Cohomology of flag bundles over compact Hermitian locally symmetric spaces

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 $p=0$.