AG Seminar

Abstract:  I will talk about my recent joint work with Piotr Pstragowski, where we explain how to produce weight filtrations on various cohomology theories without inverting residual characteristics in the coefficients. In particular, our results provide weight filtrations on all known logarithmic cohomology groups of projective sncd pairs (X,D) over Dedekind domains. For de Rham cohomology (resp. Hodge filtered de Rham cohomology, resp. Hodge cohomology), we show that our construction recovers the décalaged pole-order filtration originally defined by Deligne. Since our weight filtration is an invariant of the open part U=X\D, this proves (without assuming resolution of singularities) that the décalaged pole-order filtration is an invariant of U, extending a theorem of Deligne from characteristic zero to positive and mixed characteristic. To demonstrate the computability of these invariants, I will present explicit examples. If time permits, I will also explain how to extend our methods to singular schemes.

Abstract: Let X be a smooth algebraic variety over a perfect field k of characteristic p, equipped with a flat lift over W_2​(k). Drinfeld discovered that the de Rham complex of X, viewed as an object in the derived category of sheaves on X, carries a canonical endomorphism called the Sen operator, which acts on the cohomology sheaves by multiplication by the degree. This construction refines the celebrated theorem of Deligne and Illusie. In this talk, I will present a reformulation of Drinfeld’s construction in a form more amenable to computation.

The talk is based on joint work with Arthur Ogus, Nick Rozenblyum, and Gleb Terentiuk.

Abstract: Motivated by Dieudonné theory, V. Drinfeld and E. Lau introduced a "decompletion" of the ring of Witt vectors W(R) of a derived p-complete ring R such that (R/p)_{red} is perfect, extending a construction of T. Zink. I will explain various characterizations of this decompletion (called the sheared Witt vectors) and some examples. (Joint work in progress with Bhargav Bhatt, Vadim Vologodsky, and Mingjia Zhang.)

Abstract: We will discuss a proof that the integral Hodge conjecture is false for a very general abelian variety of dimension ≥ 4. Associated to any regular matroid is a degeneration of principally polarized abelian varieties. We introduce a new combinatorial invariant of regular matroids, which obstructs the algebraicity of the minimal curve class, on the very general fiber of the associated degeneration. In concert with a result of Voisin, one deduces (via the intermediate Jacobian) the stable irrationality of a very general cubic threefold. This is joint work with Olivier de Gaay Fortman, and Stefan Schreieder.

Abstract: The classical Hurwitz moduli spaces classify covers f:X'-->X, where X' and X are smooth proper curves of genera g' and g, and the ramification pattern is fixed. A natural question is if this moduli space possesses a natural modular compactfication. Of course, the first guess is to imitate the Deligne-Mumford compactification by taking appropriate covers of nodal (or log smooth) curves. In the tame case this works fine, and indeed the moduli space of admissible (or log smooth covers) was constructed by Harris-Morrison and refined by Abramovich-Vistoli. In particular, Abramovich-Vistoli defined a proper moduli stack, which is smooth, thereby indicating that this compactification is the "right" one.

In the case of positive characteristics things do not work well as inseparable maps over some components can, and in fact, must show up, as was shown by an example of Abramovich-Oort. For this reason, the question was considered too wild and intractable. Nevertheless, the case of covers of degree p is now established in a few important cases in a work in progress of M. Hippold,  and it seems that the general case will be solved. The story goes as follows: in a series of papers with my former students we established an analogue of semistable reduction for covers of curves of degree p. Such reduction naturally comes equipped with an additional datum of a relative differential form on each inseparable component of the reduction cover. This suggested the correct guess for the proper logarithmic functor, and Hippold proved that it is representable (as a log scheme), and (in some cases) log smooth over Spec(Z_p) with the log structure given by p.

In my talk I will outline the key ingredients of this story, though most of the time we will discuss the local/non-archimedean picture behind the semistable reduction of wild covers. This is the case where a few concrete examples can and will be discussed.

For those who cannot join us in person, you can also join us via Zoom.

