Abstract
We define and study the existence of very stable Higgs bundles on Riemann surfaces, how it implies a precise formula for the multiplicity of the very stable components of the global nilpotent cone and its relationship to mirror symmetry. The main ingredients are the Bialynicki-Birula theory of {mathbb {C}}^*-actions on semiprojective varieties, {mathbb {C}}^* characters of indices of {mathbb {C}}^*-equivariant coherent sheaves, Hecke transformation for Higgs bundles, relative Fourier–Mukai transform along the Hitchin fibration, hyperholomorphic structures on universal bundles and cominuscule Higgs bundles.
Highlights
Drinfeld [21] and Laumon [46] call a vector bundle on a smooth complex projective curve C of genus g a very stable bundle if there is no non-zero nilpotent Higgs field ∈ H 0(C; End(E)⊗ K ) on E
In the case of type (1, . . . , 1) fixed points, where we have a complete classification of very stable Higgs bundles in Theorem 1.2, we find a candidate for the mirror, which is supported by the following results sketched below
One can construct the Bialynicki-Birula partition on the total space of the projectivised equivariant universal bundle (which always exists over the stable locus, and extends over the whole of M using Simpson’s framed moduli Higgs moduli space [62, Theorem 4.10], see (4.19) for k = 1) on M × C which will contain the information on the filtration in Proposition 3.4 which determines the limiting Higgs bundle
Summary
Drinfeld [21] and Laumon [46] call a vector bundle on a smooth complex projective curve C of genus g a very stable bundle if there is no non-zero nilpotent Higgs field ∈ H 0(C; End(E)⊗ K ) on E. Corollary 1.4 The multiplicity of N , the moduli space of semi-stable rank n degree d bundles, in the nilpotent cone is mN = 23g−335g−5 . Of the Lagrangian upward flow WE+ ⊂ M will be a hyperholomorphic vector bundle on M if and only if E is very stable, in which case we expect the mirror to be of rank m FE. Theorem 1.5 Let Eδ ∈ MsT be a very stable Higgs bundle of type We prove in Proposition 8.1 that for cominuscule Higgs bundles where bi are nonvanishing sections except at cominuscule roots it is a polynomial In this case we offer Conjecture 8.2 that under precise conditions these are very stable and find a potential mirror in terms of the minuscule representations of the Langlands dual group, satisfying the expectation of (1.6). We relate this to considerations of Donagi– Pantev and to Drinfeld–Laumon in the geometric Langlands correspondence
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
