1,721,150 research outputs found
Lie Methods in Deformation Theory
Full text linkDeformation theory is an important subject in algebra and algebraic gemetry, whose origin dates back to Kodaira, Spencer, Kuranishi, Gerstenhaber and Grothendieck. In the last 30 year a new approach, based on some ideas from rational homotopy theory, has permitted non only to solve some long standing open problems, but also to clarify the general theory and to relate apperently different features. This approach works over a field of characteristic 0 and the central role is played by the notions of differential graded Lie algebra and L-infinity algebra
Deformations of algebraic schemes via Reedy–Palamodov cofibrant resolutions
Full text linkLet X be a Noetherian separated and finite dimensional scheme over a field K of characteristic zero. The goal of this paper is to study deformations of X over a differential graded local Artin K-algebra by using local Tate–Quillen resolutions, i.e., the algebraic analogous of the Palamodov's resolvent of a complex space. The above goal is achieved by describing the DG-Lie algebra controlling deformation theory of a diagram of differential graded commutative algebras, indexed by a direct Reedy category
Endomorphisms of Koszul complexes: formality and application to deformation theory
Full text linkWe study the differential graded Lie algebra of endomorphisms of the Koszul resolution of a regular sequence on a unitary commutative K-algebra R and we prove that it is homotopy abelian over K but not over R (except trivial cases). We apply this result to prove an annihilation theorem for obstructions of (derived) deformations of locally complete intersection ideal sheaves on projective schemes
Formal deformation theory in left-proper model categories
No full textWe develop the notion of deformation of a morphism in a left-proper model category. As an application we provide a geomet-ric/homotopic description of deformations of commutative (non-positively) graded differential algebras over a local DG-Artin ring
Formal deformation theory in left-proper model categories
Full text linkWe develop the notion of deformation of a morphism in a left-proper model category. As an application we provide a geomet-ric/homotopic description of deformations of commutative (non-positively) graded differential algebras over a local DG-Artin ring
Homotopy abelianity of the DG-Lie algebra controlling deformations of pairs (variety with trivial canonical bundle, line bundle)
Full text linkWe investigate the deformations of pairs (X,L), where L is a line bundle on a smooth projective variety X, defined over an algebraically closed field of characteristic 0. In particular, we prove that the DG-Lie algebra controlling the deformations of the pair (X,L) is homotopy abelian whenever X has trivial canonical bundle, and so these deformations are unobstructed
On the adjoint map of homotopy abelian Dg-lie algebras
Full text linkWe prove that a differential graded Lie algebra is homotopy abelian if its adjoint map into its cochain complex of derivations is trivial in cohomology. The converse is true for cofibrant algebras and false in general
DEFORMATIONS OF POLYSTABLE SHEAVES ON SURFACES: QUADRATICITY IMPLIES FORMALITY
Full text linkWe study relations between the quadraticity of the Kuranishi family of a coherent sheaf on a complex projective scheme and the formality of the DG-Lie algebra of its derived endomorphisms. In particular, we prove that for a polystable coherent sheaf of a smooth complex projective surface the DG-Lie algebra of derived endomorphisms is forif i
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