Zoom link: 899 7344 1856

Passcode: 240363

Abstract: For a smooth variety X over a field k and a smooth k-group scheme G, Grothendieck and Serre predicted that every generically trivial G-torsor over X trivializes Zariski locally on X. I will explain a resolution of the Grothendieck--Serre problem, the main new case being when k is imperfect, in which pseudo-reductive and quasi-reductive groups play a central role. The argument is built on new purity and extension theorems for torsors valid for pseudo-finite, pseudo-proper, and pseudo-complete groups, and it also rests on several other new results on algebraic groups in positive characteristic. The talk is based on joint work with Alexis Bouthier and Federico Scavia. 

I will explain how to think about Hecke operators acting on

moduli spaces of bundles on a curve over a local non-archimedian

field. I will also explain how to define a homomorphism from the algebra of

such operators  to the usual spherical Hecke algebra (to be defined at

the talk). Computing this homomorphism explicitly is a special case of some

general problem related to p-adic (or motivic) integration on singular

schemes.

We introduce the notion of a minimal energy local system on a curve. These local systems come from complex variations of Hodge structures, and most complex variations of Hodge structures are of minimal energy. We describe their Hodge filtrations and discuss how they are related to the topology of real (relative) character varieties.

There are many attempts to find a "deeper base" for arithmetic than the natural numbers, incorporating more combinatorics. One approach is to replace the natural number $n$ with the $q$-analogue $(n)_q := 1 + q + \dotsb + q^{n-1}$; such $q$-analogues have recently been linked to prismatic cohomology and formal group laws. Another is to replace the category of finite sets by the category of finite $G$-sets for a group $G$, then decategorify. I will discuss recent work linking these two approaches, which in particular clarifies some long-standing issues with the symbol ``$f^{(n)_q}$''. In technical language, our construction takes a $\lambda$-ring equipped with a compatible formal group law and produces an $S^1$-Tambara functor.

Cohomology of algebraic varieties over bases of mixed and positive characteristic p generally exhibit different behavior depending on whether the cohomological degree in question is larger or smaller than the prime number p. For example, if X is a smooth proper scheme over Z_p then for n<p-1 the n-th singular cohomology of its generic fiber with coefficients in Z_p is isomorphic to its de Rham cohomology, and for n<p the n-th de Rham cohomology of the mod p reduction of X naturally decomposes into a direct sum of its Hodge cohomology groups. Both of these properties fail in larger cohomological degrees, and I will describe a mechanism for producing examples where they fail based on the construction of Steenrod operations on cohomology of topological spaces. This is joint work with Shizhang Li.

We give a friendly introduction to the theory of higher Du Bois singularities for local complete intersections, and explain their connections with K-theory.

We develop a theory for higher Du Bois and higher rational singularities for general varieties (beyond local complete intersection) and investigate their properties.

Clausen and Scholze showed that a condensed form of Whitehead's problem has a positive answer.  Recently, Bergfalk, Lambie-Hanson and Saroch gave a set theoretic formulation and proof of this theorem.  We give an alternative set theoretic proof which shows a close connection with a proof of Shelah's theorem that V=L implies that all Whitehead groups are free.

There is a well-known theory of moduli of vector bundles on curves. Analogous construction for stacky curves turns out to be subtler: for example, the naïvely defined slope no longer gives bounded semistable locus. In this talk, I will speak about an alternative way of defining semistability, prove algebraicity of the moduli stack of vector bundles and boundedness of its semistable locus, and use the rapidly developing theory of Beyond GIT to show that the semistable locus admits a good moduli space. This work is joint with Chiara Damiolini, Vicky Hoskins and Lisanne Taams.

A global Torelli theorem does not hold for Calabi–Yau 3-folds: the best

counterexample, due independently to Ottem and Rennemo and to Borisov,

Caldararu, and Perry, gives pairs of CY3s in the same deformation family

that have equivalent derived categories of coherent sheaves, hence

isomorphic polarized Hodge structures on H^3(X,Z), but are not birationl.

One might then ask whether a derived Torelli theorem holds. I will discuss

an example of Aspinwall, Morrison, and Szendrői, which was expected to

provide a counterexample; it turns out not to, but fits into the whole

picture in a surprising way.  This is joint work with Ben Tighe,

arXiv:2407.11176.

