Deformations of generalized calibrations and compact non-Kahler manifolds with vanishing first Chern class

TL;DR

This paper explores the deformation theory of generalized calibrations in special geometric structures, showing moduli spaces are finite dimensional and providing examples including nearly parallel G_2 and certain Hermitian manifolds.

Contribution

It introduces a framework for understanding deformations of generalized calibrations with reduced structure groups and identifies conditions for special holonomy in Hermitian manifolds.

Findings

Moduli spaces of these calibrations are finite dimensional.

Examples include Hopf fibrations and nearly parallel G_2 manifolds.

Certain connected sums admit Hermitian structures with special holonomy.

Abstract

We investigate the deformation theory of a class of generalized calibrations in Riemannian manifolds for which the tangent bundle has reduced structure group U(n), SU(n), G_2 and Spin(7). For this we use the property of the associated calibration form to be parallel with respect to a metric connection which may have non-vanishing torsion. In all these cases, we find that if there is a moduli space, then it is finite dimensional. We present various examples of generalized calibrations that include almost hermitian manifolds with structure group U(n) or SU(n), nearly parallel G_2 manifolds and group manifolds. We find that some Hopf fibrations are deformation families of generalized calibrations. In addition, we give sufficient conditions for a hermitian manifold (M,g,J) to admit Chern and Bismut connections with holonomy contained in SU(n). In particular we show that any connected sum of $k \geq 3$ copies of $S^3 \times S^3$ admits a hermitian structure for which the restricted holonomy of a Bismut connection is contained in SU(3).