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 √Čtudes 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.
Reviewer: St.Papadima


32G05 Deformations of complex structures
22E25 Nilpotent and solvable Lie groups
55P62 Rational homotopy theory
17B70 Graded Lie (super)algebras