Étale rigidity for motivic spectra' is the statement that (for certain base schemes S) any p-complete A^1-invariant étale hypersheaf of spectra on Sm_S is ‚small‘,

i.e. comes from the small étale site S_et. This is a deep result proven by Bachmann, building on work of Suslin-Voevodsky, Ayoub, and Cisinski-Déglise.

In this talk, I will explain how to generalize this rigidity result to the unstable setting: I will show that certain p-complete A^1-invariant étale hypersheaves of anima are in fact coming from the small étale oo-topos.

If time permits, I will use this rigidity result to prove an étale version of Morel’s theorem that strongly A^1-invariant Nisnevich sheaves of abelian groups are strictly A^1-invariant. 

Clausen and Scholze showed that a condensed form of Whitehead's problem has a positive answer.  Recently, Bergfalk, Lambie-Hanson and Saroch gave a set theoretic formulation and proof of this theorem.  We give an alternative set theoretic proof which shows a close connection with a proof of Shelah's theorem that V=L implies that all Whitehead groups are free.

Abstract: Grothendieck’s section conjecture predicts that rational points on hyperbolic curves over some number fields are controlled by sections of a certain short exact (\’{e}tale) fundamental groups. In this talk we formulate two analogues of Grothendieck’s section conjecture over the complex numbers, with the first one being more toplogical and the second one Hodge theoretic. Furthermore, just like the arithmetic setting, we prove some injectivity results for special family of curves known as Kodaira fibrations in the topological setting, and for any family of curves in the Hodge theoretic setting.

Abstract:  In this talk, I will report about joint work with Can Yaylali (TU Darmstadt) towards A^1-homotopy theory of rigid analytic spaces [arXiv:2407.09606]. In the beginning, I will recall the motivating algebraic theory such as Morel-Voevodsky's seminal work on unstable A^1-homotopy of schemes, Voevodsky's stable version, and Ayoub's proof of a six-functor formalism for that. We seek to study a rigid analytic analogue using the rigid affine line A^1 as an interval. For this purpose, I will give some background on rigid analytic spaces where there are (at least) two canonical interval objects for doing homotopy theory, the closed unit disc B^1 and the rigid affine line A^1. The B^1-homotopy category has already been defined and studied by Ayoub and a full six-functor formalism was established by Ayoub-Gallauer-Vezzani. One drawback of the B^1-invariant theory is that analytic K-theory for rigid analytic spaces (as defined and studied by Kerz-Saito-Tamme) is not representable since it is not B^1-invariant. Thus we study an A^1-invariant version with coefficients in any presentable category. For the stable theory, we can prove the existence of a partial six-functor formalism for analytifications of schemes and algebraic morphisms between them by using the results of Ayoub's thesis. Furthermore, using coefficients in condensed categories, we render analytic K-theory representable in the unstable category and identify it with Z x BGL, in analogy to the case of schemes.

Abstract: Multiplicative affine Springer fibers are group-theoretic analogues of Lie-algebra-valued affine Springer fibers. They parametrize certain Higgs bundles that are valued in the Vinberg monoid. Since 2011, there has been an ongoing investigation into these fibers and their applications to orbital integrals, the Hitchin fibration and the Fundamental Lemma. In my talk, I will introduce parabolic multiplicative affine Springer fibers and discuss their properties, as well as their applications to global Springer theory. 

Abstract:  Recently, Annala, Hoyois, and Iwasa have defined and studied the 𝐏¹-homotopy theory, a generalization of 𝐀¹-homotopy theory that does not require 𝐀¹ to be contractible, but only requires pointed 𝐏¹ to be invertible. I will recall basic facts in their theory, and construct the 𝐏¹-motivic Gysin map. If time permits, I will also explain some applications such as Atiyah duality and Steinberg relation in 𝐏¹-homotopy theory. 

Abstract:  The goal of my talk is to explain a geometric proof of Hrushovski's

generalization of the Lang-Weil estimates on the number of points in the 

intersection of a correspondence with the graph of Frobenius. 

This is a joint work with K. V. Shuddhodan.

