zbMATH — the first resource for mathematics

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].

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