arXiv 1005.31612010-05math.AG

曲线上抛物型向量丛模空间的Brauer群

Brauer group of a moduli space of parabolic vector bundles over a curve

Indranil Biswas, Arijit Dey

备注：: Journal of K-Theory (to appear)

摘要：

设${\mathcal P}{\mathcal M}^α_s$是亏格为$g(X) \geq 2$的紧致连通Riemann曲面$X$上秩为$n \geq 2$的稳定抛物向量丛和次数为$d$的固定行列式的模空间.如果是$g(X) = 2$，那么我们假设是$n > 2$。设$m$表示$d$、$n$和拟抛物滤子的所有连续子的维数的最大公约数。证明了上同调Brauer群${\rm Br}({\mathcal P}{\mathcal M}^α_s)$同构于循环群${\mathbb Z}/ m{\mathbb Z}$ .我们还证明了${\rm Br}({\mathcal P}{\mathcal M}^α_s)$是由${\mathcal P}{\mathcal M}^α_s$上的投射丛的Brauer类生成的，该投射丛是通过限制$X\times {\mathcal P}{\mathcal M}^α_s$上的泛投射丛而得到的。证明了$X\times {\mathcal P}{\mathcal M}^α_s$上存在泛向量丛当且仅当$m=1$ .

英文摘要：

Let ${\mathcal P}{\mathcal M}^α_s$ be a moduli space of stable parabolic vector bundles of rank $n \geq 2$ and fixed determinant of degree $d$ over a compact connected Riemann surface $X$ of genus $g(X) \geq 2$. If $g(X) = 2$, then we assume that $n > 2$. Let $m$ denote the greatest common divisor of $d$, $n$ and the dimensions of all the successive quotients of the quasi-parabolic filtrations. We prove that the cohomological Brauer group ${\rm Br}({\mathcal P}{\mathcal M}^α_s)$ is isomorphic to the cyclic group ${\mathbb Z}/ m{\mathbb Z}$. We also show that ${\rm Br}({\mathcal P}{\mathcal M}^α_s)$ is generated by the Brauer class of the projective bundle over ${\mathcal P}{\mathcal M}^α_s$ obtained by restricting the universal projective bundle over $X\times {\mathcal P}{\mathcal M}^α_s$. We also prove that there is a universal vector bundle over $X\times {\mathcal P}{\mathcal M}^α_s$ if and only if $m=1$.

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摘要：

英文摘要：

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