Abstract: Given a symplectic resolution Y, we are interested in the equivariant

quantum cohomology ring.  This ring depends on a parameter which lives

in an open subset of the second cohomology of Y.  I will explain a

general program (joint with Lenoid Rybnikov) for compactifying this

parameter space.  When Y is the resolution of an affine Grassmannian

slice, then (conjecturally) these quantum cohomology rings are

trigonometric Gaudin algebras and the compactified parameter space is

the Deligne-Mumford compactification of the moduli space of marked

genus 0 curves.  In this case, we study the monodromy of the

eigenvectors of these algebras as we move around the real locus of this parameter space

Abstract: Work of Schneider, Neukirch and Milne provides formulas for integer values of zeta functions associated to smooth proper schemes over a finite field in terms of Euler characteristics on p-adic cohomology theories.  Later, Geisser extended this to general finite type schemes over a finite field under the assumption of strong resolution of singularities.  We will discuss a simplification of Milne’s contribution using the Nygaard filtration, as well as progress towards removing the assumption of resolution of singularities from Geisser’s proof.

Abstract: The celebrated Serre--Tate theorem says that deformations of an abelian variety are naturally parameterized in terms of deformation of the abelian variety's Barsotti--Tate group. In particular, this says that the natural functor from Mumford's moduli spaces of principally polarized abelian varieties to the moduli stack of Barsotti--Tate groups is formally étale. In this talk I will discuss joint work with Naoki Imai and Hiroki Kato which shows a similar result holds true for integral canonical models of arbitrary Shimura varieties of abelian type (at hyperspecial level), and how this uniquely characterizes such models (at individual level). This involves the construction of a 'syntomic realization functor' on such integral canonical models.

Abstract: Classically, Dieudonné theory offers a linear algebraic classification of finite group schemes and p-divisible groups over a perfect field of characteristic p>0. In this talk, I will discuss generalizations of this story from the perspective of p-adic cohomology theory (such as crystalline cohomology, and the newly developed prismatic cohomology due to Bhatt--Scholze) of classifying stacks. Time permitting, I will discuss some applications.

Abstract: A complex variety Z is a rational homology manifold, also called rationally smooth, if the local cohomologies at every point are those of a sphere of dimension 2dim⁡(Z). While smooth varieties are rational homology manifolds, several singular varieties also satisfy this condition. These varieties exhibit interesting geometric properties, including Poincaré duality. We study a Hodge-theoretic weakening of this notion, which captures, for example, the difference between higher Du Bois and higher rational singularities—two classes of singularities that have recently attracted significant interest.

Abstract: I will describe a general local-to-global approach to studying D-modules on moduli spaces associated to

algebraic varieties and describe some applications, particularly to uniformization of Bun_G, the moduli stack of

principal G-bundles on a curve. As a consequence, I will describe the cohomology of Bun_G and give a derived

version of the Verlinde formula.

Abstract: Many important cohomology theories in algebraic geometry, such as Hodge cohomology, lack A^1-invariance. This limits their accessibility to the powerful methods of A^1-homotopy theory developed by Morel--Voevodsky. Together with Ryomei Iwasa and Marc Hoyois, we have introduced the theory of (non-A^1-invariant) stable motivic homotopy theory to remedy this, making all cohomology theories in algebraic geometry accessible to the methods of motivic homotopy theory. In this talk, I will explain our construction. As a specific application, I will highlight Atiyah duality, which is a vast generalization of Poincaré duality for smooth projective S-schemes. I will also explain how it allows proving that, e.g. the logarithmic de Rham cohomology of a projective SNC pair (X,D) is an invariant of the open part X\D. If time permits, I will also mention some recent refinements of the above result I have obtained in collaboration with Piotr Pstragowski.  

Absolute purity is a conjecture of Cisinski-Déglise and Grothendieck which, to me, is the crux conjecture for motives of regular schemes. I have no idea how to prove it, but I will explain what is at stake, the applications, some reductions and a new proof of the geometric case of smoooth schemes over a discrete valuation ring obtained in joint work with Bachmann and Morrow, relying on prismatic hoolabaloo.