Higgs Bundles and Local Systems on Elliptic Curves

Trozzo, Marco (2016) Higgs Bundles and Local Systems on Elliptic Curves, [Dissertation thesis], Alma Mater Studiorum Università di Bologna. Dottorato di ricerca in Matematica, 27 Ciclo. DOI 10.6092/unibo/amsdottorato/7772.
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If S is an elliptic curve, the total space X of the cotangent bundle of S is the moduli space of rank one and degree zero Higgs bundles on S and the corresponding character variety Y is C* x C*. The punctual Hilbert scheme X^[n] of X can be identified with the moduli space of stable marked Higgs bundles on S and there is a natural isomorphism of graded vector spaces between the rational cohomology groups of the Hilbert schemes of X and Y that exchanges the perverse Leray filtration on X^[n] with the halved weight filtration on Y^[n]. We prove that there is a diffeomorphism between the Hilbert schemes that induces the given isomorphism in cohomology. We also give a complete description of Higgs bundles corresponding to subschemes of length n ≤ 3. Moreover, we discuss a conjecture by Simpson on the compactification of the moduli space of Higgs bundles and on the dual boundary complex of the character variety, proving a result going in the direction of Simpson’s conjecture.

Tipologia del documento
Tesi di dottorato
Trozzo, Marco
Dottorato di ricerca
Scuola di dottorato
Scienze matematiche, fisiche ed astronomiche
Settore disciplinare
Settore concorsuale
Parole chiave
higgs bundles, hilbert scheme, nonabelian hodge theory, elliptic curve
Data di discussione
21 Dicembre 2016

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