An index theorem for anti-self-dual orbifold-cone metrics

Recently, Atiyah and LeBrun proved versions of the Gauss–Bonnet and Hirzebruch signature theorems for metrics with edge-cone singularities in dimension four, which they applied to obtain an inequality of Hitchin–Thorpe type for Einstein edge-cone metrics. Interestingly, many natural examples of edge-cone metrics in dimension four are anti-self-dual (or self-dual depending upon choice of orientation). On such a space there is an important elliptic complex called the anti-self-dual deformation complex, whose index gives crucial information about the local structure of the moduli space of anti-self-dual metrics. In this paper, we compute the index of this complex in the orbifold case, and give several applications.     

Toroidal orbifolds, gerbes and group cohomology

In this paper we compute the integral cohomology of certain semi-direct products of the form , arising from a linear action on the -torus, where is a finite group. The main application is the complete calculation of torsion gerbes for six-dimensional examples arising in string theory.

     

Formality theorem for gerbes

The main result of the present paper is an analogue of Kontsevich formality theorem in the context of the deformation theory of gerbes. We construct an _L∞$L_{\infty }$ deformation of the Schouten algebra of multi-vectors which controls the deformation theory of a gerbe.     

Bivariant Chern character and longitudinal index

In this paper we consider a family of Dirac-type operators on fibration P→B equivariant with respect to an action of an étale groupoid. Such a family defines an element in the bivariant K theory. We compute the action of the bivariant Chern character of this element on the image of Connes' map Φ in the cyclic cohomology. A particular case of this result is Connes' index theorem for étale groupoids [A. Connes, Noncommutative Geometry, Academic Press, 1994] in the case of fibrations.     

Skew Derivations and Deformations of a Family of Group Crossed Products

We obtain deformations of a crossed product of a polynomial algebra with a group, under some conditions, from universal deformation formulas. We show that the resulting deformations are nontrivial by a comparison with Hochschild cohomology. The universal deformation formulas arise from actions of Hopf algebras generated by automorphisms and skew derivations, and are universal in the sense that they apply to deform all algebras with such Hopf algebra actions.     

Hochschild cohomologies and deformations of the pointwise superproduct

We consider the associative superalgebra of smooth compactly supported functions on ℝ ⁿ taking values in a Grassmann algebra. We find the lower Hochschild cohomologies and also the deformations of this superalgebra under certain additional conditions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 323–346, March, 2009.     

Twisted longitudinal index theorem for foliations and wrong way functoriality

For a Lie groupoid G with a twisting σ (a PU(H)-principal bundle over G), we use the (geometric) deformation quantization techniques supplied by Connes tangent groupoids to define an analytic index morphism in twisted K-theory. In the case the twisting is trivial we recover the analytic index morphism of the groupoid.For a smooth foliated manifold with twistings on the holonomy groupoid we prove the twisted analog of the Connes–Skandalis longitudinal index theorem. When the foliation is given by fibers of a fibration, our index coincides with the one recently introduced by Mathai, Melrose, and Singer.We construct the pushforward map in twisted K-theory associated to any smooth (generalized) map f:W→M/F and a twisting σ on the holonomy groupoid M/F, next we use the longitudinal index theorem to prove the functoriality of this construction. We generalize in this way the wrong way functoriality results of Connes and Skandalis when the twisting is trivial and of Carey and Wang for manifolds.     

An algebraic version of a theorem of Kurihara

Let E/Q be an elliptic curve and let p be an odd supersingular prime for E. In this article, we study the simplest case of Iwasawa theory for elliptic curves, namely when E(Q) is finite, ш(E/Q) has no p-torsion and the Tamagawa factors for E are all prime to p. Under these hypotheses, we prove that E(Q_n) is finite and make precise statements about the size and structure of the p-power part of ш(E/Q_n). Here Q_n is the n-th step in the cyclotomic Z_p-extension of Q.     

On the Integrability of Deformation Quantized Toda Lattice

In the present paper, we are concerned with deformation quantization of irregular Poisson structures. Translating Toda lattice equation into Hamiltonian formalism equation, we also study the global integrability of deformation quantized Toda lattice.     

