Atiyah Class Research Articles

On jets, extensions and characteristic classes II

In this paper we define the generalized Atiyah classes $c_{\mathcal {J}}(\mathcal {E})$ and $c_{\mathcal {O}_X}(\mathcal {E})$ of a quasi-coherent sheaf $\mathcal {E}$ with respect to a pair $(\mathcal {I},d)$, where $\mathcal {I}$ is a left and right $\mathcal {O}_X$-module and $d$ a derivation. We relate this class to the structure of left and right modules on the first order jet bundle $\mathcal {J}^1_{\mathcal {I}}(\mathcal {E})$. In the main result of the paper we show $c_{\mathcal {O}_X}(\mathcal {E})=0$ if and only if there is an isomorphism $\mathcal {J}^1_{\mathcal {I}}(\mathcal {E})^{left} \cong \mathcal {J}^1_{\mathcal {I}}(\mathcal {E})^{right}$ as $\mathcal {O}_X$-modules. We also give explicit examples where $c_{\mathcal {O}_X}(\mathcal {E})\neq 0$ using jet bundles of line bundles on the projective line. Hence the classes $c_{\mathcal {J}}(\mathcal {E})$ and $c_{\mathcal {O}_X}(\mathcal {E})$ are nontrivial. The classes we introduce generalize the classical Atiyah class.

Symplectic structures on moduli spaces of framed sheaves on surfaces

We provide generalizations of the notions of Atiyah class and Kodaira-Spencer map to the case of framed sheaves. Moreover, we construct closed two-forms on the moduli spaces of framed sheaves on surfaces. As an application, we define a symplectic structure on the moduli spaces of framed sheaves on some birationally ruled surfaces.

The Atiyah class and complex structure stabilization in heterotic Calabi-Yau compactifications

Holomorphic gauge fields in N=1 supersymmetric heterotic compactifications can constrain the complex structure moduli of a Calabi-Yau manifold. In this paper, the tools necessary to use holomorphic bundles as a mechanism for moduli stabilization are systematically developed. We review the requisite deformation theory -- including the Atiyah class, which determines the deformations of the complex structure for which the gauge bundle becomes non-holomorphic and, hence, non-supersymmetric. In addition, two equivalent approaches to this mechanism of moduli stabilization are presented. The first is an efficient computational algorithm for determining the supersymmetric moduli space, while the second is an F-term potential in the four-dimensional theory associated with vector bundle holomorphy. These three methods are proven to be rigorously equivalent. We present explicit examples in which large numbers of complex structure moduli are stabilized. Finally, higher-order corrections to the moduli space are discussed.

Hochschild cohomology and Atiyah classes

In this paper we prove that on a smooth algebraic variety the HKR-morphism twisted by the square root of the Todd genus gives an isomorphism between the sheaf of poly-vector fields and the sheaf of poly-differential operators, both considered as derived Gerstenhaber algebras. In particular we obtain an isomorphism between Hochschild cohomology and the cohomology of poly-vector fields which is compatible with the Lie bracket and the cupproduct. The latter compatibility is an unpublished result by Kontsevich. Our proof is set in the framework of Lie algebroids and so applies without modification in much more general settings as well.

On jets, extensions and characteristic classes I

In this paper, we give general de nitions of non-commutative jets in the local and global situation using square zero extensions and derivations. We study the functors Exank(A; I), where A is any k-algebra, and I is any left and right A-module and use this to construct ane non-commutative jets. We also study the Kodaira-Spencer class KS(L) and relate it to the Atiyah class.

Atiyah Classes and Closed Forms on Moduli Spaces of Sheaves

Let X be a smooth n -dimensional projective variety, and let Y be a moduli space of stable sheaves on X . By using the local Atiyah class of a universal family of sheaves on Y , which is well defined even when such a universal family does not exist, we are able to construct natural maps f: H^i(X, Ω_X^j) → H^{k+i-n} (Y, Ω_Y^{k+j-n}), for any i, j = 1,…,n and any k ≥ \max\{n-i, n-j\} . In particular, for k = n-i , the map f associates a closed differential form of degree j-i on the moduli space Y to any element of H^i(X,Ω_X^j) . This method provides a natural way to construct closed differential forms on moduli spaces of sheaves. We remark that no smoothness hypothesis is made on the moduli space Y . As an application, we describe the construction of closed differential forms on the Hilbert schemes of points of X .

