HKT manifolds: Hodge theory, formality and balanced metrics

TL;DR

This paper explores Hodge theory, formality, and balanced metrics on compact HKT manifolds, revealing similarities to Kähler geometry and providing obstructions to certain structures.

Contribution

It establishes Hodge-theoretic properties for HKT manifolds, proves formality of the associated DGA for SL(n,H)-manifolds, and studies balanced HKT structures on solvmanifolds.

Findings

Hodge theory for specific complexes on HKT manifolds mirrors Kähler behavior

Formality of the differential graded algebra for compact HKT SL(n,H)-manifolds

Obstructions to the existence of certain HKT structures on complex manifolds

Abstract

Let $(M,I,J,K,\Omega)$ be a compact HKT manifold and denote with $\partial$ the conjugate Dolbeault operator with respect to $I$, $\partial_J:=J^{-1}\overline\partial J$, $\partial^\Lambda:=[\partial,\Lambda]$ where $\Lambda$ is the adjoint of $L:=\Omega\wedge-$. Under suitable assumptions, we study Hodge theory for the complexes $(A^{\bullet,0},\partial,\partial_J)$ and $(A^{\bullet,0},\partial,\partial^\Lambda)$ showing a similar behavior to K\"ahler manifolds. In particular, several relations among the Laplacians, the spaces of harmonic forms and the associated cohomology groups, together with Hard Lefschetz properties, are proved. Moreover, we show that for a compact HKT $\mathrm{SL}(n,\mathbb{H})$-manifold the differential graded algebra $(A^{\bullet,0},\partial)$ is formal and this will lead to an obstruction for the existence of an HKT $\mathrm{SL}(n,\mathbb{H})$-structure $(I,J,K,\Omega)$ on a compact complex manifold $(M,I)$. Finally, balanced HKT structures on solvmanifolds are studied.