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Springer fibers and compactified Jacobians. (Fibres de Springer et jacobiennes compactifiées.) (French) Zbl 1129.14301
Ginzburg, Victor (ed.), Algebraic geometry and number theory. In Honor of Vladimir Drinfeld’s 50th birthday. Basel: Birkhäuser (ISBN 978-0-8176-4471-0/hbk). Progress in Mathematics 253, 515-563 (2006).
Let $$F$$ be a non-Archimedean local field of equal characteristics and $$E$$ a finite-dimensional vector space over $$F$$. The affine Grassmannian of $$\mathcal{O}_F$$-lattices in $$E$$ contains a closed subscheme $$X_{\gamma}$$ of lattices preserved by a fixed regular semisimple topologically nilpotent endomorphism $$\gamma$$ of $$E$$. This $$X_{\gamma}$$ is called the affine Springer fiber at $$\gamma$$. It is acted on by a free abelian group $$\Lambda_{\gamma}$$, so that the quotient $$Z_{\gamma}=X_{\gamma}/\Lambda_{\gamma}$$ exists and is a projective scheme over the residue field of $$F$$. The author associates with $$\gamma$$ a singular projective curve $$C_{\gamma}$$ with the complete local ring at the singular point isomorphic to $$\mathcal{O}_F[\gamma]$$. It is proved that $$Z_{\gamma}$$ is the universal cover of the compactified Jacobian $$\overline P(C_{\gamma})$$ [see A. Altman and S. Kleiman, Bull. Am. Math. Soc. 82, 947–949 (1976; Zbl 0336.14008)]. Some results of Altman and Kleiman on the geometry of compactified Jacobians [cf. also A. B. Altman, A. Iarrobino and S. L. Kleiman, in: Real and complex singularities, Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976, 1–12 (1977; Zbl 0415.14014)] are deduced from this fact together with a certain group scheme action on $$Z_{\gamma}$$ with an open orbit. Deformations of $$Z_{\gamma}$$ and $$X_{\gamma}$$ under variation of $$\gamma$$ are studied. A version of the Goresky–Kottwitz–MacPherson conjecture on the purity of the $$l$$-adic cohomology groups is formulated and proved in the homogeneous case in the same manner as for ordinary Springer fibers [T. A. Springer, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31, No. 2, 271–282 (1984; Zbl 0581.20048)].
For the entire collection see [Zbl 1113.00007].

##### MSC:
 14G20 Local ground fields in algebraic geometry 14F20 Étale and other Grothendieck topologies and (co)homologies 14K30 Picard schemes, higher Jacobians