Lefschetz Complex Conditions for Complex Manifolds

Abstract

For any compact complex manifold M with a compatible symplectic formω, we consider the homomorphisms L1,0: H1,0(M)→ H{n, n−1(M) and L0, 1: H0, 1(M)→ Hn − 1, n(M) given by the cup product with [ω]n − 1, n being the complex dimension of M andH*, *(M) the Dolbeault cohomology of M. We say that Mhas Lefschetz complex type (1, 0) (resp. (0, 1)) if L1, 0 (resp.L0, 1) is injective. Such conditions can be considered as complexversions of the (real) Lefschetz condition studied by Benson and Gordonin [Topology27 (1988), 513–518]for symplectic manifolds. Within the class of compactcomplex nilmanifolds, we prove that the injectivity of L1, 0characterizes those complex structures which are Abelian in the sense ofBarberis et al. [Ann. Global Anal. Geom.13 (1995), 289–301]. In contrast, complex tori are the only nilmanifolds having Lefschetz complex type (0, 1).