Abstract
For any compact complex manifold M with a compatible symplectic formω, we consider the homomorphisms L 1,0: H 1,0(M)→ H {n, n−1(M) and L 0, 1: H 0, 1(M)→ H n − 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 L 1, 0 (resp.L 0, 1) is injective. Such conditions can be considered as complexversions of the (real) Lefschetz condition studied by Benson and Gordonin [Topology 27 (1988), 513–518]for symplectic manifolds. Within the class of compactcomplex nilmanifolds, we prove that the injectivity of L 1, 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).
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Cordero, L.A., Fernández, M. & Ugarte, L. Lefschetz Complex Conditions for Complex Manifolds. Annals of Global Analysis and Geometry 22, 355–373 (2002). https://doi.org/10.1023/A:1020527112556
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DOI: https://doi.org/10.1023/A:1020527112556