Fundamentalgroup and crystals

• Applicant: Professorin Dr. Hélène Esnault
• Subject Area: Mathematics
• Term: from 2014 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 262461915

• Final Report Year: 2017

Final Report Abstract

We had three main themes on the relation between the fundamental group of varieties, notably over finite fields, and various types of isocrystals: Gieseker’s conjecture: vanishing of the étale fundamental group should force the infinitesimal crystals to be trivial. Recall we prove the conjecture on smooth projective varieties over an algebraically closed field of characteristic p > 0 with Mehta, this was the starting point of the proposal. We progressed with V. Srinivas (Tata Institute of Fundamental Research, Mumbai) proving a version of Gieseker’s conjecture on singular projective varieties over finite fields and a relative version. de Jong’s conjecture: vanishing of the étale fundamental group on smooth projective varieties over an algebraically closed field of characteristic p > 0 should force the isocrystals to be trivial. This is a very profound conjecture and we do not have yet a complete answer. Yet with Atsushi Shiho (Tokyo University) we could prove it under some restriction on the geometry of the variety (Annales de l’Insititut Fourier). On the way we proved a vanishing theorem on the crystalline Chern class of locally free (or convergent) isocrystals. This is the starting point of new studies, notably of Bhatt-Lurie. Deligne’s conjecture: on a smooth variety over a finite field we could prove, with Tomoyuki Abe (KAVLI, Tokyo University) the existence of ℓ-adic companions to overconvergent F-isocrystals. The method and the result were already used in a number of applications. Moreover Kedlaya proved afterwards the same theorem a using different method. Simpson’s conjecture: rigid complex local systems on smooth projective varieties are integral. We could use in a first proof isocrystals and their ℓ-adic companions to prove it with Michael Groechenig, and found later on a shorter proof solely based on Drinfeld’s ℓ-adic a companions. The result has a number of applications, e.g. in Köhler geometry.

Publications

• (2016). Some fundamental groups in arithmetic geometry. Proceedings of Symposia in Pure Mathematics 97 (2) 169-179
  Esnault, Hélène
  
• (2018) Convergent isocrystals on simply connected varieties. Ann. inst. Fourier (Annales de l’institut Fourier) 68 (5) 2109–2148
  Esnault, Hélène; Shiho, Atsushi
  
• (2019) A Relative Version of Gieseker’s Problem on Stratifications in Characteristic p>0. International Mathematics Research Notices 2019 (18) 5635–5648
  Esnault, Hélène; Srinivas, Vasudevan
  
• (2019) Chern classes of crystals. Trans. Amer. Math. Soc. (Transactions of the American Mathematical Society) 371 (2) 1333–1358
  Esnault, Hélène; Shiho, Atsushi
  
• Simply connected varieties in characteristic p > 0 , with an appendix by Jean-Benoît Bost. Compositio math. 152 (2016), 255–287
  Hélène Esnault with V. Srinivas
  
• A remark on Deligne’s finiteness theorem. Int. Math. Res. Not. 16 (2017), 4962–4970
  Hélène Esnault
  
• Survey on some aspects of Lefschetz theorems in algebraic geometry. Revista Matemática Complutense, 30 (2) (2017), 217–232
  Hélène Esnault
  
• (2018): D-modules and finite monodromy. In: Sel. Math. New Ser. 24 (1), S. 145–155
  Hélène Esnault with M. Kisin