Lan Project

More details on the project will be provided here as they become available.

The working group will be led by Koji Shimizu, Rose Lopez, and Jeremy Taylor. To participate in this working group, send an email to one of these organizers.

Both students and faculty are encouraged to participate in the working groups.

The project meets at 1:30-5 pm at Evans 959.

Title

Logarithmic Riemann--Hilbert correspondences for rigid varieties

Abstract

I will give a survey on certain recent works with Hansheng Diao, Ruochuan Liu, and Xinwen Zhu in the smooth case, and (in progress) with David Sherman in the ideally log smooth case, on the logarithmic Riemann--Hilbert correspondences for rigid varieties, which can be viewed as p-adic analogues of the classical correspondences over complex numbers in works of Deligne and Illusie--Kato--Nakayama, among others. I will start with some review of the classical story and some background materials in the first part, and present our main results in the second part. The talks will focus on the overall pictures---some technical definitions, constructions, and examples will be left to the afternoon sessions of the workshop.

Project

The main goal of the project is to understand basic materials on this topic.

Basics in log geometry

References:

K. Kato, Logarithmic structures of Fontaine--Illusie.

L. Illusie, An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology.

A. Ogus, Lectures on logarithmic algebraic geometry.

H. Diao, K.-W. Lan, R. Liu, and X. Zhu, Logarithmic adic spaces: some foundational results.

(a) Understand the precise definitions of log smooth morphisms of fs log schemes and locally noetherian fs log adic spaces. Given some examples of morphisms X ->Y and Z -> Y whose fiber products in the coherent, fine, and fs categories are not the same.

(b) Consider the toric monoid P = \sigma \cap Z^3, where \sigma is the cone spanned by the four vectors (\pm 1, \pm 1, 1). Describe the stratification of X_P = Spec(k[P]) formed by X_Q = Spec(k[Q]), where Q runs over the faces of P and where the closed immersions X_Q -> X_P are defined by the ideals generated by I_Q = P \setminus Q. Verify that the locally closed strata X_Q^\circ = X_Q \setminus \cup_{Q' \subsetneq Q} X_{Q'} are smooth. Consider the log structure M_{X_P} on X_P induced by P -> k[P]. Describe the pullback of M_{X_P} to each X_Q^\circ. Describe d: O_{X_P} -> \Omega^{log, 1}_{X_P} (base field omitted) and its pullback to each X_Q^\circ.

(c) Redo (b) with X_P replaced with Spa(k<P>, k^+<P>), where k is a p-adic field and k^+ is its ring of integers, and with X_Q replaced with the strata defined by the ideals generated by I_Q. Explain whether X_Q^\circ is affinoid, for each face Q of P.

(d) Write down a finite Kummer étale cover of the X_P's in (b) and (c).

Classical Riemann--Hilbert correspondence over the complex numbers

References:

P. Deligne, Equations différentielles à points singuliers réguliers.

P. Deligne, Théorie de Hodge: II.

C. A. M. Peters and J. H. M. Steenbrink, Mixed Hodge Structures, especially section 4.

H. Esnault and E. Viehweg, Lectures on Vanishing Theorems, sections 2 and 3.

K. Kato and C. Nakayama, Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over C.

L. Illusie and K. Kato and C. Nakayama, Quasi-unipotent logarithmic Riemann--Hilbert correspondences.

A. Ogus, On the logarithmic Riemann--Hilbert correspondence.

(a) Write down some examples of connections over the affine line with regular singularities and without regular singularities.

(b) Consider U = Spec(k[T, T^{-1}]) -> X = \Spec(k[T]). Consider the finite étale covering U_m = Spec(k[T^{1/m}, T{-1/m}]), for some positive integer m invertible in k. What is the normalization of X in U_m? Let \pi: X_m -> X denote the canonical morphism, with restriction \pi_U to U_m. Let E = \pi_{U, *}(O_{U_m}), equipped with the connection \nabla induced by d: O_{U_m} -> \Omega_{U_m}^1 (base field omitted). When k = C, describe the local system L over U^{an} corresponding to (E, \nabla) under Deligne's classical Riemann-Hilbert correspondence. Describe the canonical extension of (E, \nabla) to X characterized by the condition that the eigenvalues of residues along the boundary divisor D = (X - U)_{red} (with the subscript "red" meaning "reduced") have real parts in [0, 1). What exactly are the eigenvalues of residues in this case?

(c) Describe the X in (b) as a log scheme with its log structure induced by D. Explain what the associated log complex analytic space X^{an} is, and describe (X^{an})^{log} -> X^{an} as constructed in Kato-Nakayama. Also, understand the meaning of unipotent and quasi-unipotent local systems on (X^{an})^{log}, as in Illusie-Kato-Nakayama. Does the local system L over U^{an} extend to (X^{an})^{log}? Is the extension unipotent? Is it quasi-unipotent? If so, describe the corresponding log connection over X^{an}_{k\'et}, and its pushforward to X^{an}_{an}. Furthermore, consider the ideally smooth log scheme D^\partial, which is the same underlying scheme D equipped with the nontrivial log structure pulled back from X. Is the above correspondence over X^{an} compatible with pullbacks to D^{an}?

Geometric de Rham period sheaves

References:

P. Scholze, p-adic Hodge theory for rigid-analytic varieties, and its corrigendum.

R. Liu and X. Zhu, Rigidity and a Riemann--Hilbert correspondence for p-adic local systems.

H. Diao, K.-W. Lan, R. Liu, and X. Zhu, Logarithmic Riemann--Hilbert correspondences for rigid varieties.

K.-W. Lan, R. Liu, and X. Zhu, De Rham comparison and Poincaré duality for rigid varieties.

(a) Let (k, k^+) = (Q_p, Z_p). For each positive integer k, let k_m = k(\zeta_m) and k^+_m denote its ring of integers. Consider the tower X_m = Spa(k_m<(1/m)P>, k_m^+<(1/m)P>) -> X = Spa(k<P>, k^+<P>) for suitable choices of P, such as P = Z_{\geq 0} or the cone in (1)(b) above, describe the perfectoid space \widehat{X} associated with the log affinoid perfectoid object \varprojlim_m X_m in X_{prok\'et}. Describe the values over \widehat{X} of the period sheaves introduced in Diao-Lan-Liu-Zhu.

(b) What is the pullback Y_m -> Y of the tower X_m -> X to a close subspace Y of X defined by an ideal I of P? Is it a log affinoid perfectoid object in Y_{prok\'et}? Note that the formations of fiber products in the worlds of schemes and analytic adic spaces behave differently. The former introduces nonreduced schemes in general, but the latter does not!

Some further examples

(a) For k = C or Q_p, consider the Tate curve E over the open punctured unit disc D. Describe the rank-two local system obtained by taking the first relative cohomology of this E over D_r with coefficients in C when k = C, and in Q_p when k = Q_p. When k = C, describe the corresponding integrable connection over D_r^* via the Riemann--Hilbert correspondence, and describe its canonical extension (characterized by eigenvalues of residues with real parts in [0, 1) as before) over the disc D_r, equipped with the log structure defined by the origin. Describe the compatibility of these with their pullbacks to the origin of D_r (equipped with the nontrivial log structure pulled back from D_r). Describe (without proof) a similar picture when k = Q_p.

(b) Can you describe the canonical extensions and their pullbacks to the origin in (a) as higher direct images of some log smooth morphism?