Algebraic geometry and cohomology in mixed characteristic
15-19 May 2023 @ Northwestern
Occasion
This workshop is being organized in celebration of Bhargav Bhatt’s Nemmers Prize.
Speakers
Emelie Arvidsson (Utah)
Daniel Bragg (Utah)
Hélène Esnault (FU Berlin)
Sarah Frei (Dartmouth)
Daniel Halpern-Leistner (Cornell)
Luc Illusie (Orsay)
Kiran Kedlaya (San Diego)
Jacob Lurie (IAS)
Akhil Mathew (Chicago)
Shubhodip Mondal (MPIM)
Wiesia Nizioł (Jussieu)
Arthur Ogus (Berkeley)
Martin Olsson (Berkeley)
Alex Petrov (Clay/MPIM)
Karl Schwede (Utah)
Jakub Witaszek (Princeton)
Bogdan Zavyalov (IAS)
* = to be confirmed
Organizers
Benjamin Antieau, Bao Le Hung, and Yuchen Liu.
Registration
Please register here. Note: no additional funding is possible.
Warning: there might be a spam email going out to some people about conference hotel booking.
We will announce details from our @northwestern.edu
addresses when the time comes.
Funding
Some support is available for reimbursing participants’ lodging and travel expenses. This support is primarily for students and postdocs. Decisions were to be made by 28 February 2023, but will be a few days late; they will be sent out on 13 March 2023.
Lodging
All participants are asked to book their own hotel, whether or not they will be reimbursed. The talks conference will take place at the Hilton Orrington; the link gives access to a special rate for rooms available until 1 May 2023.
Travel
Northwestern is accessible via Amtrak through Chicago Union Station and via plane through Chicago O’Hare or Midway airports. One can take a cab from any of these stations or take public transportation on the Red or Purple ‘L’ Lines or on the UP-N Line of the Metra.
Parking
Parking is available at an Evanston Public Parking Garage nearby to the hotel and elsewhere around Evanston.
Room
Talks will take place in the Orrington Room at the Hilton Orrington.
WiFi
Hilton Meeting, Password: meetings23
Reception
The reception will take place from 1700 to 1900 Monday evening on campus in Harris 108 and is open to everyone attending the conference.
Tentative schedule
M | Tu | W | Th | F |
---|---|---|---|---|
0900 Coffee | 0900 Coffee | 0900 Coffee | 0900 Coffee | 0900 Coffee |
0930 Kedlaya | 0930 Lurie | 0930 Schwede | 0930 Nizioł | 0930 Halpern-Leistner |
1030 Coffee | 1030 Coffee | 1030 Coffee | 1030 Coffee | 1030 Coffee |
1130 Bragg | 1130 Esnault | 1130 Mathew | 1130 Arvidsson | 1100 Illusie |
1230 Lunch | 1230 Lunch | Free afternoon | 1230 Lunch | 1200 Break |
1415 Frei | 1430 Ogus | 1430 Witaszek | 1215 Petrov | |
1515 Coffee | 1530 Coffee | 1530 Coffee | ||
1545 Mondal | 1630 Zavyalov | 1630 Olsson | ||
1700 Reception |
Titles and abstracts
Arvidsson, Properties of log canonical singularities in positive characteristic. We will investigate if some well known properties of log canonical singularities over the complex numbers still hold true over perfect fields of positive characteristic and over excellent rings with perfect residue fields. We will discuss both pathological behavior in characteristic p as well as some positive results for threefolds. We will see that the pathological behavior of these singularities in positive characteristic is closely linked to the failure of Kodaira-type vanishing theorems in positive characteristic. Additionally, we will explore how these questions are related to the moduli theory of varieties of general type.
This is based on joint work with F. Bernasconi and Zs. Patakfalvi, as well as joint work with Q. Posva.
Bragg, A Stacky Murphy’s Law for the Stack of Curves. We show that every Deligne-Mumford gerbe over a field occurs as the residual gerbe of a point of the moduli stack of curves. Informally, this means that the moduli space of curves fails to be a fine moduli space in every possible way. We also show the same result for a list of other natural moduli problems. This is joint work with Max Lieblich.
Esnault, Crystallinity properties of complex rigid local systems. Joint work in progress with Michael Groechenig.
We prove in all generality that on a smooth complex quasi-projective variety $X$, rigid connections yield $F$-isocrystals on almost all good reductions $X_{\mathbb F_q}$ and that rigid local systems yield crystalline local systems on $X_K$ for $K$ the field of fractions of the Witt vectors of a finite field $\mathbb F_q$, for almost all $X_{\mathbb F_q}$. This improves our earlier work where, if $X$ was not projective, we assumed a strong cohomological condition (which is fulfilled for Shimura varieties of real rank $\geq 2$), and we obtained only infinitely many $\mathbb F_q$ of growing characteristic. While the earlier proof was via characteristic $p$, the new one is purely $p$-adic and uses $p$-adic topology. This is work in progress and we shall discuss the projective case during the lecture.
