Holonomy of the Bismut connection on Vaisman solvmanifolds

RAQUEL VILLACAMPA; ADRIÁN ANDRADA

Rio de Janeiro

Congreso; International Congress of Mathematicians 2018; 2018

Instituto de Matemática Pura y Aplicada

A Hermitian manifold $(M,J,g)$ is called locally conformal K\"ahler (LCK) if each point of $M$ has a neighborhood where the metric $g$ is conformal to a Kahler metric with respect to $J$. Equivalently, there exists a closed $1$-form $\theta$ on $M$ such that $d\omega=\theta\wedge\omega$, where $\omega$ denotes the fundamental $2$-form associated to $(J,g)$, defined by $\omega(\cdot,\cdot)=g(J\cdot,\cdot)$. The $1$-form $\theta$ is called the Lee form. If the Lee form is parallel with respect to the Levi-Civita connection associated to $g$, the LCK structure is called Vaisman. The family of Vaisman manifolds is very important and they have been deeply studied since the seminal work of I. Vaisman in the '80s. On the other hand, any Hermitian manifold $(M^{2n},J,g)$ admits a unique connection $\nabla^b$ which satisfies $\nabla^bJ=0$, $\nabla^bg=0$ and its torsion $T^b$ es totally skew-symmetric, that is, $c(X,Y,Z)=g(X,T^b(Y,Z))$ is a $3$-form on $M$. The connection $\nabla^b$ is called the Bismut connection and it has holonomy contained in $U(n)$.In this talk we study the holonomy of the Bismut connection on Vaisman solvmanifolds, that is, on compact quotients $\Gamma\backslash G$, where $G$ is a simply connected solvable Lie group and $\Gamma$ is a discrete subgroup of $G$. We assume that the Vaisman structure on $\Gamma\backslash G$ is induced by a left-invariant Vaisman structure on $G$.In this case, we prove that the (restricted) holonomy of $\nabla^b$ reduces to a one-dimensional subgroup of $U(n)$, which is not contained in $SU(n)$. To show this, we exhibit first some general results about the Bismut connection on arbitrary Vaisman manifolds, and then we use the characterization of unimodular solvable Lie algebras that admit Vaisman structures, given in \cite{AO}.This is a joint work with Raquel Villacampa (Zaragoza, España).