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22
Configurations in abelian categories. IV. Invariants and changing stability conditions
, 2007
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Derived Quot schemes
"... (0.1) A typical moduli problem in geometry is to construct a “space ” H parametrizing, up to isomorphism, objects of some given category Z (e.g., manifolds, vector bundles etc.). This can be seen as a kind of a nonAbelian cohomology problem and the construction usually consists of two steps of oppo ..."
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Cited by 42 (1 self)
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(0.1) A typical moduli problem in geometry is to construct a “space ” H parametrizing, up to isomorphism, objects of some given category Z (e.g., manifolds, vector bundles etc.). This can be seen as a kind of a nonAbelian cohomology problem and the construction usually consists of two steps of opposite nature, namely applying
Moduli of objects in dgcategories
, 2006
"... To any dgcategory T (over some base ring k), we define a D −stack MT in the sense of [HAGII], classifying certain T opdgmodules. When T is saturated, MT classifies compact objects in the triangulated category [T] associated to T. The main result of this work states that under certain finiteness ..."
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Cited by 35 (4 self)
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To any dgcategory T (over some base ring k), we define a D −stack MT in the sense of [HAGII], classifying certain T opdgmodules. When T is saturated, MT classifies compact objects in the triangulated category [T] associated to T. The main result of this work states that under certain finiteness conditions on T (e.g. if it is saturated) the D −stack MT is locally geometric (i.e. union of open and geometric substacks). As a consequence we prove the algebraicity of the group of autoequivalences of saturated dgcategories. We also obtain the existence of reasonable moduli for perfect complexes on a smooth and proper scheme, as
Lectures on noncommutative geometry
, 2005
"... These Lectures are based on a course on noncommutative geometry given by the author in 2003. The lectures contain some standard material, such as Poisson and Gerstenhaber algebras, deformations, Hochschild cohomology, Serre functors, etc. We also discuss many less known as well as some new results ..."
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Cited by 18 (2 self)
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These Lectures are based on a course on noncommutative geometry given by the author in 2003. The lectures contain some standard material, such as Poisson and Gerstenhaber algebras, deformations, Hochschild cohomology, Serre functors, etc. We also discuss many less known as well as some new results such as noncommutative ChernWeil theory, noncommutative symplectic geometry, noncommutative differential forms and doubletangent bundles.
Deformation theory of objects in homotopy and derived categories I: General theory
, 2008
"... 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 E over a DG category we define four deformation functors Def h (E) , coDef h (E) , Def(E) , coDef(E). The first two functors describe ..."
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Cited by 11 (6 self)
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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 E 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 codeformations) of E in the homotopy
Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes
"... Abstract A quasismooth derived enhancement of a DeligneMumford stack X naturally endows X with a functorial perfect obstruction theory in the sense of BehrendFantechi. We apply this result to moduli of maps and perfect complexes on a smooth complex projective variety. For moduli of maps, for X = ..."
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Cited by 11 (1 self)
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Abstract A quasismooth derived enhancement of a DeligneMumford stack X naturally endows X with a functorial perfect obstruction theory in the sense of BehrendFantechi. We apply this result to moduli of maps and perfect complexes on a smooth complex projective variety. For moduli of maps, for X = S an algebraic K3surface, g ∈ N, and β = 0 in H 2 (S, Z) a curve class, we construct a derived stack RM red g,n (S; β) whose truncation is the usual stack M g,n (S; β) of pointed stable maps from curves of genus g to S hitting the class β, and such that the inclusion M g (S; β) → RM red g (S; β) induces on M g (S; β) a perfect obstruction theory whose tangent and obstruction spaces coincide with the corresponding reduced spaces of OkounkovMaulikPandharipandeThomas [OP2, MP, MPT]. The approach we present here uses derived algebraic geometry and yields not only a full rigorous proof of the existence of a reduced obstruction theory not relying on any result on semiregularity maps but also a new global geometric interpretation. We give two further applications to moduli of complexes. For a K3surface S we show that the stack of simple perfect complexes on S is smooth. This result was proved with different methods by Inaba ([In]) for the corresponding coarse moduli space. Finally, we construct a map from the derived stack of stable embeddings of curves (into a smooth complex projective variety X) to the derived stack of simple perfect complexes on X with vanishing negative Ext's, and show how this map induces a morphism of the corresponding obstruction theories when X is a CalabiYau threefold. An important ingredient of our construction is a perfect determinant map from the derived stack of perfect complexes to the derived stack of line bundles whose tangent morphism is given by Illusie's trace map for perfect complexes.
RiemannRoch theorems and elliptic genus for virtually smooth Schemes
"... Abstract. For a proper scheme X with a fixed 1perfect obstruction theory E • , we define virtual versions of holomorphic Euler characteristic, χ−ygenus, and elliptic genus; they are deformation invariant, and extend the usual definition in the smooth case. We prove virtual versions of the Grothend ..."
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Abstract. For a proper scheme X with a fixed 1perfect obstruction theory E • , we define virtual versions of holomorphic Euler characteristic, χ−ygenus, and elliptic genus; they are deformation invariant, and extend the usual definition in the smooth case. We prove virtual versions of the GrothendieckRiemannRoch and HirzebruchRiemannRoch theorems. We show that the virtual χ−ygenus is a polynomial, and use this to define a virtual topological Euler characteristic. We prove that the virtual elliptic genus satisfies a Jacobi modularity property; we state and prove a localization theorem in the toric equivariant case. We show how some of our results apply to moduli spaces of stable
Cohomology operations and the Deligne conjecture
 Czechoslovak Math. J
"... Abstract. The aim of this note, which raises more questions than it answers, is to study natural operations acting on the cohomology of various types of algebras. It contains a lot of very surprising partial results and examples. ..."
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Cited by 8 (5 self)
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Abstract. The aim of this note, which raises more questions than it answers, is to study natural operations acting on the cohomology of various types of algebras. It contains a lot of very surprising partial results and examples.
Intrinsic brackets and the L∞deformation theory of bialgebras
, 2004
"... Abstract. We show that there exists a Lie a bracket on the cohomology of any type of (bi)algebras over an operad or a prop, induced by an L∞structure on the defining cochain complex, such that the associated L∞master equation captures deformations. This in particular implies the existence of a Lie ..."
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Cited by 8 (4 self)
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Abstract. We show that there exists a Lie a bracket on the cohomology of any type of (bi)algebras over an operad or a prop, induced by an L∞structure on the defining cochain complex, such that the associated L∞master equation captures deformations. This in particular implies the existence of a Lie bracket on the GerstenhaberSchack cohomology [7] of a bialgebra that extends the classical intrinsic bracket [6] on the Hochschild cohomology, giving an affirmative answer to an old question about the existence of such a bracket. We also explain how the results of [25] provide explicit formulas for this bracket. Conventions. We assume a certain familiarity with operads and props, see [21, 17, 22, 23]. The reader who wishes only to know how the intrinsic bracket on the GerstenhaberSchack cohomology looks might proceed directly to Section 6 which is almost independent on the rest of the paper and contains explicit calculations. We also assume some knowledge of the concept of strongly homotopy Lie algebras (also called L∞algebras), see [11, 14]. We will make no difference between an operad P and the prop P generated by this operad. This means that for us operads are particular cases of props. As usual, bialgebra will mean a Hopf algebra without (co)unit and antipode. To distinguish these bialgebras from other types of “bialgebras ” we will sometimes call them also Assbialgebras. All algebraic objects will be defined over a fixed field k of characteristic zero although, surprisingly, our constructions related to Assbialgebras make sense over the integers. 1. Introduction and