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Lie description of higher obstructions to deforming submanifolds. (English) Zbl 1174.13021

Let \(\chi:L\rightarrow M\) be a morphism of differential graded Lie algebras. It introduces a functor \(Def_{\chi}: \{\text{local Artinian rings}\} \rightarrow Sets\), which have the same formal properties as \(Def_L\) - a functor studied by Goldman-Millson, Kontsevich and the author. The tangent and the obstruction spaces of \(Def_L\) are respectively the first and second cohomology group of the suspension of the mapping cone of \(\chi\). This construction applies to Hilbert and Brill- Noether functors.
The approach is used in some concrete examples. Let \(Z\) be a smooth closed submanifold of a compact Kähler manifold \(X\) and \(\omega\) a closed differential \((p,q)\)-form on \(X\) such that \(\omega_{|Z}=0\). Then the obstructions of embedded deformations of \(Z\) inside \(X\) are contained in the kernel of the contraction map \(H^1(Z,N_{Z|X})\rightarrow H^{q+1}(Z,\Omega^{p-1}_Z)\).

MSC:

13D10 Deformations and infinitesimal methods in commutative ring theory
14D15 Formal methods and deformations in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14H51 Special divisors on curves (gonality, Brill-Noether theory)
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