Repository logo

Geometry of the Hitchin Morphism and Spectral Correspondences

dc.contributor.advisorRayan, Steven
dc.contributor.committeeMemberSteele, Tom
dc.contributor.committeeMemberGroechenig, Michael
dc.contributor.committeeMemberGhezelbash, Masoud
dc.contributor.committeeMemberSamei, Ebrahim
dc.contributor.committeeMemberSzmigielski, Jacek
dc.creatorBanerjee, Kuntal 2024
dc.description.abstractWe 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.
dc.subjectSpectral correspondence
dc.subjectspectral curve
dc.subjecttwisted pair
dc.subjectHiggs bundle
dc.subjectco-Higgs bundle
dc.subjectmoduli space
dc.subjectHitchin fibration
dc.subjectprojective line
dc.subjectelliptic curve
dc.subjectpush-pull formula
dc.subjectcartographic group
dc.subjectGalois group
dc.subjectmonodromy group
dc.subjectHitchin morphism
dc.subjectalgebraic surface
dc.subjectcomplex surface
dc.subjectspectral cover
dc.titleGeometry of the Hitchin Morphism and Spectral Correspondences
dc.type.materialtext and Statistics of Saskatchewan of Philosophy (Ph.D.)


Original bundle
Now showing 1 - 1 of 1
Thumbnail Image
778.03 KB
Adobe Portable Document Format
License bundle
Now showing 1 - 1 of 1
No Thumbnail Available
2.27 KB
Plain Text