We realize the infinitesimal Abel–Jacobi map as a morphism of formal deformation theories, realized as a morphism in the homotopy category of differential graded Lie algebras. The whole construction is carried out in a general setting, of which the classical Abel–Jacobi map is a special example.
We analyse infinitesimal deformations of pairs $(X,sF)$ with $sF$ a coherent sheaf on a smooth projective variety $X$ over an algebraically closed field of characteristic $0$. We describe a differential graded Lie algebra controlling the deformation problem, and we prove an analog of a Mukai-Artamkin Theorem about the trace map.
We 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.
© Università degli Studi di Roma "La Sapienza" - Piazzale Aldo Moro 5, 00185 Roma
