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Oberseminar "Algebra und Theoretische Mathematik"

Time and Venue:

Thursday, 9.15-10.15. By default, talks will be in-person and will take place in lecture hall 2/39-733 (new: C46.733) (Math building). Talks will be 60 minutes.

Sommersemester 2023:

27.04.2023 Ruijie Yang, Bonn
Higher multiplier ideals
For any effective Q-divisor D on a complex manifold X, there is a multiplier ideal associated to the pair (X,D), which is an ideal sheaf measuring the singularity of the pair and has many applications in algebraic geometry and commutative algebra. In this talk, I will discuss the construction of a new family of ideal sheaves associated to (X,D), indexed by an integer indicating the Hodge level, such that the lowest level recovers the usual multiplier ideals. This family of ideals is closely related to, but different from, the theory of Hodge ideals developed by Popa and Mustata. We establish their local and global properties systematically using various types of D-modules. If time permits, we will discuss some application to the Riemann-Schottky problem. This is based on the joint work with Christian Schnell.
04.05.2023 Franco Giovenzana, Paris
Deformations of Lagrangian Subvarieties
We will discuss open problems and conjectures around the deformation theory of Lagrangian subvarieties of irreducible symplectic manifolds.
11.05.2023 Ana María Botero, Bielefeld
Toroidal b-divisors and applications in differential and arithmetic geometry
We define toroidal b-divisors on a quasi projective variety over a field. These can be seen as conical functions on a balanced polyhedral space. We show the existence of an intersection pairing for so called nef toroidal b-divisors, which gives rise to a Monge-Ampére type measure on the polyhedral space. We then show some applications of this theory. On the one hand side, b-divisors are used to encode singularities of psh metrics and we derive Chern-Weil type formulae for such metrics on line bundles. On the other hand, using a Hilbert-Samuel formula, we compute asymptotic dimension formulae of spaces of automorphic forms on mixed Shimura varieties. Finally, if time permits, we connect our work to the notion of adelic line bundles of Yuan and Zhang, and outline some current research directions.
06.07.2023 Bernd Sturmfels, Leipzig
Positve del Pezzo geometry
Positive geometry is an emerging field at the interface of combinatorics and algebraic geometry with particle physics. This lecture explores the positive geometries associated with del Pezzo surfaces and their moduli spaces. Scattering amplitudes offer new vistas on configurations of six and seven points in the plane, their Weyl group symmetries of types E6 and E7, and on 27 lines on real and tropical cubic surfaces. This is joint work with Nick Early, Alheydis Geiger, Marta Panizzut and Claudia Yun.


Wintersemester 2022/23:

29.09.2022 Ben Tighe, Chicago
Hodge Theory of Singular Symplectic Varieties with Isolated Singularities
The total Lie algebra $mathfrak g$ of a compact Kahler manifold $X$ is the Lie algebra generated by all $mathfrak{sl}_2$-representations corresponding to classes satisfying Hard Lefschetz. For hyperkahler manifolds, Looijenga-Lunts and Verbitsky showed that the canonical Hodge structure on the cohomology ring $H^*(X)$ is completely determined by its structure as a $mathfrak g$-representation. We extend this result to the total Lie algebra on the intersection cohomology of primitive symplectic varieties with at worst isolated singularities, which also gives a new algebraic proof in the hyperkahler setting. As an application, we show that the cohomology of Q-factorial primitive symplectic varieties with isolated singularities all carry pure Hodge structures. Time permitting, I can discuss how our methods may work in the general case.
10.11.2022 Lars Halvard Halle, Bologna
Degenerations of Hilbert schemes and relative VGIT
This talk will be a report on joint work with K. Hulek and Z. Zhang. First I will explain how some central results in VGIT can be extended to a relative setting. After this, I will discuss an application of relative VGIT to the study of certain degenerations of Hilbert schemes of points.
8.12.2022 Already at 9 am! Salvatore Floccari, Hannover
Sixfolds of generalized Kummer type and K3 surfaces
I will present a construction which associates to any sixfold K of generalized Kummer type a hyper-Kähler manifold Y deformation equivalent to a Hilbert scheme of lenght 3 subscheme on a K3 surface, relating the most well studied deformation types of hyper-Kähler manifolds in dimension 6. Our construction is reminiscent of the classical construction of Kummer K3 surfaces, in the sense that Y is obtained as resolution of the quotient of K by a group of symplectic automorphisms. As a consequence we are able to show that any projective sixfold K as above determines a well-defined K3 surface. We use this construction to prove that the Kuga-Satake correspondence is algebraic for infinitely many new families of K3 surfaces of general Picard rank 16.
8.12.2022 Not the usual time: 10am - 11 am! Cécile Gachet, Nice
Finite quotients of abelian varieties, étale in codimension 2, with a Calabi-Yau resolution
Let A be an abelian variety, and let G be a finite group acting freely in codimension 2 on A. It is quite natural to ask whether the quotient A/G has a crepant resolution. If it does, this crepant resolution can also be characterized among K-trivial varieties by a certain vanishing of its second Chern class. In the 90ies, Oguiso classified such quotients and such resolutions in dimension 3: there are exactly two of them.
In this talk, we prove that there are no such objects in dimension 4 and 5. For that purpose, we use some computational group theory (implemented in GAP), as well as a correspondence between the Hodge numbers of a crepant resolution and the orbifold Hodge numbers of the quotient that is being resolved, due to Batyrev and Dais.
15.12.2022 Klaus Altmann, Berlin
The structure of full exceptional sequences of line bundles on toric varieties of Picard rank two or: Infecting Z^2
For a smooth projective toric variety of Picard rank two we classify all exceptional sequences of invertible sheaves which have maximal length. In particular, we prove that unlike non-maximal sequences, they
(a) remain exceptional under lexicographical reordering
(b) satisfy strong height constraints in the Picard lattice
(c) are full, that is, they generate the derived category of the variety.
Moreover, we determine the exceptional poset of those sequences and show how this poset governs the property "strongly exceptional". (This is joint work with Frederik Witt, Stuttgart; see arXiv:2112.14637 [math.AG])
5.1.2023 Mads Villadsen, Stony Brook
Generic vanishing and Chen-Jiang decompositions
The Generic Vanishing theorem of Green and Lazarsfeld describes the behaviour of the cohomology of direct images of canonical bundles to abelian varieties when twisted by line bundles of degree zero. I will discuss Chen-Jiang decompositions of these direct images, first introduced by J. Chen and Z. Jiang for generically finite morphisms to abelian varieties, which explain their generic vanishing behaviour and certain positivity properties in detail. In particular, I will discuss how to prove the existence of these decompositions using classical variational Hodge theory.
12.1.2023 Aline Zanardini, Leiden
On the GIT stability of pencils of plane curves
In this talk I will present a possible approach to the problem of classifying pencils of plane curves up to projective equivalence via geometric invariant theory. In particular, I will explain how the stability of a pencil is related to the stability of its generators, to the log canonical threshold, and to the multiplicities of a base point. And I will also discuss some relevant examples.
19.1.2023 Yichen Qin, Berlin
Hodge numbers of motives attached to Kloosterman moments and Airy moments.
Fresán, Sabbah, and Yu constructed motives $mathrm{M}_{n+1}^k$ over $mathbb{Q}$ encoding symmetric power moments of Kloosterman sums in $n$ variables. When $n=1$, they use the irregular Hodge filtration on the exponential mixed Hodge structures associated with $mathrm{M}_{2}^k$ to compute the Hodge numbers of $mathrm{M}_{2}^k$, which turn out to be either $0$ or $1$. As consequence, they proved that the $L$-functions of $mathrm{M}_{2}^k$ extend meromorphically to $mathbb{C}$ and satisfy certain functional equations.
In this talk, I will explain how to compute the (irregular) Hodge numbers of some motives $mathrm{M}_{n+1}^k$ when $n>1$ and some other related motives attached to Airy moments constructed by Sabbah and Yu. In particular, these Hodge numbers come from some generating series, and we find many cases where the Hodge numbers can be bigger than $1$.


