Compactification of the stacks of Drinfeld's shtukas
The study and success of Langlands' correspondence for \(\text{Gl}_r\) in positive characteristic are based on the study of the arithmetic and geometry of stacks of Drinfeld's shtukas; c.f. for \(r=2\), the works of V. G. Drinfeld, and for general \(r\) the works L. Lafforgue. Let \(X\) be a smooth, geometrically connected, projective algebraic curve over a finite field \({\mathbb F}_q\). A rank-\(r\) shtuka over \(S\) is a diagram of rank \(r\) vector bundles over \(X\times S\), \({\mathcal E} \hookrightarrow {\mathcal E}' \hookleftarrow {\mathcal E}''\overset \tau \simeq {\mathcal E}^{\sigma}\), where \({\mathcal E}'/{\mathcal E}\) and \({\mathcal E}'/{\mathcal E}''\) are supported for the graph of two morphisms \(0: S\to X\) and \(\infty :S\to X\). \(0\) and \(\infty\) define a morphism \( \text{Cht}^{r,d } \to {X\times X} \), \( \text{Cht}^{r,d}\) being the stack of these objects (\(\deg({\mathcal E} )=d\)). This stack is smooth but neither proper nor of type finite. To avoid these problems one considers a polygon, \(p\), and the stack of finite type, \(\text{Cht}^{r,d, \bar p\leq p}\), of shtukas with Harder-Narasimhan polygon \(\bar p\) and \(\bar p \leq p\), c.f. [\textit{L. Lafforgue}, ``Chtoucas de Drinfeld et conjecture de Ramanujan-Petersson'', Astérisque 243 (1997; Zbl 0899.11026)]. However, \(\text{Cht}^{r,d, \bar p\leq p}\), is not proper, L. Lafforgue introduces a compactification, \(\overline{\text{Cht}^{r,d, \bar p\leq p}}\), of these stacks. This is the stack of iterated shtukas. Grosso modo, an iterated shtuka is a shtuka, \({\mathcal E} \hookrightarrow {\mathcal E}' \hookleftarrow {\mathcal E}''\overset \tau\Leftarrow {\mathcal E}^\sigma\), where \(\tau\) is allowed to be a complete morphism. Verification of the properness of this stack is not direct and it takes several pages [cf. \textit{L. Lafforgue}, J. Am. Math. Soc. 11, No. 4, 1001--1036 (1998; Zbl 1045.11041) and ``Cours à l'Institut Tata sur les chtoucas de Drinfeld et la correspondence de Langlands'', Prepublication I.H.E.S/M/02/45]. In this work the author presents these stacks in a new and elegant way. To do so, he uses the variation of GIT quotients [cf. \textit{I. V. Dolgachev} and \textit{Y. Hu}, Publ. Math., Inst. Hautes Études Sci. 87, 5--56 (1998; Zbl 1001.14018) and \textit{M. Thaddeus}, J. Am. Math. Soc. 9 (3), 691--723 (1996; Zbl 0874.14042)]. The author considers an effective divisor, \(N\subset X\), and the stack \(\overline{\text{Cht}_N^{r,d}}\) [cf. \textit{L. Lafforgue}, Invent. Math. 147, No. 1, 1--241 (2002; Zbl 1038.11075)]; the finite group \(Gl_r({\mathcal O}_N)\) acts on this stack and the quotient \(\overline{\text{Cht}_N^{r,d}}/Gl_r({\mathcal O}_N)\) is the coarse space associated with the stack \(\overline{\text{Cht}^{r,d}}\times_{X\times X}(X\setminus N)^2\). Let \({\mathcal Y}\) be the stack which is a \(\text{PGL}(h)\)-torsor over the stack \[ \overline{\text{Cht}_N^{r,d}}\times_{\text{Vec}^{r,d }}\text{Vec}^{r,d,p\leq \bar p}\times_{{\mathbb A}^{r-1}/{\mathbb G}_m^{r-1}} {\mathbb A}^{r-1} \] which is given by the surjective morphisms \({\mathcal O}_X^h\to {\mathcal E}\) such that the induced morphism \(H^0(X,{\mathcal O}_X^h)\to H^0(X,{\mathcal E})\) is an isomorphism. Let \({\mathcal Z}_0\) be the scheme \({\mathbb P}( \Hom(\wedge^r {\mathcal O}^h_{\text{Pic}^d(X)},p_*{\mathcal M}))\), \({\mathcal M}\) being a universal line bundle on \(X\times \text{Pic}^d(X)\) and \[ {\mathcal Z}={\mathcal Z}_0\times \prod_{1\leq i\leq m} {\mathbb P}(\bigoplus_{1\leq s\leq r} \Hom(\wedge^s k^h, \wedge^s {\mathcal O}^r_{N_i})^{\otimes r!/s}), \] where \(N=\sqcup_{1\leq i\leq m} N_i\), \(m\geq 0\) and the ``\(N_i\)'' are effective divisors without multiplicities. There exists a natural morphism, \({\mathcal Y}\to {\mathcal Z}\), which is equivariant for the action of \(\text{Gl}_r({\mathcal O}_N)\times \text{PGL}(h) \times {\mathbb G}_m^{r-1} \) over \({\mathcal Y}\) and \({\mathcal Z}\), respectively. By choosing certain \(\text{PGL}(h) \times {\mathbb G}_m^{r-1} \)-linearizations of an ample line bundle over \( {\mathcal Z}\) and \(p\) a fairly convex polygon, by using the variation of GIT quotients the author concludes that for \(q>r(r-1)/2\) the morphism \({\mathcal Y}^{ss}\to {\mathcal Z}^{ss}\times (X\setminus N)^2\) is proper and therefore \[ \overline{\text{Cht}^{r,d, \bar p\leq p}}\times_{X\times X}(X\setminus N)^2\to (X\setminus N)^2 \] and \[ \overline{\text{Cht}^{r,d, \bar p\leq p}}\to {X\times X} \] are proper.
scientific article; zbMATH DE number 2138003
scientific article