Compatible almost complex structures on the Hard Lefschetz condition
Dexie Lin
TL;DR
The paper addresses whether Cirici–Wilson's criterion, which provides a sufficient condition for a compact almost Kähler manifold to satisfy the Hard Lefschetz condition, is also necessary. It constructs infinite families of compatible almost complex structures on the standard torus $T^{2n+2}$ that satisfy Hard Lefschetz while violating the expected equality between Betti and variant Hodge numbers, specifically obtaining $b^1>2h^{1,0}$ (i.e., $b^1>2\ell^{1,0}$). It also shows, for each $0<k\le n$, infinite families with $h^{1,0}=\ell^{1,0}=n-k+1$, and discusses constraints like $\ell^{1,0}=\ell^{0,1}$ in dimension four as necessary but not sufficient. The results delineate the boundary between HL-type topological constraints and cohomological invariants in almost Kähler geometry, highlighting the richness of compatible almost complex structures on tori and their geometric implications.
Abstract
For a compact Kähler manifold, it is well-established that its de Rham cohomology satisfies the Hard Lefschetz condition, which is reflected in the equality between the Betti numbers and the Hodge numbers. A special subclass of symplectic manifolds also adheres to this condition. Cirici and Wilson \cite{CW20} employ the variant Hodge number to propose a sufficient criterion for compact almost Kähler manifolds to satisfy this condition. In this paper, we show that this condition is only sufficient by presenting examples of compact almost Kähler manifolds that fulfill the Hard Lefschetz condition while violating the equality between the variant Hodge numbers and Betti numbers, that is, \[b^1>2h^{1,0}.\] This phenomenon contrasts with the behavior observed in compact Kähler manifolds.