Holonomy of the Bismut Connection on Hermitian Solvmanifolds

ADRIÁN ANDRADA

Campinas

Workshop; IV School and Workshop on Lie Theory; 2015

Universidad Federal de Campinas

Let M = \Gamma\G be a solvmanifold, that is, a compact quotient of a simply connnected solvable Lie group G by a discrete subgroup \Gamma. We consider invariant Hermitian structures (J, g) on M that satisfy the two following conditions: (i) the complex structure J is abelian (i.e. [JX, JY ]=[X, Y ] for X, Y left-invariant vector fields on G), and (ii) the Hermitian metric g is balanced (i.e. the fundamental 2-form \omega := g(J·, ·) is co-closed). Such a Hermitian structure is called abelian balanced. Given such a structure on M, we study its associated Bismut connection, that is, the only linear connection \nabla on M satisfying \nabla g = \nabla J = 0 and such that its torsion T is skew-symmetric (i.e. g(T(X, Y ), Z) defines a 3-form on M).As our main result, we prove that when (J, g) is abelian balanced, the holonomy of \nabla reduces to SU(n), where dim M = 2n. Moreover, if the center of G has dimension 2k, this holonomy further reduces to SU(n ( k).We also exhibit different ways of producing examples of solvable Lie groups equipped with left-invariant abelian balanced Hermitian structures. This is a joint work with Raquel Villacampa.