28 research outputs found
Derived Quot schemes
Realizing a part of the Derived Deformation Theory program, we construct a
"derived" analog of the Grothendieck's Quot scheme parametrizing subsheaves in
a given coherent sheaf F on a smooth projective variety X. This analog is a
differential graded manifold RQuot_h(F) (so it is always smooth in an
appropriate sense) whose tangent space at a point represented by a subsheaf K
in F, is a cochain complex quasiisomorphic to RHom(K, F/K).Comment: 46 pages, AMS-TeX. Revised version, to appear in Ann. Sci. EN
Linear Koszul Duality II - Coherent sheaves on perfect sheaves
In this paper we continue the study (initiated in a previous article) of
linear Koszul duality, a geometric version of the standard duality between
modules over symmetric and exterior algebras. We construct this duality in a
very general setting, and prove its compatibility with morphisms of vector
bundles and base change.Comment: Final version, to appear in JLMS. The numbering differs from the
published version, and is the one used in our papers [MR2] and [MR3] from the
bibliograph
Deformation theory of objects in homotopy and derived categories I: general theory
This is the first paper in a series. We develop a general deformation theory
of objects in homotopy and derived categories of DG categories. Namely, for a
DG module over a DG category we define four deformation functors \Def
^{\h}(E), \coDef ^{\h}(E), \Def (E), \coDef (E). The first two functors
describe the deformations (and co-deformations) of in the homotopy
category, and the last two - in the derived category. We study their properties
and relations. These functors are defined on the category of artinian (not
necessarily commutative) DG algebras.Comment: Alexander Efimov is a new co-author of this paper. Besides some minor
changes, Proposition 7.1 and Theorem 8.1 were correcte
Del Pezzo surfaces with 1/3(1,1) points
We classify del Pezzo surfaces with 1/3(1,1) points in 29 qG-deformation
families grouped into six unprojection cascades (this overlaps with work of
Fujita and Yasutake), we tabulate their biregular invariants, we give good
model constructions for surfaces in all families as degeneracy loci in rep
quotient varieties and we prove that precisely 26 families admit
qG-degenerations to toric surfaces. This work is part of a program to study
mirror symmetry for orbifold del Pezzo surfaces.Comment: 42 pages. v2: model construction added of last remaining surface,
minor corrections, minor changes to presentation, references adde
The stringy instanton partition function
We perform an exact computation of the gauged linear sigma model associated to a D1-D5 brane system on a resolved A 1 singularity. This is accomplished via supersymmetric localization on the blown-up two-sphere. We show that in the blow-down limit the partition function reduces to the Nekrasov partition function evaluating the equivariant volume of the instanton moduli space. For finite radius we obtain a tower of world-sheet instanton corrections, that we identify with the equivariant Gromov-Witten invariants of the ADHM moduli space. We show that these corrections can be encoded in a deformation of the Seiberg-Witten prepotential. From the mathematical viewpoint, the D1-D5 system under study displays a twofold nature: the D1-branes viewpoint captures the equivariant quantum cohomology of the ADHM instanton moduli space in the Givental formalism, and the D5-branes viewpoint is related to higher rank equivariant Donaldson-Thomas invariants