Minimal Models of Nilmanifolds

In this paper we first determine minimal models of nilmanifolds associated with given rational nilpotent Lie algebras. Then we study some properties of nilmanifolds through their associated Lie algebras and minimal models. In particular, we will see that a minimal model of a nilmanifold is formal if and only if it is a torus, and thus a non-toral nilmanifold has no complex structure which is birationally isomorphic to a Kahler manifold. 0. Introduction A nilmanifold is a compact homogeneous space of nilpotent Lie group. The nilmanifolds are known to give counterexamples relating to Kahler structure: non-Kähler almost Kahler manifolds, non-Kähler symplectic manifolds, compact complex manifolds of which the Frölicher spectral sequence does not degenerate at El, and so forth. There is a series of papers in this area starting with Thurston's paper [12] on non-Kähler symplectic manifolds (cf. [1, 4, 5, 7, 10]). In this paper, instead of some specific nilmanifolds, we discuss general nilmanifolds in terms of their associated rational nilpotent Lie algebras. This way clarifies as well as generalizes the arguments in the related problems; for instance, Kodaira-Thurston's first example of non-Kähler symplectic manifold can be characterized as a 4-dimensional nilmanifold with its associated nilpotent Lie algebra g, where g has a basis {X{ ,X2,X3,X4} for which the only nonzero bracket multiplication is [Xx, X2] = —X3. In §1 we determine explicitly minimal models of nilmanifolds associated with given rational nilpotent Lie algebras. In §2 it will be shown that a minimal model of a nilmanifold is formal if and only if it is a torus. Applying a result of Deligne, Griffiths, Morgan, and Sullivan [6], we see that a non-toral nilmanifold has no birational Kahler structure (see §2 for definition). In §3 we briefly discuss symplectic structures of nilmanifolds through their associated Lie algebras and Received by the editors September 10, 1987 and, in revised form, June 10, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 14M17, 32M10, 53C30; Secondary 14M17, 53C15, 57T20.