On the deformation theory of pair (X, E)
Si Li
TL;DR
This paper develops an absolute deformation-obstruction theory for pairs (X, E), where X is a smooth projective scheme and E is a perfect complex, extending previous relative theories and fitting into an exact triangle.
Contribution
It constructs the absolute deformation-obstruction theory for pairs (X, E) using relative theories, linking obstructions of E, (X,E), and X in a unified framework.
Findings
Obstruction theories for E, (X,E), and X form an exact triangle.
Provides a new framework for deformation theory of pairs in algebraic geometry.
Extends previous relative obstruction theories to absolute cases.
Abstract
Huybrechts and Thomas recently constructed relative obstruction theory of objects of the derived category of coherent sheaves over smooth projective family. In this paper, we use this construction to obtain the absolute deformation-obstruction theory of the pair (X, E), with X smooth projective scheme and E perfect complex, and show that the obstruction theories for E, (X,E), and X fit into exact triangle as derived objects on the moduli space.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology