Cohomology of flag bundles over compact Hermitian locally symmetric spaces

TL;DR

This paper describes the cohomology of flag bundles over compact Hermitian locally symmetric spaces, extending previous results and providing explicit calculations of cohomology groups and Picard groups in this geometric setting.

Contribution

It offers a new comprehensive description of the cohomology and Picard groups of flag bundles over these spaces, generalizing earlier partial results and including cases where the complexification of the group is not simply connected.

Findings

Explicit cohomology descriptions for flag bundles over symmetric spaces.

Determination of Picard groups for these bundles.

Vanishing results for certain cohomology groups when G is simple.

Abstract

Let $E\to B$ be a complex analytic fiber bundle with fiber $F$, a flag variety over a compact complex manifold $B$. We shall obtain a description of the cohomology of $E$ when $B=X_\Gamma:=\Gamma\backslash X, E=Y_\Gamma:=\Gamma\backslash Y$ and $F=K/H$, a flag variety, where $Y=G/H$ and $X=G/K$, a Hermitian globally symmetric space of non-compact type with $G$ being a real, connected, non-compact, semisimple linear Lie group with no compact factors and simply connected complexification, $K\subset G$, a maximal compact subgroup, $H=Z_K(S)$, the centralizer in $K$ of a toral subgroup $S\subseteq K$ containing $Z(K)$, the center of $K$ and $\Gamma\subset G$, a uniform and torsionless lattice in $G$. We also obtain a description of the Picard group of $Y_\Gamma$ and $X_{\Gamma}$, for which the complexification of $G$ need not be simply connected. Moreover when $G$ is simple, we obtain the values of $ q$ for which $H^{p,q}(X_\Gamma)$ vanishes when $p=0,1$. This extends the results of R. Parthasarathy from $1980$, who considered (partially) the case $p=0$.