**Wintersemester 2018/19**

**Unless otherwise specified talks take place in room 119 (Arnimallee 3) at 16:15**

**Schedule**

**27th August 2018 Makiko Mase (Tokyo)**

**On dualities among families of K3 surfaces associated to strange duality of singularties**

**18th October 2018****Sebastian Posur (Siegen) **

**Categorical abstraction as a tool for computing with equivariant vector bundles**

Motivated by the difficult task of finding low rank indecomposable vector bundles on projective space, we discuss a construction strategy that exploits categorical abstraction as a powerful computational tool. We will make use of skeletal versions of the tensor category of representations of finite groups over a splitting field, categorically internalized versions of the exterior algebra, and an equivariant version of the BGG correspondence, which is an exact equivalence between the bounded derived category of coherent sheaves on projective space and the stabilization of the category of finitely presented modules over the graded exterior algebra. These computational methods are all provided by our computer algebra project CAP (Categories, Algorithms, Programming).

**25th October 2018****Nathan Ilten-Gee (Vancouver) **

**Deformations of smooth complete toric varieties: obstructions and the cup product**

**Abstract:** Let X be a smooth complete toric variety. I will explicitly describe the obstruction space and the cup product map in combinatorial terms. Using this, I give an example of a smooth projective toric threefolds for which the cup product map does not vanish, showing that in general, smooth complete toric varieties may have obstructed deformations. This is joint work with Charles Turo.

**1st November 2018****Dominic Bunnett (FU Berlin) **

Moduli of hypersurfaces in weighted projective space

**Abstract:** The moduli space of smooth hypersurfaces in projective space was constructed by Mumford in the 60’s using his newly developed classical (a.k.a. reductive) Geometric Invariant Theory. I wish to generalise this construction to hypersurfaces in weighted projective space (or more generally orbifold toric varieties). The automorphism group of a toric variety is in general non-reductive and I will use new results in non-reductive GIT, developed by F. Kirwan et al., to construct a moduli space of quasismooth hypersurfaces. I will give geometric characterisations of notions of stability arising from non-reductive GIT.

**8th November 2018****NO TALK due to NoGags Leipzig!**

**15th November 2018****Matej Filip (Mainz) **

**Abstract:** We will define basic notions of logarithmic geometry. For a log scheme with incoherent log structure on a well-behaved subset of relative codimension two we then prove the log Hodge to de Rham degeneration. This leads to smoothing of some toroidal crossing spaces which are generalization of normal crossing spaces.

**and**

**Jan Christophersen (Oslo)**

**Vanishing cotangent cohomology for Plücker algebras**

**Abstract:** Using representation theory I will show the vanishing of higher cotangent cohomology modules for the homogeneous coordinate ring of Grassmannians in the Plücker embedding. As a biproduct we answer a question of Wahl about the cohomology of the square of the ideal sheaf for the case of Plücker relations. Joint work with Nathan Ilten.

**22nd November **

**No talk!**

**29th November 2018****Achim Henning: talk postphoned--17th January 2019 **

**6th December 2018 **

**Irem Portakal (Magdeburg)**

**Torus actions on Kazhdan-Lusztig varieties**

**Abstract:** Matrix Schubert Varieties were first studied by Fulton combinatorially in terms of Rothe diagrams. It turns out to be possible to describe the natural effective torus action on these varieties in terms of bipartite graphs. This brings about an easy description for the rigidity of these varieties in toric case. More generally we construct a so-called Kazhdan-Lusztig variety by intersecting a matrix Schubert variety with an affine space. We consider the restricted torus action and observe that it can be understood via directed graphs. Thereupon we present some computational approaches for the classification of toric Kazhdan-Lusztig varieties. This is a joint-work with Maria Donten-Bury and Laura Escobar.

**13th December 2018****Frederik Witt (Stuttgart)**

**Toric Higgs sheaves**

**Abstract:** Higgs bundles appear in various apparently unrelated contexts. After giving a short introduction to Higgs bundles in general I will discuss an equivariant version over toric varieties. This is based on joint work with Klaus Altmann (Berlin) and Jan Christophersen (Oslo).

**20th December 2018****No talk due to Magdeburg-Kolloquium**

**10th January 2019**

**Severin Barmeier (Bonn)**

**Diagrams of algebras, categories of coherent sheaves and deformations**

**Abstract:** Given a complex algebraic variety X, the restriction of its structure sheaf to a finite cover of affine open sets can be viewed as a diagram of (commutative) algebras. Deformations of a diagram obtained in this way correspond precisely to deformations of the category of (quasi)coherent sheaves as an Abelian category (after W. Lowen and M. Van den Bergh).

We describe the higher deformation theory explicitly via L-infinity algebras for X covered by two affine opens and explain the connection to "classical" deformations of the complex structure and deformation quantizations by means of examples. This is joint work with Y. Frégier.

**17th January 2019**

**Achim Hennings (Siegen)**

**Nicht-degenerierte Polynome, log-plurikanonische Formen und äquisinguläre Deformationen in Dimension 3**

**Abstract:** Es sollen die log-plurikanonischen Formen auf einer (Newton-) nicht-degenerierten singulären Hyperfläche berechnet werden (wie von M. Morales angegeben). Die erhaltene Darstellung durch Residuen ermöglicht die Fortsetzung auf Deformationen. Es soll eine Anwendung auf die Berechnung äquisingulärer Deformationen in Dimension 3 aufgezeigt werden.

**24th January 2019**

**Kristin Shaw (Oslo)**

**Bounding Betti numbers of patchworked real hypersurfaces by Hodge numbers**

**Abstract:** Almost 150 years ago Harnack proved a tight upper bound on the number of connected components of a real planar algebraic curve of degree d. For hypersurfaces in higher dimensional projective space we can ask the analogou questions, yet we know very little about the topology of real algebraic hypersurfaces. For examples, we do not know the maximal number of connected components of real quintic surfaces in projective space.

In this talk I will explain our proof of a conjecture of Itenberg which, for a particular class of real algebraic projective hypersurfaces, bounds all Betti numbers, not only the number of connected components, in terms of the Hodge numbers of the complexification. The real hypersurfaces we consider arise from Viro’s patchworking construction, which is a powerful combinatorial method for constructing topological types of real algebraic varieties. To prove the bounds conjectured by Itenberg we develop a real analogue of tropical homology and use spectral sequences to compare it to the usual tropical homology of Itenberg, Katzarkov, Mikhalkin, Zharkov. Their homology theory gives the Hodge numbers of a complex projective variety from its tropicalization. Lurking in the spectral sequence of the proof are the keys to having combinatorial control of the topology of the real hypersurface produced from a patchwork.

This is joint work in progress with Arthur Renaudineau.

**31st January 2019**

**No talk!**

**7th February 2019**

**No talk!**