Frei, Symplectic involutions of Kummer-type fourfolds. The middle cohomology of hyperkahler fourfolds of Kummer type was studied by Hassett and Tschinkel, who showed that a large portion is generated by cycle classes of fixed loci of symplectic involutions. In recent joint work with Katrina Honigs, we study Kummer-type fourfolds over arbitrary fields which arise as moduli spaces of stable sheaves on an abelian surface. We have extended the results of Hassett and Tschinkel and characterized the Galois action on the even cohomology. We do this by giving an explicit description of the symplectic involutions on the fourfolds. This has natural consequences for derived equivalences between Kummer fourfolds.
Halpern-Leistner, The noncommutative minimal model program (slides). There are many situations in which the derived category of coherent sheaves on a smooth projective variety admits a semiorthogonal decomposition that reflects something interesting about its geometry. I will present a new unifying framework for studying these semiorthogonal decompositions using Bridgeland stability conditions. Then, I will formulate some conjectures about canonical flows on the space of stability conditions that imply several important conjectures on the structure of derived categories, such as the D-equivalence conjecture and Dubrovin’s conjecture.
Illusie, Non-decomposability of the de Rham complex and non-semisimplicity of the Sen operator, after A. Petrov. Let $k$ be a perfect field of characteristic $p>0$. A. Petrov has recently constructed a projective, smooth scheme $X/W(k)$, of relative dimension $p+1$, such that the Hodge to de Rham spectral sequence of $X_0 = X \otimes k$ does not degenerate at $E_1$, thus solving a question raised by Deligne-Illusie in 1987. In addition, Petrov showed that the Sen operator of Bhatt-Lurie on $X_0$ is not semisimple. The purpose of this and the next lecture is to give an idea of the main points and techniques in the proofs, as well as to discuss a number of related results pertaining to de Rham cohomology of smooth schemes over $k$ and their liftings over $W_2(k)$ or $W(k)$. I will present Petrov’s example, and give an idea of the strategy of its proof. In particular, for $X_0/k$ smooth, lifted to $X_1$ over $W_2(k)$, I will state Petrov’s basic formula relating the first possibly non-trivial extension class in the canonical filtration of the de Rham complex of $X_0/k$ to the obstruction to lifting Frobenius of $X_0$ to $X_1$ and a certain cohomology class arising from the discrepancy between $S^p$ and $\Gamma^p$.
Kedlaya, Algebraic Frobenius structures. Let $X$ be a smooth projective variety over $Q$. Let $Z$ be a strict normal crossings divisor on $X$. Then for any vector bundle $E$ on $X$ equipped with a logarithmic (for $Z$) integrable connection, for almost all primes $p$ we construct another vector bundle with logarithmic connection which interpolates the pullbacks along all analytic Frobenius lifts on $X$ (in a sense which we will make precise in the talk). In particular, when the connection is of geometric origin, we obtain an isomorphism of vector bundles with connection which constitutes an “algebraic Frobenius structure”. Conversely, in certain cases one can establish this isomorphism directly to show that E is an isocrystal, e.g., when the connection is rigid (recovering a result of Esnault-Groechenig) or superrigid (extending work of Klevdal-Patrikis). Moreover, in certain examples the algebraic Frobenius structures for different primes fit together in a manner that as yet lacks a general explanation.
Lurie, Deligne-Illusie from the Prismatic Perspective (slides). Let X be a projective algebraic variety over the field of complex numbers. A central result of Hodge theory is that every cohomology class on X can be represented (uniquely) by a harmonic differential form. This has an algebraic consequence: the Hodge-to-de-Rham spectral sequence of X degenerates. In the late 1980’s, Deligne and Illusie gave a purely algebraic proof of the latter statement, using positive characteristic algebraic geometry in place of classical Hodge theory. In this talk, I’ll describe some recent joint work with Bhargav Bhatt which offers a new point of view on the work of Deligne-Illusie, using the theory of prismatic cohomology.
Mathew, Nonconnective animated rings and affine stacks. I will discuss a theory of “derived rings” (a nonconnective generalization of animated or simplicial commutative rings) and their connection with affine stacks in the sense of Toën. Based on joint work with Bhargav Bhatt, work of Arpon Raksit, and joint work with Shubhodip Mondal.
Mondal, Unipotent homotopy theory of schemes. Building on Toen’s work on higher (affine) stacks, I will discuss a notion of homotopy theory for algebraic varieties, which we call `unipotent homotopy theory’. Over a field of char. $p>0$, I will explain how our unipotent homotopy group schemes recover (1) unipotent completion of the Nori fundamental group scheme, (2) $p$-adic ‘etale homotopy groups, and (3) certain formal group laws arising from algebraic varieties constructed by Artin–Mazur. Time permitting, I will discuss unipotent homotopy types of Calabi–Yau varieties and show that the unipotent homotopy group schemes $\pi^U_i$ of Calabi–Yau varieties (of dimension n) are derived invariant for all $i$; the case $i = n$ corresponds to a recent result of Antieau–Bragg. This is joint work with Emanuel Reinecke.
