$L_{\infty}$-algebra of an unobstructed deformation functor

Details $L_{\infty}$-algebra of an unobstructed deformation functor $L_{\infty}$-algebra of an unobstructed deformation functor

1999-07-06

arxiv.org/abs/math/9907031v2

Intern. Math. Research Notices 3 (2000) 147-164

Mathematics

Algebraic Geometry

LaTeX, 15 pages; small changes

Scientific paper

This is a comment on the Kuranishi method of constructing analytic deformation spaces. It is based on a simple observation that the Kuranishi map can always be inverted in the category of $L_{\infty}$-algebras. The $L_{\infty}$-structure obtained by this inversion is used to define an ''unobstructed'' deformation functor which is always representable by a smooth pointed moduli space. The singular nature of the original Kuranishi deformation space emerges in this setting merely as a result of the truncation of this ``naive'' $L_{\infty}$-algebra controlling the deformations to a usual differential Lie algebra.

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