Sommersemester 2022:

12.5.2022 Batu Güneysu, Chemnitz
The Chern Character of Fredholm Modules over Differential Graded Algebras and the Duistermaat-Heckmann Localization Formula on Loop Spaces
In this talk, I am going to report on recent results that connect constructions from noncommutative geometry and a mathematically well-defined realization of the supersymmetric path integral on a spin manifold. In the first part of the talk I am going to explain at an abstract level how to construct the Chern character of theta-summable Fredholm module over a locally convex differential graded algebra and explain its basis constructions like e.g. homotopy invariance. In the second part, I am going to describe how these results produce a mathematically well-defined realization of the supersymmetric path integral on a compact spin manifold and a natural localization formula on the corresponding loop space. This is joint work with Sergio Cacciatori and Matthias Ludewig.
19.5.2022 Hendrik Süß, Jena
The normalised volume of singularities
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.
9.6.2022 Dan Bath, Leuven
Logarithmic Comparison Theorems for Hyperplane Arrangements, Twisted or Otherwise.
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.
30.6.2022 Leonid Monin, Leipzig
Toric bundles.
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.
30.6.2022 Angel David Rios Ortiz, Leipzig
Infinitesimal deformations of the tangent bundle of Hyperkahler manifolds.
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.
14.7.2022 Avi Steiner, Mannheim
"Symmetrizing" logarithmic derivations with respect to matroid duality
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.


Wintersemester 2021/22:

26.10.2021 Camilla Felisetti, Trento
Topology of Lagrangian fibrations and P=W phenomena on irreducible symplectic varieties.
Irreducible holomorphic symplectic (IHS) varieties can be thought as a generalization of hyperkähler manifolds allowing singularities. Among them we can find for example moduli spaces of sheaves on K3 and abelian surfaces, which have been recently shown to play a crucial role in non abelian Hodge theory. After recalling the main features of IHS varieties, I will present several results concerning their intersection cohomology and the perverse filtration associated with a Lagrangian fibration, establishing a compact analogue of the celebrated P=W conjecture by de Cataldo, Hausel and Migliorini for varieties which admit a symplectic resolution. The talk is based on joint works with Mirko Mauri, Junliang Shen and Qizheng Yin.
28.10.2021- at 11.30 h, not the usual time, and IN PERSON! Mathieu Dutour Sikirić, Zagreb
Variations on Perfect form Theory.
Perfect forms were introduced by Korkine and Zolotarev for solving the lattice sphere packing problem. Their theory was later extended by Voronoi who provided an algorithm for their enumeration. In the first part of the talk we will summarize the classical theory and its application to computing cohomology groups of the general linear group. Then we will hint at possible generalizations of perfect form theory with T-space, and central compactification being described in some details. We aill also show the potential applicability to computing an invariant complex for the symplectic group. In the last part of the talk, we come back to the classical theory and show how one can enumerate short configuration of vectors in small rank.
16.11.2021 Laura Pertusi, Milano
Serre-invariant stability conditions and cubic threefolds.
Stability conditions on the Kuznetsov component of a Fano threefold of Picard rank 1, index 1 and 2 have been constructed by Bayer, Lahoz, Macrì and Stellari, making possible to study moduli spaces of stable objects and their geometric properties. In this talk we investigate the action of the Serre functor on these stability conditions. In the index 2 case and in the case of GM threefolds, we show that they are Serre-invariant. Then we prove a general criterion which ensures the existence of a unique Serre-invariant stability condition and applies to some of these Fano threefolds. Finally, we apply these results to the study of moduli spaces in the case of a cubic threefold X. In particular, we prove the smoothness of moduli spaces of stable objects in the Kuznetsov component of X and the irreducibility of the moduli space of stable Ulrich bundles on X. These results come from joint works with Song Yang and with Soheyla Feyzbakhsh and in preparation with Ethan Robinett.
30.11.2021 Guillem Blanco, Leuven
Yano's conjecture.


Sommersemester 2021:

12.04.2021 Enrico Fatighenti, Toulouse
Fano varieties from homogeneous vector bundles.
The idea of classifying Fano varieties using homogeneous vector bundles was behind Mukai's classification of prime Fano 3-folds. In this talk, we give a survey of some recent progress along the same lines, including a biregular rework of the non-prime Mori-Mukai 3-folds classification and some examples of higher-dimensional Fano varieties with special Hodge-theoretical properties.
26.04.2021 Noémie Combe, Leipzig
Complement of the discriminant variety, Gauss–skizze operads and hidden symmetries
In this talk, the configuration space of marked points on the complex plane is considered. We investigate a decomposition of this space by so-called Gauss-skizze i.e. a class of graphs being forests. These Gauss-skizze, reminiscent of Grothendieck's dessins d'enfant, provide a totally different real geometric insight on this complex configuration space, which under the light of classical complex geometry tools, remains invisible. Topologically speaking, this stratification is shown to be a Goresky–MacPherson stratification. We prove that for Gauss-skizze, classical tools from deformation theory, ruled by a Maurer--Cartan equation can be used only locally. We show as well, that the deformation of the Gauss-skizze is governed by a Hamilton--Jacobi differential equation. Finally, a Gauss-skizze operad is introduced which can be seen as an enriched Fulton--MacPherson operad, topologically equivalent to the little 2-disc operad. The combinatorial flavour of this tool allows not only a new interpretation of the moduli space of genus 0 curves with n marked points, but gives a very geometric understanding of the Grothendieck--Teichmuller group.
10.05.2021 Daniele Agostini, Leipzig
On the irrationality of moduli spaces of K3 surfaces
In this talk, we consider quantitative measures of irrationality for moduli spaces of polarized K3 surfaces of genus g. We show that the degree of irrationality is bounded polynomially in terms of g, so that these spaces become more irrational, but not too fast. The key insight is that the irrationality is bounded by the coefficients of a certain modular form. This is joint work with Ignacio Barros and Kuan-Wen Lai.
17.05.2021 Enrica Mazzon, Bonn
Toric geometry and integral affine structures in non-archimedean mirror symmetry
The SYZ conjecture is a conjectural geometric explanation of mirror symmetry. Based on this, Kontsevich and Soibelman proposed a non-archimedean approach to mirror symmetry. This led to the notion of essential skeleton and the construction of non-archimedean SYZ fibrations by Nicaise-Xu-Yu. In this talk, I will introduce these objects and report on recent results extending the approach of Nicaise-Xu-Yu. This yields new types of non-archimedean retractions. For families of quartic K3 surfaces and quintic 3-folds, the new retractions relate nicely with the results on the dual complex of toric degenerations and on the Gromov-Hausdorff limit of the family. This is based on a work in progress with Léonard Pille-Schneider.
31.05.2021 Robin van der Veer, Leuven
MLE, tropical geometry and slopes of Bernstein-Sato ideals.
Let X be a smooth subvariety of a complex torus. For general data vectors the MLE problem on X has exactly |chi(X)| solutions. We investigate what happens to these solutions when the data vector approaches a non-general value. Assuming that X is schön we relate this behaviour to special rays in the tropical variety of X. We also explain how the non-general data vectors are related a Bernstein-Sato ideal associated to X. Based on joint work with Anna-Laura Sattelberger.
28.06.2021 Andreas Höring, Nice
Stein complements in projective manifolds
Let X be a complex projective manifold, and let Y be a prime divisor in X. If Y is ample, it is well-known that the complement X setminus Y is an affine variety. Vice versa, assume that X setminus Y is affine, or more generally a Stein manifold. Then X setminus Y does not contain any curve, in particular Y has positive intersection with every curve. This leads to our main question: if X setminus Y is Stein, what can we say about the normal bundle of Y ? After some general considerations I will focus on the case where Y is the projectivised tangent bundle of some manifold M, and X is a ``canonical extension''. We will see that the Stein property leads to many restrictions on the birational geometry of M. This is work in progress with Thomas Peternell.
05.07.2021 Daniel Greb, Essen
Cycles in the K3 period domain and moduli of complex hyperkähler metrics
Hyperkähler metrics on K3 surfaces give rise to rational curves of degree 2 in the K3 period domain, socalled twistor lines. While these are used in the proofs of many deep results, their existence also implies that the group of isometries of the K3 lattice does not act properly discontinuously on the period domain, preventing a moduli space of unpolarised K3 surfaces to exist. I will report on work (very much) in progress with Martin Schwald, in which we study the cycle space of the K3 period domain. This space parametrises twistor lines as part of its real locus, but also all their degenerations and complex deformations. I will explain how many foundational problems disappear when passing to the cycle space and also indicate how the original version of Penrose's Twistor Theory (the "nonlinear graviton" construction) can be used to understand what kind of geometric structure a small complex deformation of an honest twistor line corresponds to.