Nizioł, Duality theory for p-adic pro-etale cohomology of analytic curves. It is well known that local Tate duality extends to $p$-adic etale cohomology of schemes over local fields. In a joint work with Pierre Colmez and Sally Gilles we conjecture that it also extends to p-adic pro-etale cohomology of analytic spaces and prove it for analytic curves. I will discuss this work and, if time permits, its geometric analog.
Ogus, Diffracting prisms: resolution of a quandary (slides). Let $X/k$ be a smooth scheme over a perfect field $k$ of characteristic $p > 0$ and let $W$ be the Witt ring of $k$. Inspired by Drinfeld’s ‘‘stacky’’ approach to prismatic cohomology, Bhatt, Lurie, and others have shown that a lifting of $X/k$ to $W$ induces a canonical action of the divided power version of $\Gm$ on $R\Gamma(X,\Omega^\bullet_{X/k})$. The $\bZ/p\bZ$-grading associated to the restriction of this action to $\mu_p$ generalizes the Deligne–Illusie decomposition theorem. I will explain several explicit geometric constructions of this action and how discussions with Bhatt and Vologodsky clarified the relationships among them. This will not be a formal research talk, but rather a story still in progress.
Olsson, Homological descent and base change for stacks. Classically for a morphism $f:X\rightarrow Y$ of schemes the functor \(f_!:D(X)\rightarrow D(Y)\) on derived categories of 'etale sheaves is defined using compactifications and ordinary pushforward \(f_*\) for proper morphisms. Verdier duality relates this functor to $f_*$, and this observation was used in earlier joint work with Laszlo to define $f_!$ for a morphism of algebraic stacks. However, this approach has some deficiencies; in particular, it does not readily give the proper base change theorem in the context of stacks. The base change theorem was subsequently proven by Liu and Zheng using other methods. In this talk I will explain an alternate approach to $f_!$, which in particular gives the proper base change theorem, based on homological descent - a theory whose existence was suggested by Gabber some years ago. This is joint work with Bhargav Bhatt.
Petrov, Non-decomposability of the de Rham complex and non-semisimplicity of the Sen operator II. This is a continuation of Luc Illusie’s talk. I will describe a general algebraic approach to analyzing extensions in the canonical filtration on a complex admitting a certain special type of a derived commutative algebra structure, applicable e.g. to diffracted Hodge complex of a smooth scheme over $W(k)$, or de Rham, Hodge, or etale cohomology of an abelian scheme. The approach relies on the analog of Steenrod operations for these derived commutative algebras. As an application, we will see a discrepancy between equivariant Hodge and de Rham cohomology of an abelian variety equipped with an action of a discrete group. This discrepancy gives rise to an example of a liftable variety with a non-degenerate Hodge-to-de Rham spectral sequence.
Schwede, Perfectoid signature and an application to étale fundamental groups (slides). In characteristic $p > 0$ commutative algebra, the $F$-signature measures how close a strongly $F$-regular ring is from being non-singular. Here $F$-regular singularities are a characteristic $p > 0$ analog of klt singularities. In this talk, using the perfectoidization of Bhatt–Scholze, we will introduce a mixed characteristic analog of $F$-signature and Hilbert–Kunz multiplicity. As an application, we show it can be used to provide an explicit upper bound on the size of the étale fundamental group of the regular locus of a BCM-regular singularities (related to results of Xu, Braun, Zhuang, Carvajal–Rojas, Tucker, Bhatt–Gabber–Olsson, and others in characteristic zero and characteristic p). BCM-regular singularities, as introduced by Pérez-R.G. and Ma and myself, can be thought of as a mixed characteristic analog of klt and $F$-regular singularities from characteristic zero or $p > 0$ respectively. This is joint work with Hanlin Cai, Seungsu Lee, Linquan Ma and Kevin Tucker.
Witaszek, Singularities in mixed characteristic via the Riemann-Hilbert correspondence (slides). In my talk, I will start by reviewing how various properties of characteristic zero singularities can be understood topologically by ways of the Riemann-Hilbert correspondence. After that, I will explain how similar ideas can be applied in the study of mixed characteristic singularities. This is based on a joint work (in progress) with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, and Joe Waldron.
Zavyalov, Poincare Duality in an abstract 6-functor formalism. In this talk, I will discuss Poincare Duality in the context of an abstract 6-functor formalism. Somewhat surprisingly, a 6-functor formalism satisfies an appropriate form of Poincare Duality under a minimal set of extra assumptions. Furthermore, these assumptions are essentially independent of the ``coefficient’’ categories D(X). This makes it easy to verify these assumptions in practice. In particular, this allows us to reprove previously established Poincare Duality results in a uniform and almost formal way.
Acknowledgments
The organizers thank the Nemmers Prize Fund and the Northwestern Department of Mathematics for funding. The image above is due to Bill Smith and used under CC BY 2.0.