AG Seminar

Abstract: Johannes Krah showed that the blowup of P^2 in 10 general points admits a phantom subcategory. We construct three types of objects in such a phantom: a strong generator, projections of skyscraper sheaves, and a family of objects with two nonzero cohomology sheaves. We study the deformation theory of these objects to show that the phantom contains rich geometry, such as encoding the blowdown map to P^2. We also show that there exists a co-connective dg-algebra whose derived category is a phantom.

Abstract: In this talk I will present some enhancements and generalizations of a criterion for six-functor formalisms first sketched by Voevodsky in 2001. This principle was then implemented by Ayoub in order to show that the stable motivic homotopy theory of quasi-projective schemes has the structure of a six-functor formalism, although it has later been generalized by works of Cisinski, Déglise, Hoyois, Khan, and Ravi leading to a six-functor formalism of genuine stable motivic homotopy theory on qcqs derived algebraic stacks with separated diagonals and nice stabilizers.

In the proposed framework, we produce six-functor formalism using the cohomological behaviour of smooth maps, closed immersions, and smooth proper maps (where the relevant cohomological property is expressed by a version of Atiyah duality). This is related to recent results of Dauser-Kuijper and Cnossen-Lenz-Linskens, which enhances  Mann's result (following Liu-Zheng) on the construction of six-functor formalisms using the cohomological behaviour of étale maps and proper maps.

Abstract: For each (isogeny class of) simple exceptional algebraic group G, Serre asked if there exist motives with Galois group G. We now have a complete answer to this question, due to work of many authors: all exceptional groups do in fact appear. A stronger question is the function field analogue: does there exist a motivic local system with monodromy group G? In this case, the answer was known except in the case G has type E_6. I'll explain joint work with Krämer and Maculan constructing infinitely many essentially distinct motivic local systems with monodromy group E_6.

Abstract: Given a family of complex algebraic varieties and a path in the base, flat connections on the fibres carry an operation of isomonodromic deformation: choosing a path in the base, we can deform a flat connection from one fibre to another along this path while keeping the underlying monodromy representation constant.

We solve the problem of upgrading this operation of isomonodromic deformation along a path to a functor between categories of flat connections with logarithmic singularities along a divisor D. The main tool used is the twisted fundamental groupoid \Pi_1(X,D). As applications, (1) we get that isomonodromy gives a map of moduli stacks of flat connections with logarithmic singularities, (2) we encode higher homotopical information at level 2, i.e. we get an action of the fundamental 2-groupoid of the base of our family on the categories of logarithmic flat connections on the fibres, and (3) our methods produce a geometric incarnation of the isomonodromy functors as Morita equivalences which are more primary than the isomonodromy functors themselves, and from which they can be formally extracted by passing to representation categories.

In this replacement talk, I will explain Tian's remarkable proof that any zero cycle that passes the Brauer-Manin test is global for any rationally connected surfaces over function fields of curves.

Abstract: Combinatorial (or “cut-and-paste”) K-theory is a modern approach to the study of classical scissors congruence group, and can be applied to other geometric settings as well, such as the categories of varieties and semi-algebraic sets. We present the K-theory of squares category as a framework that unifies Waldhausen K-theory as well as many instances of combinatorial K-theory in a natural way. As an application, we lift the Euler characteristic for definable sets in an o-minimal structure to a map of K-theory spectra.

Abstract: This talk will focus on descent and excision of cohomology theories of schemes. We will start with a discussion of the canonical topology on spectral schemes. Unlike on classical schemes, this topology includes many other types of covers, such as h-covers. Then I will explain that THH and TC satisfy descent with respect to the canonical topology, which generalizes the flat descent by Bhatt—Morrow—Scholze. This in turn implies the cdh descent of K-theory on spectral schemes, despite its failure on classical schemes. Furthermore, this implies the cdh pro-excision of K-theory on spectral schemes, which generalizes the derived case proven by Kelly—Saito—Tamme (the original noetherian case is due to Kerz-Strunk—Tamme). Our proof of the cdh pro-excision is quite different from the previous ones and is more algebraic in nature. The results presented here are based on discussions with Antieau, Burklund, and Krause.

Abstract: The property (T) conjecture for mapping class groups predicts that finite dimensional unitary local systems on moduli stacks of curves $\mathcal{M}_{g,n}$ for $g\geq 3$ are rigid (in the sense that they admit no infinitesimal deformations). While extensively studied for local systems with finite monodromy, a special case known as the Ivanov conjecture, much less is known when the monodromy is infinite.

We establish rigidity of local systems of conformal blocks arising from SU(2) and SO(3) modular categories, over $\mathcal{M}_g$ for $g\geq 7$ and at conformal levels $\ell$ such that $\ell+2$ is prime and at least $5$. These are natural infinite monodromy examples arising in quantum topology via the Witten-Reshetikhin-Turaev construction or alternatively in algebraic geometry via non-abelian theta functions.

The core of our argument is a proof that any infinitesimal deformation of a conformal block local system, within the space of all flat unitary local systems, necessarily remains a conformal block local system. This then implies triviality, since conformal block local systems admit no such internal deformations by a result known as Ocneanu rigidity. The proof combines the factorization property of conformal blocks with elementary Hodge theory on certain root stacks over $\overline{\mathcal{M}}_{g,n}$, over which conformal block local systems extend.