Wintersemester 2020/21:

06.11.2020 Christian Sevenheck, Chemnitz
Hodge ideals of free divisors
Hodge ideals are new invariants of singular hypersurfaces. Their definition relies on the intricate theory of Mixed Hodge modules of Morihiko Saito, but they have recently been studied from a birational point of view in a series of papers by Popa and Mustata. In this talk, I will describe some results (joint work with Luis Narvaez Macarro and Alberto Castano Dominguez) about Hodge ideals for a specific class of non-isolated singularities, namely, some free divisors. We obtain a general formula for these ideals, and an algorithm to calculate them in some particular cases.
13.11.2020 Alessandra Sarti, Poitiers
K3 surfaces with maximal finite automorphism groups
It was shown by Mukai that the maximum order of a finite group acting symplectically on a K3 surface is 960 and that the group is isomorphic to the Mathieu group $M_{20}$. Then Kondo showed that the maximum order of a finite group acting on a K3 surface is 3840 and this group contains the Mathieu group with index four. Kondo showed also that there is a unique K3 surface on which this group acts, which is a Kummer surface. I will present recent results on finite groups acting on K3 surfaces, that contain strictly the Mathieu group and I will classify them. I will show that there are exactly three groups and three K3 surfaces with this property. This is a joint work with C. Bonnafé.
20.11.2020 Sönke Rollenske, Marburg
Stratifications in the moduli space of stable surfaces, (based on joint work with B. Anthes, M. Franciosi, R. Pardini)
The Gieseker moduli space of surfaces of general type admits a modular compactification, the moduli space of stable surfaces. Our knowledge about the "new" surfaces in the boundary is still limited and I will discuss different possibilities to organise them, in particular a Hodge-theoretic approach proposed by Green, Griffiths, Laza, and Robles.
Everything will be illustrated with many pictures.
25.11.2020 - at 15.30 h, NOT THE USUAL TIME! Dan Bath, Purdue
Bernstein--Sato ideals and hyperplane arrangements.
Bernstein--Sato polynomials are a classical invariant attached to a divisor, whereas Bernstein--Sato ideals are an invariant to a factorization of a divisor. The information encoded by the Bernstein--Sato ideal is expected to be at least as rich as the data of Bernstein--Sato polynomials. For example, it has been recently proved that the zero locus of this ideal determines the nontrivial rank one local systems on the complement of said divisor (Budur, van der Veer, Wu, Zhou). For a large class of tame divisors, we study this ideal along with naturally attached D-modules. For most (i.e. tame) hyperplane arrangements, our results verify Budur's Topological Multivariable Strong Monodromy Conjecture--a connection between Bernstein--Sato ideals and the numerics of a resolution of singularities. Time permitting, we will discuss some related results for most hyperplane arrangements, including, but not limited to: combinatorial formulas for roughly half the zeroes of the Bernstein--Sato ideal; a combinatorial formula for all the zeroes in the free case; the principality of this ideal; the relationship between the Bernstein--Sato ideal and the Bernstein--Sato polynomial.
27.11.2020 Thomas Krämer, Berlin
Semicontinuity of Gauss maps and the Schottky problem
We show that the degree of the Gauss map for subvarieties of abelian varieties is semicontinuous in families, and we discuss the loci where it jumps. In the case of theta divisors this gives a finite stratification of the moduli space of ppav's whose strata include the locus of Jacobians and the Prym locus. This is joint work with Giulio Codogni.
04.12.2020 Davide Lombardo, Pisa
Endomorphism rings of Jacobians
I will discuss one of the fundamental algorithmic problems in the theory of abelian varieties, namely, the determination of the endomorphism ring of the Jacobian J of a curve given by explicit equations (over a number field). I will describe a practical solution to this problem, starting with the case of curves of genus 2: this will involve both analytic ingredients, in the form of complex uniformisation, and arithmetic ones. In particular, I will highlight the relevance to this problem of a certain local-global principle for the endomorphism algebra of J. If time permits, I will also outline how to prove this local-global principle for abelian surfaces and, under the assumption of the Mumford-Tate conjecture, for all abelian varieties.
11.12.2020 Claudio Onorati, Oslo
Birational geometry of hyper-Kaehler tenfolds of O'Grady type
