Symplectically aspherical Kähler manifolds, scalar curvature, and the fundamental group

2026/07/31 by Luca F. Di Cerbo, Alexander Dranishnikov, Ekansh Jauhari
Mathematics · #math.GT #math.AG #math.DG #msc:53C23 #msc:32J27 #msc:53C55 #msc:14F35 #msc:32Q45 #msc:55S15

paper · pdf

v2: Introduction, acknowledgments, and references updated. 24 pages, 1 figure

arxiv created 2026/08/04 · arxiv updated 2026/08/05

Abstract

We present a detailed study of closed smooth manifolds having Kähler forms that pullback to exact forms on the universal cover. We show that these manifolds, which we call symplectically aspherical Kähler manifolds, exist in abundance, even outside the aspherical setting, and have interesting topological and geometric features, such as large fundamental group á la Kollár and the absence of Kähler metrics of positive scalar curvature. Motivated by the latter, we extend the Gromov--Lawson Conjecture on aspherical manifolds to symplectically aspherical manifolds and prove it in the spin case. We also study Kähler cones on symplectically aspherical Kähler manifolds and the realizability problem of their fundamental group, and explore their other complex geometric properties.

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