nLab
Goldman-Millson quasi-abelianity theorem

Contents

Idea

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

Since the dgla A 0,*(T X)A^{0,*}(T_X) is a model for the derived global sections RΓ𝒯 XR\Gamma\mathcal{T}_X of the tangent sheaf 𝒯 X\mathcal{T}_X of XX, the Goldman-Millson quasi-abelianity theorem can be stated by saying that if XX be a smooth complex manifold with trivial canonical bundle, then RΓ𝒯 XR\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 𝕂\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

Revised on September 13, 2012 00:16:33 by Urs Schreiber (82.169.65.155)