Seminar Algebra und Topologie
Das Seminar findet normalerweise am Mittwoch in F.13.11, 16:30 - 17:30 statt.
Beteiligte Dozenten: Jens Hornbostel, Sascha Orlik, Kay Rülling, Tobias Schmidt, Britta Späth, Matthias Wendt.
Bangxin Wang: Non-semisimple quantum invariants for 3-manifolds
Since the advent of Reshetikhin-Turaev invariants for 3-manifolds, the study of quantum invariants and its categorification has witnessed fruitful developments in both topology and algebra. In this talk we step out of the semi-simple setting and see how this allows us to capture richer topological and algebraic data. Early non-semisimple constructions like Hennings invariants and Lyubashenko invariants didn’t receive enough attention due to the fact that once the first Betti number is positive, the invariants always become 0. More recently, the discovery of modified trace made it possible to construct interesting non-semisimple invariants such as CGP invariants and DGGPR invariants. In my joint work with De Renzi and Martel we constructed a graded Hennings TQFT which incorporates homological information of cobordisms, and has the potential to capture more subtle information of mapping class groups of surfaces. With Runkel and Gainutdinov we are now trying to formulate this TQFT in Lyubashenko’s language with modified trace, leading to a non-semisimple version of Turaev’s homotopy QFT, which should include BCGP TQFT as an example. The algebraic tool that we need for this goal are G-crossed ribbon categories and twisted local modules. We provide a concrete example from unrolled quantum sl2.
Christopher Lazda: Boundedness of the p-primary torsion of the Brauer groups of K3 surfaces
The transcendental Brauer group of a variety X over a field k is the image of its Brauer group inside the Brauer group of the base change of X to a separable closure of k. If X is a K3 surface, and k is finitely generated of characteristic 0, then it was shown by Skorobogatov and Zarhin that this group is finite. If k is finitely generated of characteristic p (and X is again a K3 surface), then later work of Skorobogatov and Zarhin (in the case p =/=2) and Ito (in the case p=2) showed that its prime-to-p torsion subgroup is finite. One cannot in general expect finiteness of p-torsion in characteristic p, however, I will explain how to use Madapusi-Pera's proof of the Tate conjecture for K3 surface to show that one does have such a finiteness result in the case that X is non-supersingular. Combined with known results in the supersingular case, this shows that, in general, the p-torsion will always at least be of finite exponent, that is, annihilated by a fixed power of p. This is joint work with Alexei Skorobogatov.
Manuel Blickle: Perverse F_p sheaves and F-rational singularities
In this talk I will speak about one aspect of a joint work with Bhatt, Lyubeznik, Singh and Zhang which explains how notions of singularity, in particular Cohen-Macaulay-ness and F-rationality are reflected by the perversity or perverse simplicity of certain coherent sheaves with a Frobenius action. These in turn relate via a Riemann-Hilbert-Type correspondence to etale constructible F_p sheaves. Despite the fancy words in the title, the talk should be reasonably accessible already to advanced Master students with some background in commutative algebra.
Samuel Lerbet: A real analogue of the Hodge conjecture
In complex geometry, the cycle class map plays a major organisational role. It attaches to an irreducible (smooth) subvariety of codimension c of a smooth variety X its fundamental class with integer coefficients as an analytic subvariety of X(C), and the integral Hodge conjecture predicts that its image is precisely the subgroup of Hodge classes of type (c,c). In real geometry, orientability difficulties prevent a verbatim reproduction of this construction by simply substituting real loci for complex loci, at least with integer coefficients. The invariant obtained by replacing cohomology with integer coefficients by cohomology with modulo 2 coefficients, known as the Borel-Haefliger cycle class, is also much less refined than the complex cycle class map. In this talk, we will explain how, by replacing the Chow group of codimension c cycles by the Chow-Witt group of oriented cycles, a quadratic refinement of the former, we can define a lift of the Borel-Haefliger cycle class map with values in the integral cohomology of the real locus, which we call the real cycle class map. We will then state a precise conjecture giving a lower bound on the size of its image in terms of the exponents of its cokernel, which can be seen as a real analogue of Hodge's conjecture.
Daniel Marlowe: Deriving the higher K-theory of forms
Hermitian K-theory is a quadratic refinement of algebraic K-theory, bearing the same relation to oriented Chow groups that 'vanilla' K-theory does to the usual Chow groups. Classically an invariant of exact or triangulated categories with duality, hermitian K-theory has enjoyed a renaissance in recent years due to work of Calmès et al, building on ideas of Lurie, situating it in the realm of (Poincaré) localising invariants of stable infinity categories equipped with an appropriate notion of nondegenerate quadratic functor. In this talk, we explain how to bridge the gap between the 1-categorical world and the homotopy-coherent setting. Along the way, we'll discuss nonabelian derived functors of 2-polynomial functors, Z-linear models for stable categories, and recover the usual comparison result for algebraic K-theory. As an application, we'll sketch how the genuine symmetric hermitian K-theory of a divisorial scheme recover the hermitian K-theory of its on-the-nose symmetric forms.