Symplectic structures on moduli spaces of sheaves via the Atiyah class

It is proven that the composition of the Yoneda coupling with the semiregularity map is a closed 2-form on moduli spaces of sheaves. Two examples are given when this 2-form is symplectic. Both of them are moduli spaces of torsion sheaves on the cubic 4-fold Y . The first example is the Fano scheme of lines in Y . Beauville and Donagi showed that it is symplectic but did not construct an explicit symplectic form on it. We prove that our construction provides a symplectic form. The other example is the moduli space of torsion sheaves which are supported on the hyperplane sections H ∩ Y of Y and are cokernels of the Pfaffian representations of H ∩ Y .

The Kontsevich Weight of a Wheel with Spokes Pointing Outward

This is a companion note to “Hochschild cohomology and Atiyah classes” by Damien Calaque and the author. Using elementary methods we compute the Kontsevich weight of a wheel with spokes pointing outward. The result is in terms of modified Bernoulli numbers. The same result had been obtained earlier by Torossian (unpublished) and also recently by Thomas Willwacher using more advanced methods.

The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem

We develop a formalism involving Atiyah classes of sheaves on a smooth manifold, Hochschild chain and cochain complexes. As an application we prove a version of the Riemann–Roch theorem.

THE BIG CHERN CLASSES AND THE CHERN CHARACTER

Let X be a smooth scheme over a field of characteristic 0. The Atiyah class of the tangent bundle TXof X equips TX[-1] with the structure of a Lie algebra object in the derived category D+(X) of bounded below complexes of [Formula: see text] modules with coherent cohomology [6]. We lift this structure to that of a Lie algebra object [Formula: see text] in the category of bounded below complexes of [Formula: see text] modules in Theorem 2. The "almost free" Lie algebra [Formula: see text] is equipped with Hochschild coboundary. There is a symmetrization map [Formula: see text] where [Formula: see text] is the complex of polydifferential operators with Hochschild coboundary. We prove a theorem (Theorem 1) that measures how I fails to commute with multiplication. Further, we show that [Formula: see text] is the universal enveloping algebra of [Formula: see text] in D+(X). This is used to interpret the Chern character of a vector bundle E on X as the "character of a representation" (Theorem 4). Theorems 4 and 1 are then exploited to give a formula for the big Chern classes in terms of the components of the Chern character.

The global decomposition theorem for Hochschild (co-)homology of singular spaces via the Atiyah–Chern character

We generalize the decomposition theorem of Hochschild, Kostant and Rosenberg for Hochschild (co-)homology to arbitrary morphisms between complex spaces or schemes over a field of characteristic zero. To be precise, we show that for each such morphism X → Y , the Hochschild complex H X / Y , as introduced in [R.-O. Buchweitz, H. Flenner, Global Hochschild (co-)homology of singular spaces, Adv. Math. (2007), doi: 10.1016/j.aim.2007.06.012], decomposes naturally in the derived category D ( X ) into ⊕ p ⩾ 0 S p ( L X / Y [ 1 ] ) , the direct sum of the derived symmetric powers of the shifted cotangent complex, a result due to Quillen in the affine case. Even in the affine case, our proof is new and provides further information. It shows that the decomposition is given explicitly and naturally by the universal Atiyah–Chern character, the exponential of the universal Atiyah class. We further use the decomposition theorem to show that the semiregularity map for perfect complexes factors through Hochschild homology and, in turn, factors the Atiyah–Hochschild character through the characteristic homomorphism from Hochschild cohomology to the graded centre of the derived category.

