## The homotopy invariance of the Kuranishi space.(English)Zbl 0707.32004

Deligne’s principle: “deformation theory in characteristic zero is controlled by dg Lie algebras, in a homotopy invariant way”, as foreseen by M. Schlessinger and J. Stasheff in their work on moduli spaces arising in rational homotopy theory [Univ. North Carolina Preprint (1979)] is once more illustrated here, by the deformation theory of analytic structures on complex compact manifolds M, controlled by the Kodaira-Spencer algebra. One may in general associate to a (cochain) dg Lie $$(L^*,d)$$ the Kuranishi space $$K_ L=\{y\in C^ 1(L)|$$ $$dy+[y,y]=0\},$$ where $$C^ 1(L)$$ is a complement to $$dL^ 0$$ in $$L^ 1$$. If L has a suitable topology and dim $$H^ 1L<\infty$$ (shortly, if L is analytic) then the germ $$(K_ L,0)$$ has the structure of a finite- dimensional analytic germ; for the Kodaira-Spencer algebra of M, this construction just gives the versal deformation of M. The main result establishes the fact that the analytic germ $$(K_ L,0)$$ depends only on the homotopy type in low dimensions of the underlying dg Lie $$(L^*,d)$$. The proof uses techniques from an earlier paper of the authors [Publ. Math., Inst. Hautes Études Sci. 67, 43-96 (1988; Zbl 0678.53059)]; the task here is easier, since there is no need to divide out by the gauge action of $$\exp (L^ 0)$$. As far as the computational side is concerned, one will find here one step further beyond the case of formal dg Lie algebras (previously extensively studied by the authors [loc. cit.]). For $$M=N/\Gamma$$, where N is a complex nilpotent Lie group whose Lie algebra n is defined over $${\mathbb{R}}$$, and $$\Gamma$$ is a cocompact lattice, it is shown, by adapting an old argument of K. Nomizu [Ann. Math., II. Ser. 59, 531-538 (1954; Zbl 0058.022)], that the Kodaira-Spencer algebra of M is homotopy equivalent with the finite-dimensional dg Lie $$(\Lambda n^*\otimes n,d\otimes 1)$$, where $$(\Lambda n^*,d)$$ is the dga coming from the Koszul resolution; by the above-mentioned main result, the base space of the versal deformation of M turns out to be independent of $$\Gamma$$ and may be easily computed as the Kuranishi space of $$(\Lambda n^*\otimes n,d\otimes 1)$$, namely as the germ at zero of the affine variety of the Lie algebra endomorphisms of n.