Koszul complexes and spectral sequences associated with Lie algebroids

We study some spectral sequences associated with a locally free $\mathcal O_X$-module $\mathcal A$ which has a Lie algebroid structure. Here $X$ is either a complex manifold or a regular scheme over an algebraically closed field $k$. One spectral sequence can be associated with $\mathcal A$ by choosing a global section $V$ of $\mathcal A$, and considering a Koszul complex with a differential given by inner product by $V$. This spectral sequence is shown to degenerate at the second page by using Deligne's degeneracy criterion. Another spectral sequence we study arises when considering the Atiyah algebroid $\mathcal D_E$ of a holomolorphic vector bundle $E$ on a complex manifold. If $V$ is a differential operator on $E$ with scalar symbol, i.e, a global section of $\mathcal D_E$, we associate with the pair $(E,V)$ a twisted Koszul complex. The first spectral sequence associated with this complex is known to degenerate at the first page in the untwisted ($E=0$) caseComment: 8 pages. To appear in S\~ao Paulo Journal of Mathematical Science

Semistable Higgs bundles on Calabi-Yau manifolds

We provide a partial classification of semistable Higgs bundles over a simply connected Calabi-Yau manifolds. Applications to a conjecture about a special class of semistable Higgs bundles are given. In particular, the conjecture is proved for K3 and Enriques surfaces, and some related classes of surfaces

Picard group of hypersurfaces in toric 3-folds

We show that the usual sufficient criterion for a generic hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to hypersurfaces in complete simplicial toric varieties. This sufficient condition is always satisfied by generic K3 surfaces embedded in Fano toric 3-folds.Comment: 14 pages. v2: some typos corrected. v3: Slightly changed title. Final version to appear in Int. J. Math., incorporates many (mainly expository) changes suggested by the refere

Lie algebroid cohomology and Lie algebroid extensions

We consider the extension problem for Lie algebroids over schemes over a field. Given a locally free Lie algebroid Q over a scheme X, and a coherent sheaf of Lie OX-algebras L, we determine the obstruction to the existence of extensions 0\u2192L\u2192E\u2192Q\u21920, and classify the extensions in terms of a suitable Lie algebroid hypercohomology group. In the preliminary sections we study free Lie algebroids and recall some basic facts about Lie algebroid hypercohomology

On the irreducibility of some quiver varieties

We prove that certain quiver varieties are irreducible and therefore are isomor-phic to Hilbert schemes of points of the total spaces of the bundles OP1( 12n) for n 65 1

Parafermionic Liouville field theory and instantons on ALE spaces

In this paper we study the correspondence between the $\hat{\textrm{su}}(n)_{k}\oplus \hat{\textrm{su}}(n)_{p}/\hat{\textrm{su}}(n)_{k+p}$ coset conformal field theories and $\mathcal{N}=2$ SU(n) gauge theories on $\mathbb{R}^{4}/\mathbb{Z}_{p}$. Namely we check the correspondence between the SU(2) Nekrasov partition function on $\mathbb{R}^{4}/\mathbb{Z}_{4}$ and the conformal blocks of the $S_{3}$ parafermion algebra (in $S$ and $D$ modules). We find that they are equal up to the U(1)-factor as it was in all cases of AGT-like relations. Studying the structure of the instanton partition function on $\mathbb{R}^4/\mathbb{Z}_p$ we also find some evidence that this correspondence with arbitrary $p$ takes place up to the U(1)-factor.Comment: 21 pages, 6 figures, misprints corrected, references added, version to appear in JHE

Cohomology of skew-holomorphic Lie algebroids

We introduce the notion of skew-holomorphic Lie algebroid on a complex manifold, and explore some cohomologies theories that one can associate to it. Examples are given in terms of holomorphic Poisson structures of various sorts.Comment: 16 pages. v2: Final version to be published in Theor. Math. Phys. (incorporates only very minor changes

Instantons on ALE spaces and Super Liouville Conformal Field Theories

We provide evidence that the conformal blocks of N=1 super Liouville conformal field theory are described in terms of the SU(2) Nekrasov partition function on the ALE space O_{P^1}(-2).Comment: 10 page

Gauge fixing and equivariant cohomology

The supersymmetric model developed by Witten to study the equivariant cohomology of a manifold with an isometric circle action is derived from the BRST quantization of a simple classical model. The gauge-fixing process is carefully analysed, and demonstrates that different choices of gauge-fixing fermion can lead to different quantum theories.Comment: 18 pages LaTe

A_{N-1} conformal Toda field theory correlation functions from conformal N=2 SU(N) quiver gauge theories

We propose a relation between correlation functions in the 2d A_{N-1} conformal Toda theories and the Nekrasov instanton partition functions in certain conformal N=2 SU(N) 4d quiver gauge theories. Our proposal generalises the recently uncovered relation between the Liouville theory and SU(2) quivers. New features appear in the analysis that have no counterparts in the Liouville case.Comment: 23 pages. v2: some typos correcte