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arXiv 0909.0041math.AG
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实数域上的泛向量丛

Universal vector bundle over the reals

Indranil Biswas, Jacques Hurtubise

备注：
Final version; to appear in the Transactions of the AMS
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摘要：

设X_R是定义在R上的一条几何不可约光滑射影曲线，使得X_R没有任何实点.设X= X_R \times_R C为复曲线。证明了X_R x Pic^d（X_R）$ if and only if $ χ（L）$ is odd for L in Pic^d(X_R)$上存在泛实代数线丛。存在环X x Pic^d（X）上的泛四元代数线丛当且仅当次数d是奇数。取整数r和d，使得r > 1，且d与r互质。设M_{X_R}（r，d）（M_X（r，d）$）是X_R（X）上秩为r、度为d的稳定向量丛的模空间.证明了X_R x M_{X_R}（r，d）上存在泛实代数向量丛当且仅当X（E）对E在M_{X_R}（r，d）中为奇数. X x M_X（r，d）上存在泛四元数向量丛当且仅当d为奇数.本文还研究了X_R可几何约或X_R有实点的情形。

英文摘要：

Let X_R be a geometrically irreducible smooth projective curve, defined over R, such that X_R does not have any real points. Let X= X_R\times_R C be the complex curve. We show that there is a universal real algebraic line bundle over X_R x Pic^d(X_R)$ if and only if $χ(L)$ is odd for L in Pic^d(X_R)$. There is a universal quaternionic algebraic line bundle over X x Pic^d(X) if and only if the degree d is odd. Take integers r and d such that r > 1, and d is coprime to r. Let M_{X_R}(r,d) (respectively, M_X(r,d)$) be the moduli space of stable vector bundles over X_R (respectively, X) of rank r and degree d. We prove that there is a universal real algebraic vector bundle over X_R x M_{X_R}(r,d) if and only if χ(E) is odd for E in M_{X_R}(r,d). There is a universal quaternionic vector bundle over X x M_X(r,d) if and only if the degree d is odd. The cases where X_R is geometrically reducible or X_R has real points are also investigated.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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