Abstract: 

In complex geometry, the nonabelian Hodge correspondence relates complex representations of the fundamental group of a smooth projective algebraic variety to semistable Higgs bundles with vanishing Chern classes. Its proof, however, relies essentially on analytic methods, including harmonic metrics and partial differential equations. In p-adic geometry, Faltings, Heuer, and others have constructed, by algebraic methods, embeddings from categories of fundamental-group representations into categories of Higgs bundles. Unlike in the complex setting, a general intrinsic characterization of the p-adic Higgs bundles that arise from genuine representations of the fundamental group remains unknown.

The aim of this seminar is to advance our understanding of this problem by studying stability techniques for Higgs bundles that are relevant to p-adic geometry, with particular emphasis on their foundations in positive characteristic. Our main thread will be Langer’s 2004 work on semistable sheaves in positive characteristic and his 2015 work on Bogomolov’s inequality for Higgs sheaves. Through these papers, we will systematically study the interaction among Frobenius pullback, stability, Chern classes, the Cartier transform, and Higgs–de Rham flows. We will conclude with the work of Lan, Sheng, and Zuo, examining how the stability and periodicity of Higgs bundles can be used to construct crystalline representations of the étale fundamental group of the generic fiber.

Prerequisites and intended audience: a basic background in algebraic geometry is assumed, and some familiarity with the general context of p-adic (non-abelian) Hodge theory would be helpful. The seminar is primarily intended for researchers and graduate students in p-adic geometry who would like to become familiar with stability theory for vector bundles and Higgs bundles.

The seminar will be conducted primarily in English. It will be held in person only, with no online participation or video recordings.

Lecture Titles:

Lecture 1 (Tongmu He, 7/29, Wed): Introduction to stability in positive characteristic: background on stability and Chern classes [1]

Lecture 2 (Tongmu He, 7/31, Fri): Frobenius destabilization and the Bogomolov inequality of semistable sheaves: Langer’s 2004 work [2]

Lecture 3 (Tongmu He, 8/03, Mon): Simpson filtrations: preparation for Langer’s 2015 work [5]

Lecture 4 (Daxin Xu, 8/05, Wed): Cartier descent and the Ogus–Vologodsky correspondence: fundational tools [4]

Lecture 5 (Tongmu He, 8/07, Fri): Higgs-de Rham flows and the Bogomolov inequality for semistable Higgs sheaves: Langer’s 2015 Work [6]

Lecture 6 (Heng Du, 8/10, Mon): Introduction to Fontaine–Laffaille–Faltings theory

Lecture 7 (Tongmu He, 8/12, Wed): Periodic Higgs–de Rham flows and crystalline representations: the work of Lan–Sheng–Zuo [7]

Note: The precise content of the seminar may be adjusted as it progresses and may not coincide exactly with the lecture titles currently listed.

References:

[1] D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, 2010.

[2] A. Langer, Semistable sheaves in positive characteristic, Ann. of Math. (2) 159 (2004), 251–276.

[3] A. Langer, Addendum to “Semistable sheaves in positive characteristic”, Ann. of Math. (2) 160 (2004), 1211–1213.

[4] A. Ogus and V. Vologodsky, Nonabelian Hodge theory in characteristic p, Publ. Math. Inst. Hautes Études Sci. 106 (2007), 1–138.

[5] A. Langer, Semistable modules over Lie algebroids in positive characteristic, Doc. Math. 19 (2014), 509–540.

[6] A. Langer, Bogomolov’s inequality for Higgs sheaves in positive characteristic, Invent. Math. 199 (2015), 889–920.

[7] G. Lan, M. Sheng, and K. Zuo, Semistable Higgs bundles, periodic Higgs bundles and representations of algebraic fundamental groups, J. Eur. Math. Soc. 21 (2019), 3053–3112.