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On moduli of \(G\)-bundles on a curve for exceptional \(G\). (English) Zbl 0969.14016

From the introduction: Let \(G\) be a simple and simply connected complex algebraic group. Let \({\mathcal M}_{G,X}\) be the stack of \(G\)-bundles on the smooth connected and projective complex curve \(X\) of genus \(g\). If \(\rho:G\to SL_r\) is a representation of \(G\), denote by \({\mathcal D}_\rho\) the pullback of the determinant bundle under the morphism \({\mathcal M}_{G,X}\to {\mathcal M}_{SL_r,X}\) defined by extension of the structure group. We associate to \(G\) a special number \(d(G)\) and the representation \(\rho(G)\) as follows: \[ \begin{matrix} \text{Type of }G & A_r & B_r(r\geq 3) & C_r & D_r(r\geq 4) & E_6 & E_7 & E_8 & F_4 & G_2\\ d(G) & 1 & 2 & 1 & 2 & 6 & 12 & 60 & 6 & 2\\ \rho(G) & \overline\omega_1 &\overline\omega_1 & \overline\omega_1 & \overline\omega_1 & \overline\omega_6 & \overline\omega_7 & \overline\omega_8 & \overline\omega_4 & \overline\omega_1 \end{matrix} \] Theorem 1.1. There is a line bundle \({\mathcal L}\) on \({\mathcal M}_{G,X}\) such that \(\text{Pic}({\mathcal M}_{G,X}) @>\sim>> \mathbb{Z}{\mathcal L}\). Moreover we may choose \({\mathcal L}\) in such a way that \({\mathcal L}^{\otimes d(G)} ={\mathcal D}_{\rho(G)}\).
The above theorem is proved, for classical \(G\) and \(G_2\), by Y. Laszlo and C. Sorger [Ann. Sci. Éc. Norm. Supér., IV. Sér. 30, No. 4, 499-525 (1997; Zbl 0918.14004)]. The general case had been conjectured in the cited paper. Our proof is purely algebraic in nature: The basic idea is not only to identify the space of conformal blocks \(B_{G,X}(\ell;p;0)\) with sections of \({\mathcal L}^\ell\) provided that \({\mathcal L}\) exists, but also to use the space of conformal blocks and its properties as the decomposition formulas of A. Tsuchiya, K. Ueno and Y. Yamada [Adv. Stud. Pure Math. 19, 459-566 (1989; Zbl 0696.17010)] and G. Faltings [J. Algebr. Geom. 3, No. 2, 347-374 (1994; Zbl 0809.14009)] to prove the existence of \({\mathcal L}\). Suppose \(g(X)\geq 2\). For the coarse moduli spaces \(M_{G,X}\) of semi-stable \(G\)-bundles, we show that the roots of the determinant bundle of theorem 1.1 do only exist on the open subset of regularly stable \(G\)-bundles. This will allow us to complete the following result of A. Beauville, Y. Laszlo and C. Serger [Compos. Math. 112, No. 2, 183-216 (1998)], which was proved there for classical \(G\) and \(G_2\).
Theorem 1.2. Let \(G\) be semi-simple. Then \(M_{G,X}\) is locally factorial if and only if \(G\) is special in the sense of Serre.

MSC:

14H10 Families, moduli of curves (algebraic)
14D20 Algebraic moduli problems, moduli of vector bundles
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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References:

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