We study the Ext modules in the category of left modules over a twisted algebra of a finite quiver over a ringed space $(X,mathcal O_X)$, allowing for the presence of relations. We introduce a spectral sequence which relates the Ext modules in that c
ategory with the Ext modules in the category of $mathcal O_X$-modules. Contrary to what happens in the absence of relations, this spectral sequence in general does not degenerate at the second page. We also consider local Ext sheaves. Under suitable hypotheses, the Ext modules are represented as hypercohomology groups
We consider the extension problem for Lie algebroids over schemes over a field. Given a locally free Lie algebroid Q over a scheme (X,O), and a sheaf of finitely generated Lie O-algebras L, we determine the obstruction to the existence of extensions
0 --> L --> E --> Q --> 0, 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.
We advocate that a generalized Kronheimer construction of the Kahler quotient crepant resolution $mathcal{M}_zeta longrightarrow mathbb{C}^3/Gamma$ of an orbifold singularity where $Gammasubset mathrm{SU(3)}$ is a finite subgroup naturally defines th
e field content and interaction structure of a superconformal Chern-Simons Gauge Theory. This is supposedly the dual of an M2-brane solution of $D=11$ supergravity with $mathbb{C}timesmathcal{M}_zeta$ as transverse space. We illustrate and discuss many aspects of this of constructions emphasizing that the equation $pmb{p}wedgepmb{p}=0$ which provides the Kahler analogue of the holomorphic sector in the hyperKahler moment map equations canonically defines the structure of a universal superpotential in the CS theory. The kernel of the above equation can be described as the orbit with respect to a quiver Lie group $mathcal{G}_Gamma$ of a locus $L_Gamma subset mathrm{Hom}_Gamma(mathcal{Q}otimes R,R)$ that has also a universal definition. We discuss the relation between the coset manifold $mathcal{G}_Gamma/mathcal{F}_Gamma$, the gauge group $mathcal{F}_Gamma$ being the maximal compact subgroup of the quiver group, the moment map equations and the first Chern classes of the tautological vector bundles that are in a one-to-one correspondence with the nontrivial irreps of $Gamma$. These first Chern classes provide a basis for the cohomology group $H^2(mathcal{M}_zeta)$. We discuss the relation with conjugacy classes of $Gamma$ and provide the explicit construction of several examples emphasizing the role of a generalized McKay correspondence. The case of the ALE manifold resolution of $mathbb{C}^2/Gamma$ singularities is utilized as a comparison term and new formulae related with the complex presentation of Gibbons-Hawking metrics are exhibited.
We show that very general hypersurfaces in odd-dimensional simplicial projective toric varieties verifying a certain combinatorial property satisfy the Hodge conjecture (these include projective spaces). This gives a connection between the Oda conjec
ture and Hodge conjecture. We also give an explicit criterion which depends on the degree for very general hypersurfaces for the combinatorial condition to be verified.
We consider the Noether-Lefschetz problem for surfaces in Q-factorial normal 3-folds with rational singularities. We show the existence of components of the Noether-Lefschetz locus of maximal codimension, and that there are indeed infinitely many of
them. Moreover, we show that their union is dense in the natural topology.
Relying on a notion of numerical effectiveness for Higgs bundles, we show that the category of numerically flat Higgs vector bundles on a smooth projective variety $X$ is a Tannakian category. We introduce the associated group scheme, that we call th
e Higgs fundamental group scheme of $X$, and show that its properties are related to a conjecture about the vanishing of the Chern classes of numerically flat Higgs vector bundles.
In this paper we study Higgs and co-Higgs $G$-bundles on compact Kahler manifolds $X$. Our main results are: (1) If $X$ is Calabi-Yau, and $(E,,theta)$ is a semistable Higgs or co-Higgs $G$-bundle on $X$, then the principal $G$-bundle $E$ is semistab
le. In particular, there is a deformation retract of ${mathcal M}_H(G)$ onto $mathcal M(G)$, where $mathcal M(G)$ is the moduli space of semistable principal $G$-bundles with vanishing rational Chern classes on $X$, and analogously, ${mathcal M}_H(G)$ is the moduli space of semistable principal Higgs $G$-bundles with vanishing rational Chern classes. (2) Calabi-Yau manifolds are characterized as those compact Kahler manifolds whose tangent bundle is semistable for every Kahler class, and have the following property: if $(E,,theta)$ is a semistable Higgs or co-Higgs vector bundle, then $E$ is semistable.
We show that the hypercohomology of the Chevalley-Eilenberg-de Rham complex of a Lie algebroid L over a scheme with coefficients in an L-module can be expressed as a derived functor. We use this fact to study a Hochschild-Serre type spectral sequence attached to an extension of Lie algebroids.
The Noether-Lefschetz theorem asserts that any curve in a very general surface $X$ in $mathbb P^3$ of degree $d geq 4$ is a restriction of a surface in the ambient space, that is, the Picard number of $X$ is $1$. We proved previously that under some
conditions, which replace the condition $d geq 4$, a very general surface in a simplicial toric threefold $mathbb P_Sigma$ (with orbifold singularities) has the same Picard number as $mathbb P_Sigma$. Here we define the Noether-Lefschetz loci of quasi-smooth surfaces in $mathbb P_Sigma$ in a linear system of a Cartier ample divisor with respect to a (-1)-regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether-Lefschetz loci which contain a line, defined as a rational curve that is minimal in a suitable sense.
We study the moduli space $I_{n,r}$ of rank-$2r$ symplectic instanton vector bundles on $mathbb{P}^3$ with $rge2$ and second Chern class $nge r+1, n-requiv 1(mathrm{mod}2)$. We introduce the notion of tame symplectic instantons by excluding a kind of
pathological monads and show that the locus $I^*_{n,r}$ of tame symplectic instantons is irreducible and has the expected dimension, equal to $4n(r+1)-r(2r+1)$. The proof is inherently based on a relation between the spaces $I^*_{n,r}$ and the moduli spaces of t Hooft instantons