I will present an explicit example of a particular hyper-Kaehler tenfold of O'Grady type and investigate its birational geometry in detail. This will enable us to state one of the main results of my recent joint work with G. Mongardi, namely the determination of the Kaehler cone of such manifolds. I will outline the main ingredients in the proof and, time permitting, I will discuss the monodromy group of such manifolds.
08.01.2021 Christian Lehn, Chemnitz
BBDGGHKP-decomposition in the analytic case
The classical Bogomolov-Beauville decomposition is central in the classification theory of varieties with trivial first Chern class. In higher dimensional complex geometry, it is important to consider singular varieties as well and the question comes up whether there is a generalization of the decomposition theorem to the singular setup. Here, the differential geometric techniques of the original proof completely break down, but Höring-Peternell (2017) gave a proof for projective varieties with log terminal singularities and numerically trivial canonical class as a culmination of work by Druel–Greb–Guenancia–Höring–Kebekus–Peternell over the past decade. In a joint work with Bakker and Guenancia, we prove the theorem in full generality, i.e. a common generalization of the BB and the DGGHKP decomposition. Our proof reduces to the projective case using methods from Hodge theory, locally trivial deformations, and singular Kähler-Einstein metrics.
15.01.2021 Sam Streeter, Bath
Title: Campana points and powerful values of norm forms
The theory of Campana points is of growing interest in arithmetic geometry due to its ability to interpolate between the notions of rational and integral points. Further, it naturally lends itself to studying “arithmetically interesting” solutions of equations. In this talk, I will introduce Campana points and explain the key ideas and principles behind recent results on asymptotics for Campana points of bounded height, providing evidence for a Manin-type conjecture proposed in work of Pieropan, Smeets, Tanimoto and Várilly-Alvarado. I will also indicate how these results give rise to an asymptotic formula for powerful (e.g. square-full) values of norm forms.
22.01.2021 Andrea Fanelli, Bordeaux
Rational simple connectedness and Fano threefolds.
The notion of rational simple connectedness can be seen as an algebraic analogue of simple connectedness in topology. The work of de Jong, He and Starr has already produced several recent studies to understand this notion. In a project with Laurent Gruson and Nicolas Perrin, we start the study of rational simple connectedness for Fano threefolds by explicit methods from birational geometry.
29.01.2021 Philip Boalch, Paris
Diagrams, nonabelian Hodge spaces and global Lie theory
The class of moduli spaces appearing in nonabelian Hodge theory has been significantly enriched over the past 20 years or so, by considering solutions of the 2d self-duality equations with more involved behaviour at the boundary. In brief one can relax Simpson's tameness condition, and this leads to stable meromorphic connections/Higgs fields with arbitrary order poles (on parabolic vector bundles). Much of this was motivated by examples occurring in Seiberg-Witten theory, and in the classical integrable systems literature. For example the topological Atiyah-Bott/Goldman symplectic structures were extended to this context by the speaker (Adv. Math 2001), the Corlette/Donaldson correspondence with complex connections was extended by Sabbah (Ann. Inst. Fourier 1999), and the construction of the hyperkahler moduli spaces plus the extension of the Hitchin/Simpson correspondence with Higgs bundles was carried out by Biquard and the author (Compositio 2004). Some more recent work has extended the TQFT (quasi-Hamiltonian) approach to these holomorphic symplectic varieties from the generic case to the general case, and clarified the extra deformation parameters that occur, leading to the notion of ``wild Riemann surface''. In this talk I'll review some of the simplest examples of complex dimension two, and their link to affine Dynkin diagrams (leading to the notion of ``global Weyl group''). Then I'll explain a way to extend this link by attaching a diagram to any nonabelian Hodge space on the affine line. This is an attempt to organise the vast bestiary of examples of complete hyperkahler manifolds that occur. A key idea is that all the nonabelian Hodge spaces have concrete descriptions as moduli spaces of Stokes local systems (the wild character varieties), generalising the well-known explicit presentations of the (tame) character varieties, coming from a presentation of the fundamental group. This is joint work with D. Yamakawa (Compte Rendus Math. 2020).