Jefferson Baudin: A Grauert-Riemenschneider vanishing theorem for Witt canonical sheaves
Vanishing theorems are a powerful tool in understanding the geometry of characteristic zero varieties. A particularly useful one is Grauert-Riemenschneider vanishing, which asserts that if f : Y -> X is a projective birational morphism and Y is smooth, then higher pushfowards of \omega_Y vanish. On the other hand, this vanishing theorem famously fails in positive characteristic. In this talk, we will explain how to prove a Witt vector version of Grauert-Riemenchneider vanishing, solving a question of Blickle and Esnault and later studied by several authors. If time permits, we will give applications to singularities and to the cohomology of irregular ordinary varieties.
Luca Terenzi: The six functor formalism for perverse Nori motives
Let k be a field of characteristic zero. Conjecturally, there should exist an abelian category of mixed motives containing the universal cohomology groups of k-varieties; morphisms and extensions in this category should be deeply related to algebraic cycles. The theory of Nori motives provides an unconditional (albeit quite mysterious) candidate for such a category.
The conjectural theory of mixed motives should be part of a more general theory of mixed motivic sheaves over k-varieties, governed by Grothendieck's formalism of the six operations. In the last decade there have been various attempts at extending the theory of Nori motives in this direction.
After reviewing Nori's original theory in some detail, I will present the theory of perverse Nori motives, recently introduced by Ivorra--Morel. A complete six functor formalism is now available in this setting, by work of Ivorra--Morel and of myself; the final goal of my talk is to discuss the main ideas behind its construction.
Jakub Löwit: On equivariant K-theory, affine Grassmannian and perfection
Let G be an algebraic group acting on a scheme X. Then its equivariant algebraic K-theory maps into global invariant functions on a certain associated fixed-point scheme, which is of independent interest in geometric representation theory. After recalling this comparison and its relationship to equivariant Hochschild homology, I will explain what simplifies in characteristic p after perfection. The resulting map is an equivalence in interesting cases, such as torus-equivariant algebraic K-theory of split toric varieties or affine Schubert varieties in the perfect GL_n-affine Grassmannian.
Lucy Yang: Involutions and the Brauer group in derived algebraic geometry
Classical results of Albert and Saltman (extended by Knus–Parimala–Srinivas) have established a connection between the existence of (anti-)involutions on the Azumaya algebras A used to define the Brauer group and 2-torsion Brauer classes. Moreover, the presence of more general forms of involutions is related to the behavior of Brauer classes under corestriction along quadratic extensions. In this work, we investigate how the data of an involution on A is reflected in additional structure on its (derived) category of modules and introduce the notion of an involution on a generalized Azumaya algebra. Using the theory of Poincaré infinity-categories developed by Calmès–Dotto–Harpaz–Hebestreit–Land–Moi–Nardin–Nikolaus–Steimle, we introduce involutive versions of the Picard and Brauer group and relate them to their classical and non-involutive counterparts. This is joint work in progress with Viktor Burghardt and Noah Riggenbach.
Peter Schneider: Nonexistence of projective smooth representations in natural characteristic
Let G be a p-adic Lie group. We consider the category Mod(G) of smooth G-representations over a field k of characteristic p. In joint work with C. Sorensen we studied the nonexistence of nonzero projective objects in Mod(G). In this generality we show that this nonexistence holds provided the center of G is non-discrete. Note that some such condition is necessary since for finite G one always has enough projective objects. We have a completely general result, though, if G is the group of F-points of a connected reductive group over a finite extension F of Q_p. Then Mod(G) has no nonzero projective objects provided G is nontrivial.
One might object that such a negative result is not particularly exciting. Therefore I will explain in the second half of the talk the underlying technique which at first sight looks unrelated to the problem. This is the study of derived smooth induction functors from compact open subgroups of G, which has also applications to other questions.
Juan Esteban Rodriguez Camargo: D-modules on arc-stacks
In this talk, I will discuss a work in progress with Anschütz, Bosco, le Bras and Scholze concerning a new foundational framework for the theory of the analytic de Rham stack. In particular, we prove that under some mild finite dimensional conditions the formation of the analytic de Rham stack satisfies arc-descent. This allows us to define a very good behave theory of D-modules (from a 6-functor point of view) on a very large class of arc-stacks, that notably includes all those that appear in practice in p-adic Hodge theory and in the geometrization program of Fargues-Scholze. As an application, by looking at the de Rham stack of Div^1, we give a new proof of the p-adic monodromy theory (more precisely Crew's conjecture) that only uses the theory of Banach-Colmez spaces and the analytic de Rham stack.
