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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
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