Pure and Applied Mathematics Quarterly

Volume 15 (2019)

Number 4

Special Issue in Honor of Simon Donaldson: Part 2 of 2

Guest Editor: Richard Thomas (Imperial College London)

The $\partial \overline{\partial}$-lemma for general Clemens manifolds

Pages: 1001 – 1028

DOI: https://dx.doi.org/10.4310/PAMQ.2019.v15.n4.a2


Robert Friedman (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)


We show that the $\partial \overline{\partial}$‑lemma holds for the non-Kähler compact complex manifolds of dimension three with trivial canonical bundle constructed by Clemens as deformations of Calabi–Yau threefolds contracted along smooth rational curves with normal bundle of type $(-1,-1)$, at least on an open dense set in moduli. The proof uses the mixed Hodge structure on the singular fibers and an analysis of the variation of the Hodge filtration for the smooth fibers.

2010 Mathematics Subject Classification

14J32, 32G20, 32J17

Received 11 November 2017

Published 20 March 2020