Geometry of the Hitchin Morphism and Spectral Correspondences
Date
2024-05-07
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ORCID
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Thesis
Degree Level
Doctoral
Abstract
We explore a strong categorical correspondence between isomorphism classes of sheaves of arbitrary rank on a given algebraic curve and twisted pairs on another algebraic curve, mostly from a linear-algebraic standpoint. We aim to generalize the language of classical spectral correspondence by the annihilating polynomials of pairs. In a particular application, we realize a generic elliptic curve as a spectral cover of the complex projective line and then construct examples of cyclic twisted pairs and co-Higgs bundles on the same curve. Afterwards, by appealing to a composite push-pull projection formula, we conjecture an iterated version of spectral correspondence. We prove this conjecture for a particular class of spectral covers of the complex projective line through Galois-theoretic arguments. The proof relies upon a classification of Galois groups into primitive and imprimitive types. In this context, we revisit a classical theorem of Ritt.
We move to examining the image of Hitchin morphism on algebraic varieties in general. In our context we work with rank 2 bundles on algebraic surfaces. Unlike the case of curves, Hitchin morphism on the space of twisted Higgs sheaves is not necessarily surjective though we study its properness, in case of twist by line bundles. We apply this idea to write down a proof of surjective property of Hitchin morphism. We also present illustrative examples in case of co-Higgs bundles following results by Rayan and Colmenares.
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Keywords
Spectral correspondence, spectral curve, twisted pair, Higgs bundle, co-Higgs bundle, moduli space, semistability, Hitchin fibration, projective line, elliptic curve, push-pull formula, cartographic group, Galois group, monodromy group, Hitchin morphism, algebraic surface, complex surface, spectral cover
Citation
Degree
Doctor of Philosophy (Ph.D.)
Department
Mathematics and Statistics
Program
Mathematics