Of interest to people who study both hyperplane arrangements and commutative algebra are the homological properties of the module of logarithmic derivations of a hyperplane arrangement A. I will introduce the "ideal of pairs", which is a sort of "symmetrization" of this module of logarithmic derivations with respect to matroid duality. This is an ideal which simultaneously "sees" many of the homological properties of both the arrangement and its dual.
We will present recently constructed (joint with Charles Wang) Landau-Ginzburg models for the (small) quantum cohomology of exceptional cominuscule homogeneous spaces, as well as---if time allows---Plücker coordinate cluster structures and Newton-Okounkov bodies for these mirrors. To keep the discussions at an accessible level, we will introduce the relevant concepts using familiar examples of homogeneous spaces, such as Grassmannians and quadrics, and we will compare our results to the results known in these cases.
In this talk, I will try to give a presentation of 2 of the main theories that appear in my PhD work: the descent theory and the theory of Berkovich spaces. If there is enought time left, I will try to give some of my own results.
The tangent bundle of a hyperkahler manifold has very good deformation properties and therefore it is a natural candidate for constructing moduli spaces of (hyperholomorphic) sheaves. In this talk I will give an introduction to this topic and after I will focus on Hilbert schemes of points on a K3 surface. The main result is that in this case the tangent bundle turns out to be rigid. Joint work with Alessio Bottini. Please register with Christian Lehn. Corona restrictions at TU Chemnitz require a proof according to 3G-rule as well as wearing a FFP2 mask during the talk.
Toric bundles are certain fiber bundles with toric variety as fibers. Classical toric varieties admit a combinatorial description via fans. In my talk I will give a similar description of toric bundles and will show how one can use it to get some geometric and topological information about them. In particular, I will describe the intersection theory of toric bundles, and will present the combinatorial criterion for a toric bundle to be Fano.
In my talk I will discuss the notion of normalised volume for log terminal singularities. For the special case of toric singularities this turns out to be closely related to the notions of Mahler volume and Santaló point in convex geometry. I will explain how well-known facts from convex geometry can be utilised to deduce non-trivial statements about toric singularities. The Mahler conjecture might have a short appearance as well.
In theoretical physics, a gauged linear sigma model (GLSM) is a supersymmetric gauge theory which admits multiple phases with different physical behaviors. Such theories have been described in mathematical terms by means of variations of GIT, often yielding pairs of non isomorphic but derived equivalent Calabi—Yau pairs. We provide a GLSM description for a pair (X, Y), where X is Fano and Y is of general type, and with an argument based on window categories and matrix factorization we construct an embedding of the derived category of Y in the derived category of X. Joint work with Enrico Fatighenti, Michal Kapustka and Giovanni Mongardi.
In the 1990s, Terao and Yuzvinsky conjectured that reduced hyperplane arrangements satisfy the Logarithmic Comparison Theorem, asserting that the logarithmic de Rham complex computes the cohomology of the arrangement's complement. Essentially, this replaces the Brieskorn algebra in Brieskorn's Theorem with the logarithmic de Rham complex. We prove this conjecture by, among other things, sharply bounding the Castelnuovo--Mumford regularity of logarithmic j-forms of a central, essential, reduced arrangement. Time permitting we will discuss how to extend this untwisted Logarithmic Comparison Theorem to a twisted version. Here the twisted logarithmic de Rham complex computes the cohomology of the arrangement's complement with coefficients the rank one local system corresponding to the twist. Unlike the twisted Orlik--Solomon algebra, which can only compute a subset of the rank one local systems on the complement, this generalization computes all such rank one local systems.
From a block-diagonal (n+1)×(m+1)×(m+1) tensor symmetric in the last two entries one obtains two varieties: an intersection of symmetric determinantal hypersurfaces X in n-dimensional projective space, and an intersection of quadrics C in m-dimensional projective space. Under mild technical assumptions, we characterize the accidental singularities of X in terms of C. We apply our methods to algebraic curves and show how to construct theta characteristics of certain canonical curves of genera 3, 4, and 5, generalizing a classical construction of Cayley.
In my talk I will discuss the notion of normalised volume for log terminal singularities. For the special case of toric singularities this turns out to be closely related to the notions of Mahler volume and Santaló point in convex geometry. I will explain how well-known facts from convex geometry can be utilised to deduce non-trivial statements about toric singularities. The Mahler conjecture might have a short appearance as well.