OPERADS AND JET MODULES

Let A be an algebra over an operad in a cocomplete closed symmetric monoidal category. We study the category of A-modules. We define certain symmetric product functors of such modules generalizing the tensor product of modules over commutative algebras, which we use to define the notion of a jet module. This in turn generalizes the notion of a jet module over a module over a classical commutative algebra. We are able to define Atiyah classes (i.e., obstructions to the existence of connections) in this generalized context. We use certain model structures on the category of A-modules to study the properties of these Atiyah classes. The purpose of the paper is not to present any really deep theorem. It is more about the right concepts when dealing with modules over an algebra that is defined over an arbitrary operad, i.e., the aim is to show how to generalize various classical constructions, including modules of jets, the Atiyah class and the curvature, to the operadic context. For convenience of the reader and for the purpose of defining the notations, the basic definitions of the theory of operads and model categories are included.

A Semiregularity Map for Modules and Applications to Deformations

We construct a general semiregularity map for algebraic cycles as asked for by S. Bloch in 1972. The existence of such a semiregularity map has well known consequences for the structure of the Hilbert scheme and for the variational Hodge conjecture. Aside from generalizing and extending considerably previously known results in this direction, we give new applications to deformations of modules that encompass, for example, results of Artamkin and Mukai. The formation of the semiregularity map here involves powers of the cotangent complex, Atiyah classes, and trace maps, and is defined not only for subspaces of manifolds but for perfect complexes on arbitrary complex spaces. It generalizes in particular Illusie's treatment of the Chern character to the analytic context and specializes to Bloch's earlier description of the semiregularity map for locally complete intersections as well as to the infinitesimal Abel–Jacobi map for submanifolds.

Rozansky–Witten invariants via Atiyah classes

Recently, L. Rozansky and E. Witten associated to any hyper-Kähler manifold X a system of ’weights‘ (numbers, one for each trivalent graph) and used them to construct invariants of topological 3-manifolds. We give a simple cohomological definition of these weights in terms of the Atiyah class of X (the obstruction to the existence of a holomorphic connection). We show that the analogy between the tensor of curvature of a hyper-Kähler metric and the tensor of structure constants of a Lie algebra observed by Rozansky and Witten, holds in fact for any complex manifold, if we work at the level of cohomology and for any Kähler manifold, if we work at the level of Dolbeault cochains. As an outcome of our considerations, we give a formula for Rozansky–Witten classes using any Kähler metric on a holomorphic symplectic manifold.

Ehresmann connection for the canonical foliation on a manifold over a local algebra

For a canonical foliation on a manifoldMA over a local algebra, theA-affine horizontal distribution complementary to the leaves, similar to the horizontal distribution of a higher order connection on the fiber bundle ofA-jets, is defined. In the case of a complete manifoldMA, theA-affine horizontal distribution is proved to be an Ehresmann connection in the sense of Blumental-Hebda. It is shown that theA-affine horizontal distribution onMA exists if and only if the Atiyah class of a certain foliated principal bundle vanishes.

Super Atiyah Classes and Obstructions to Splitting of Supermoduli Space

The first obstruction to splitting a supermanifold S is one of the three components of its super Atiyah class, the two other components being the ordinary Atiyah classes on the reduced space M of the even and odd tangent bundles of S. We evaluate these classes explicitly for the moduli space of super Riemann surfaces (super moduli space) and its reduced space, the moduli space of spin curves. These classes are interpreted in terms of certain extensions arising from line bundles on the square of the varying (super) Riemann surface. These results are used to give a new proof of the non- projectedness of Mg,1, the moduli space of super Riemann surfaces with one puncture.

Moduli spaces of sheaves on K3 surfaces and symplectic stacks

We view the moduli space of semistable sheaves on a K3 surface as a global quotient stack, and compute its cotangent complex in terms of the universal sheaf on the Quot scheme. Relevant facts on the classical and reduced Atiyah classes are reviewed. We also define the notion of a symplectic stack, and show that it includes all moduli stacks of semistable sheaves on K3 surfaces.