Recent developments in the Hermitian geometry of solvmanifolds with abelian complex structures

ADRIÁN ANDRADA

Roma

Workshop; New perspectives in differential geometry: special metrics and quaternionic geometry; 2015

Istituto Nazionale di Alta Matematica

A complex structure on a solvmanifold $M=\Gamma\backslash G$ is called abelian when it is induced by a left invariant complex structure J on the solvable Lie group G such that [JX,JY]=[X,Y] for any X,Y in the Lie algebra of G.In this talk we will study solvmanifolds equipped with an abelian complex structure J and a compatible invariant Hermitian metric g. We will focus on two cases:(i) the Hermitian structure (J,g) is locally conformally KÄhler (LCK);(ii) the Hermitian structure (J,g) is balanced.In the first case, we prove that the LCK structure is actually Vaisman and G is locally isomorphic to the product of a Heisenberg group with R (joint work with M. Origlia) In the second case, we prove that the holonomy of the Bismut connection reduces to SU(n), where dim M=2n. Moreover, if the center of G has dimension 2k, then the holonomy further reduces to SU(n-k) (joint work with R. Villacampa).