Cohomology of flag bundles over compact Hermitian locally symmetric spaces
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909250632024064 |
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| author | Biswas, Pritthijit Sankaran, Parameswaran |
| author_facet | Biswas, Pritthijit Sankaran, Parameswaran |
| contents | 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 $B=X_Γ:=Γ\backslash X, E=Y_Γ:=Γ\backslash Y$ 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 $Γ\subset G$, a uniform and torsionless lattice in $G$. We also obtain a description of the Picard group of $Y_Γ$ and $X_Γ$, for which the complexification of $G$ need not be simply connected. Moreover when $G$ is simple, we obtain the values of $ q$ for which $H^{p,q}(X_Γ)$ vanishes when $p=0,1$. This extends the results of R. Parthasarathy from $1980$, who considered (partially) the case $p=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08180 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cohomology of flag bundles over compact Hermitian locally symmetric spaces Biswas, Pritthijit Sankaran, Parameswaran Differential Geometry 14M15, 17B20, 32M10, 32M15, 32Q15 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 $B=X_Γ:=Γ\backslash X, E=Y_Γ:=Γ\backslash Y$ 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 $Γ\subset G$, a uniform and torsionless lattice in $G$. We also obtain a description of the Picard group of $Y_Γ$ and $X_Γ$, for which the complexification of $G$ need not be simply connected. Moreover when $G$ is simple, we obtain the values of $ q$ for which $H^{p,q}(X_Γ)$ vanishes when $p=0,1$. This extends the results of R. Parthasarathy from $1980$, who considered (partially) the case $p=0$. |
| title | Cohomology of flag bundles over compact Hermitian locally symmetric spaces |
| topic | Differential Geometry 14M15, 17B20, 32M10, 32M15, 32Q15 |
| url | https://arxiv.org/abs/2407.08180 |