Manetti, Marco Lie description of higher obstructions to deforming submanifolds. (English) Zbl 1174.13021 Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6, No. 4, 631-659 (2007). 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 \(H^1(Z,N_{Z|X})\rightarrow H^{q+1}(Z,\Omega^{p-1}_Z)\). Reviewer: Dorin-Mihail Popescu (Bucureşti) Cited in 16 Documents 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) Keywords:differential graded Lie algebras; deformation functor; tangent and obstruction spaces; compact Kähler manifolds PDFBibTeX XMLCite \textit{M. Manetti}, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6, No. 4, 631--659 (2007; Zbl 1174.13021) Full Text: arXiv