes and Dixmier-Douady classes in terms of analogues of connections and curvatures. We also describe a prequantization result for both $S^1$-bundles and $S^1$-gerbes extending the well-known result of Weil and Kostant. In particular, we give an explicit construction of $S^1$-central extensions with prescribed curvature-like data.
We solve the part of the Donaldson-Thomas theory of Calabi-Yau threefolds which comes from super-rigid rational curves. As an application, we prove a version of the conjectural Gromov-Witten/Donaldson-Thomas correspondence for contributions from super-rigid rational curves. In particular, we prove the full GW/DT correspondence for the quintic threefold in degrees one and two.
Let $G$ be a split connected semisimple group over a field. We give a conjectural formula for the motive of the stack of $G$-bundles over a curve $C$, in terms of special values of the motivic zeta function of $C$. The formula is true if $C=\pp^1$ or $G=\sln$. If $k=\cc$, upon applying the Poincaré or Serre characteristic, the formula reduces to results of Teleman and Atiyah-Bott on the gauge group. If $k=\ffq$, upon applying the counting measure, it reduces to the fact that the Tamagawa number of $G$ over the function field of $C$ is $|π_1(G)|$.
We introduce the notion of symmetric obstruction theory and study symmetric obstruction theories which are compatible with C*-actions. We prove that the contribution of an isolated fixed point under a C*-action to equivariant Donaldson-Thomas type invariants is +/- 1. As an application, we compute weighted Euler characteristics of all Hilbert schemes of points on any 3-fold. Moreover, we calculate the zero-dimensional Donaldson-Thomas invariants of any projective Calabi-Yau 3-fold. This proves a conjecture of Maulik-Nekrasov-Okounkov-Pandharipande.
We prove that Donaldson-Thomas type invariants are equal to weighted Euler characteristics of their moduli spaces. In particular, such invariants depend only on the scheme structure of the moduli space, not the symmetric obstruction theory used to define them. We also introduce new invariants generalizing Donaldson-Thomas type invariants to moduli problems with open moduli space. These are useful for computing Donaldson-Thomas type invariants over stratifications.
We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids. Cofoliations on stacks arise from flat connections on groupoids. Connections on groupoids generalize connections on gerbes and bundles in a natural way. A flat connection on a groupoid is an integrable distribution of the morphism space compatible with the groupoid structure and complementary to both source and target fibres. A cofoliation of a stack determines the flat groupoid up to etale equivalence. We show how a cofoliation on a stack gives rise to a refinement of the Hodge to De Rham spectral sequence, where the E1-term consists entirely of vector bundle valued cohomology groups. Our theory works for differentiable, holomorphic and algebraic stacks.