nLab Goldman-Millson quasi-abelianity theorem

Idea

Let $X$ be a smooth complex manifold with trivial canonical bundle, and let $A^{0,*}(T_X)$ be the differential graded Lie algebra of smooth differential forms of type $(0,*)$ with coefficients in the tangent sheaf of $X$. The Goldman-Millson quasi-abelianity theorem GM90 states that the dgla $A^{0,*}(T_X)$ is quasi-abelian, i.e., it is quasi-isomorphic to an abelian dgla.

Since the dgla $A^{0,*}(T_X)$ is a model for the derived global sections $R\Gamma\mathcal{T}_X$ of the tangent sheaf $\mathcal{T}_X$ of $X$, the Goldman-Millson quasi-abelianity theorem can be stated by saying that if $X$ be a smooth complex manifold with trivial canonical bundle, then $R\Gamma\mathcal{T}_X$ is quasi-abelian. In this form the Goldman-Millson theorem can be generalized to smooth projective manifolds over an arbitrary characteristic zero algebraically closed field? $\mathbb{K}$, as done by Iacono and Manetti in IM10.

A description of the Goldman-Millson quasi-abelianity theorem within the framework of higher category theory is given in FM10.

References

• W. M. Goldman, J. J. Millson. The homotopy invariance of the Kuranishi space. Illinois J. Math. 34 (1990) 337-367.

• D. Iacono, M. Manetti._An algebraic proof of Bogomolov-Tian-Todorov theorem_ Deformation Spaces. Vol. 39 (2010), p. 113-133; arXiv:0902.0732

• D. Fiorenza, E. Martinengo. A short note on ∞-groupoids and the period map for projective manifolds Publications of the nLab. Vol. 2 (2012); arXiv:0911.3845