SMF

Decomposition of spectral covers

Decomposition of spectral covers

R. Donagi
  • Année : 1993
  • Tome : 218
  • Format : Papier, Électronique
  • Langue de l'ouvrage :
    Anglais
  • Class. Math. : 14D20
  • Pages : 145-175
  • DOI : 10.24033/ast.243

A G-principal Higgs bundle over a variety X (with values in an arbitrary line bundle on X) determines a family of spectral covers $\widetilde {\mathrm {X}}_\rho $ of X, one for each irreducible representation $\rho $ of G. We show that each of the $\mathrm {Pic}(\widetilde {\mathrm {X}}_\rho )$ is isogenous to the sum, with multiplicities, of a finite collection of abelian varieties, obtained as isotypic pieces for the action of the Weyl group W on $\mathrm {Pic}(\widetilde {\mathrm {X}})$, where $\widetilde {\mathrm {X}}$ is the cameral, or W-Galois, cover of X, independent of $\rho $. The piece $\mathrm {Prym}(\widetilde {\mathrm {X}})$, corresponding to the reflection representation of W, is distinguished : it occurs in $\mathrm {Pic}(\widetilde {\mathrm {X}}_\rho )$ for each $\rho $ (this characterizes Prym for ical G but not for exceptional groups such as G$_2$, E$_6$), and is essentially the moduli space of Higgs bundles with spectral data $\widetilde {\mathrm {X}}$. Various Prym identities are recovered as the case X$ =\mathbb {P}^1$, G simply laced, studied previously by Kanev.

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