Mehdi Lejmi

Existence of HKT metrics on hypercomplex manifolds of real dimension 8

Abstract A hypercomplex manifold M is a manifold equipped with three complex structures I , J , K satisfying quaternionic relations. Such a manifold admits a canonical torsion-free connection preserving the quaternion action, called the Obata connection. A quaternionic Hermitian metric is a Riemannian metric which is invariant with respect to unitary quaternions. Such a metric is called hyperkahler with torsion (HKT for short) if it is locally obtained as the Hessian of a function averaged with quaternions. An HKT metric is a natural analogue of a Kahler metric on a complex manifold. We push this analogy further, proving a quaternionic analogue of the result of Buchdahl and of Lamari that a compact complex surface M admits a Kahler structure if and only if b 1 ( M ) is even. We show that a hypercomplex manifold M with the Obata holonomy contained in S L ( 2 , H ) admits an HKT structure if and only if H 1 ( O ( M , I ) ) is even-dimensional.

Quaternionic Bott-Chern Cohomology and existence of HKT metrics

We study quaternionic Bott-Chern cohomology on compact hypercomplex manifolds and adapt some results from complex geometry to the quaternionic setting. For instance, we prove a criterion for the existence of HKT metrics on compact hypercomplex manifolds of real dimension 8 analogous to the one given by Teleman [35] and Angella-Dloussky-Tomassini [3] for compact complex surfaces.

Seiberg–Witten invariants on manifolds with Riemannian foliations of codimension 4

Abstract We define Seiberg–Witten equations on closed manifolds endowed with a Riemannian foliation of codimension 4. When the foliation is taut, we show compactness of the moduli space under some hypothesis satisfied for instance by closed K -contact manifolds. Furthermore, we prove some vanishing and non-vanishing results and we highlight that the invariants may be used to distinguish different foliations on diffeomorphic manifolds.

On the Chern–Yamabe Flow

On a closed balanced manifold, we show that if the Chern scalar curvature is small enough in a certain Sobolev norm, then a slightly modified version of the Chern–Yamabe flow (Angella et al. in On Chern–Yamabe Problem, 2015) converges to a solution of the Chern–Yamabe problem. We also prove that if the Chern scalar curvature, on closed almost-Hermitian manifolds, is close enough to a constant function in a Hölder norm, then the Chern–Yamabe problem has a solution for generic values of the fundamental constant.

Cohomologies on Hypercomplex Manifolds

We review some cohomological aspects of complex and hypercomplex manifolds and underline the differences between both realms. Furthermore, we try to highlight the similarities between compact complex surfaces on one hand and compact hypercomplex manifolds of real dimension 8 with holonomy of the Obata connection in \(\mathrm{SL}(2, \mathbb{H})\) on the other hand.