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Formality of Kapranov’s brackets in Kähler geometry via pre-Lie deformation theory. (English) Zbl 1404.53090
Summary: We recover some recent results by V. Dotsenko et al. [Mosc. Math. J. 16, No. 3, 505–543 (2016; Zbl 1386.18054)] on the Deligne groupoid of a pre-Lie algebra, showing that they follow naturally by a pre-Lie variant of the PBW theorem. As an application, we show that Kapranov’s \(L_\infty\) algebra structure on the Dolbeault complex of a Kähler manifold is homotopy abelian and independent on the choice of Kähler metric up to an \(L_\infty\) isomorphism, making the trivializing homotopy and the \(L_\infty\) isomorphism explicit.

MSC:
53C55 Global differential geometry of Hermitian and Kählerian manifolds
17B55 Homological methods in Lie (super)algebras
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
18G55 Nonabelian homotopical algebra (MSC2010)
13D10 Deformations and infinitesimal methods in commutative ring theory
32C36 Local cohomology of analytic spaces
58H15 Deformations of general structures on manifolds
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