Very stable Higgs bundles, equivariant multiplicity and mirror symmetry

Hausel, Tamás; Hitchin, Nigel

Very stable Higgs bundles, equivariant multiplicity and mirror symmetry

Hausel T, Hitchin N. 2022. Very stable Higgs bundles, equivariant multiplicity and mirror symmetry. Inventiones Mathematicae. 228, 893–989.

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2022_InventionesMahtematicae_Hausel.pdf 1.07 MB [Published Version]

DOI

10.1007/s00222-021-01093-7

Journal Article | Published | English

Scopus indexed

Author

Hausel, Tamas^ISTA ; Hitchin, Nigel

Corresponding author has ISTA affiliation

Department

Hausel Group

Grant

IST Austria Open Access Fund

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 C∗-actions on semiprojective varieties, C∗ characters of indices of 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.

Publishing Year

2022

Date Published

2022-05-01

Journal Title

Inventiones Mathematicae

Publisher

Springer Nature

Acknowledgement

We would like to thank Brian Collier, Davide Gaiotto, Peter Gothen, Jochen Heinloth, Daniel Huybrechts, Quoc Ho, Joel Kamnitzer, Gérard Laumon, Luca Migliorini, Alexander Minets, Brent Pym, Peng Shan, Carlos Simpson, András Szenes, Fernando R. Villegas, Richard Wentworth, Edward Witten and Kōta Yoshioka for interesting comments and discussions. Most of all we are grateful for a long list of very helpful comments by the referee. We would also like to thank the organizers of the Summer School on Higgs bundles in Hamburg in September 2018, where the authors and Richard Wentworth were giving lectures and where the work in this paper started by considering the mirror of the Lagrangian upward flows W+E investigated in [17]. The second author wishes to thank EPSRC and ICMAT for support. Open access funding provided by Institute of Science and Technology (IST Austria).

Volume

228

Page

893-989

ISSN

0020-9910

eISSN

1432-1297

IST-REx-ID

10704

Cite this

Hausel T, Hitchin N. Very stable Higgs bundles, equivariant multiplicity and mirror symmetry. Inventiones Mathematicae. 2022;228:893-989. doi:10.1007/s00222-021-01093-7

Hausel, T., & Hitchin, N. (2022). Very stable Higgs bundles, equivariant multiplicity and mirror symmetry. Inventiones Mathematicae. Springer Nature. https://doi.org/10.1007/s00222-021-01093-7

Hausel, Tamás, and Nigel Hitchin. “Very Stable Higgs Bundles, Equivariant Multiplicity and Mirror Symmetry.” Inventiones Mathematicae. Springer Nature, 2022. https://doi.org/10.1007/s00222-021-01093-7.

T. Hausel and N. Hitchin, “Very stable Higgs bundles, equivariant multiplicity and mirror symmetry,” Inventiones Mathematicae, vol. 228. Springer Nature, pp. 893–989, 2022.

Hausel T, Hitchin N. 2022. Very stable Higgs bundles, equivariant multiplicity and mirror symmetry. Inventiones Mathematicae. 228, 893–989.

Hausel, Tamás, and Nigel Hitchin. “Very Stable Higgs Bundles, Equivariant Multiplicity and Mirror Symmetry.” Inventiones Mathematicae, vol. 228, Springer Nature, 2022, pp. 893–989, doi:10.1007/s00222-021-01093-7.

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