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.

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.

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