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.