A Brody theorem for orbifolds

We study the Kobayashi pseudodistance for orbifolds, proving an orbifold version of Brody’s theorem and classifying which one-dimensional orbifolds are hyperbolic.     

An index theorem for Toeplitz operators on odd-dimensional manifolds with boundary

We establish an index theorem for Toeplitz operators on odd-dimensional spin manifolds with boundary. It may be thought of as an odd-dimensional analogue of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. In particular, there occurs naturally an invariant of η type associated to K¹ representatives on even-dimensional manifolds, which should be of independent interests. For example, it gives an intrinsic interpretation of the so called Wess-Zumino term in the WZW theory in physics.     

Cohomology of exact categories and (non-)additive sheaves

We use (non-)additive sheaves to introduce an (absolute) notion of Hochschild cohomology for exact categories as Ext's in a suitable bisheaf category. We compare our approach to various definitions present in the literature.     

An integral structure in quantum cohomology and mirror symmetry for toric orbifolds

We introduce an integral structure in orbifold quantum cohomology associated to the K-group and the -class. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the Landau-Ginzburg model under mirror symmetry. By assuming the existence of an integral structure, we give a natural explanation for the specialization to a root of unity in Y. Ruan's crepant resolution conjecture [Yongbin Ruan, The cohomology ring of crepant resolutions of orbifolds, in: Contemp. Math., vol. 403, Amer. Math. Soc., Providence, RI, 2006, pp. 117-126].     

Support Varieties for Selfinjective Algebras

Support varieties for any finite dimensional algebra over a field were introduced in (Proc. London Math. Soc. 88 (3) (2004) 705–732) using graded subalgebras of the Hochschild cohomology ring. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, the variety of periodic modules are lines and for symmetric algebras a generalization of Webbs theorem is true. As a corollary of a more general result we show that Webbs theorem generalizes to finite dimensional cocommutative Hopf algebras.Received November 2003Mathematics Subject Classifications (2000) Primary: 16E40, 16G10, 16P10, 16P20; Secondary: 16G70.     

On Maximal Diagonalizable Lie Subalgebras of the First Hochschild Cohomology

Let A be a basic connected finite dimensional algebra over an algebraically closed field, with ordinary quiver without oriented cycles. Given a presentation of A by quiver and admissible relations, Assem and de la Peña have constructed an embedding of the space of additive characters of the fundamental group of the presentation into the first Hochschild cohomology group of A. We compare the embeddings given by the different presentations of A. In some situations, we characterise the images of these embeddings in terms of (maximal) diagonalizable subalgebras of the first Hochschild cohomology group (endowed with its Lie algebra structure).     

An algebraic certificate for Budan’s theorem

The existing algorithms to construct the real closure of an ordered field involve very high complexities. These algorithms are based on Sturm’s theorem which we suspect to be one reason for the complexities since all known proofs of Sturm’s theorem use Rolle’s theorem which is problematic in a constructive context.Therefore we propose to replace the use of Sturm’s theorem by Budan’s theorem. In this paper we present as a first step in this direction an algebraic certificate for Budan’s theorem. An algebraic certificate is a certain kind of proof of a statement. In particular, it is an algorithm which produces, from an arbitrary data in the premise of the statement, explicit (in)equalities which express the conclusion.     

An index theorem for open manifolds whose ends are warped products

We give an index formula for the Dirac operator acting onL2-sections of the spinor bundle of an open spin manifold with a co-compact cylindrical end over which the metric is a warped product.     

On Yau’s theorem for effective orbifolds

In 1978 Yau (Yau, 1978) confirmed a conjecture due to Calabi (1954) stating the existence of Kähler metrics with prescribed Ricci forms on compact Kähler manifolds. A version of this statement for effective orbifolds can be found in the literature (Joyce, 2000; Boyer and Galicki, 2008; Demailly and Kollár, 2001). In this expository article, we provide details for a proof of this orbifold version of the statement by adapting Yau’s original continuity method to the setting of effective orbifolds in order to solve a Monge–Ampère equation. We then outline how to obtain Kähler–Einstein metrics on orbifolds with negative first Chern class by solving a slightly different Monge–Ampère equation. We conclude by listing some explicit examples of Calabi–Yau orbifolds, which consequently admit Ricci flat metrics by Yau’s theorem for effective orbifolds.