Wintersemester 2019/20:

14.10.2019 Corinne Bedussa, Leuven
Lagrangian fibrations on hyperkähler manifolds: a non-Archimedean perspective
The geometry of Lagrangian fibrations on hyperkähler manifolds is very special and it attracted a lot of attention since the pioneering work of Matsushita. In particular it has been conjectured that the base of every such fibration is a complex projective space. While this result has been proved by Hwang when the base is assumed to be smooth, the general case remains widely open. In this talk, I will give an introduction to the theory of Lagrangian fibrations on hyperkähler manifolds. Then I will outline a possible approach to study the geometry of the base. This will rely on ideas coming from a non-Archimedean interpretation of Mirror Symmetry proposed by Kontsevich and Soibelman. In this setting the role of the base is played by a topological space called essential skeleton, which can be realized as the dual intersection complex of some special degenerations of hyperkähler manifolds (minimal dlt). This will enable us to use techniques coming from Berkovich geometry and birational geometry and the key idea is to let the classical and the non-Archimedean perspective interact to gain a deeper understanding of the problem.
21.10.2019 Paul Görlach, MPI Leipzig
Injectivity of linear systems on projective varieties
In this talk, I will discuss the problem of finding the smallest dimension of a projective space to which a given complex algebraic variety X admits an injective morphism. Generalizing previous work by E. Dufresne and J. Jeffries obtained in the context of separating invariants, I will show that a linear subsystem of a strict power of a line bundle can only give rise to an injective morphism if its dimension is at least 2*dim(X). I will discuss the ramifications of this result, with a focus on toric examples.
4.11.2019 Franco Giovenzana, TU Chemnitz
On arithmetic and birational properties of the irreducible holomorphically symplectic LLSvS variety
Given a smooth cubic fourfold (not containing a plane) Y in $P^5$, one can associate naturally to it two irreducible holomorphically symplectic varieties: its Fano variety of lines F and the so-called LLSvS variety Z. The arithmetic properties of Y and birational properties of F are known to be deeply connected thanks to a previous work of Addington. In this talk I will discuss a similar link between Y and the LLSvS variety. These results are fruit of a joint work with Nicolas Addington.
11.11.2019 Yajnaseni Dutta, Universität Bonn
Positivity aspects of studying families of algebraic varieties
The canonical sheaf (for smooth varieties this is the top exterior power of the sheaf of differential forms) is an important object to study algebraic varieties. Many of the properties of families of varieties are encoded in the positivity of the pushforwards of (pluri) canonical bundles. We will focus on a conjecture of Popa and Schnell regarding the so-called global generation of these pushforwards. In the simplest case, this is an extension of the celebrated conjecture of Takao Fujita. Their conjecture was partially motivated by a result of Kawamata showing similar generations for a specific class of morphisms where Hodge theory works rather well. I will discuss some examples to motivate such positivity properties and discuss a few techniques involved. Then, I will present an overview of results in this direction. Part of this is from a joint work with Takumi Murayama.
18.11.2019 Annalisa Grossi, Universität Augsburg
Automorphisms of O’Grady’s sixfolds
In 2000 O’Grady introduced a new example of hyperkahler manifolds in dimension six as the symplectic resolution of a certain fiber of a moduli space of sheaves on an abelian surface. In this talk I give you a criterion to determine when an automorphism of an O’Grady six type manifold is induced by an automorphism of the abelian surface. Moreover I give you a classification of non-symplectic and prime order automorphisms of O’Grady six type manifolds using lattice-theoretic tools.
2.12.2019 Oksana Yakimova, Universität Jena
Poisson-commutative subalgebras of the symmetric algebra S(g)
The symmetric algebra S(g) of a reductive Lie algebra g is equipped with the standard Poisson structure, i.e., the Lie-Poisson bracket. Poisson-commutative subalgebras of S(g) attract a great deal of attention, because of their relationship to integrable systems and to geometric representation theory. There are two classical constructions of such subalgefbras, the inductive procedure of Gelfand-Tsetlin and the ``argument shift method'' of Mishchenko-Fomenko. We will discuss recent results on these well-known objects and present a generalisation of the Gelfand-Tsetlin approach to chains of almost arbitrary symmetric subalgebras.
16.1.2020 Davide Barco, Universita degli studi di Padova.
A topological approach to the Fourier transform of an elementary D-module
20.01.2020 Andreas Hohl, Universität Augsburg
Riemann–Hilbert and D-modules of pure Gaussian type
The Riemann–Hilbert correspondence is a classical result in the theory of D- modules. It states that there is an equivalence between the category of regular holonomic D-modules and the category of perverse sheaves, and as such gives a “dictionary” to translate between algebraic analysis and topology. A lot of effort has been put into finding a generalization to (not necessarily regular) holonomic D-modules, culminating in the enhanced Riemann–Hilbert correspondence of A. D’Agnolo and M. Kashiwara. I will give an overview of the main ingredients of their theory, such as ind-sheaves and the Stokes phenomenon. I will then outline how this new framework can be used to treat systems with irregular singularities (in particular, systems of so-called pure Gaussian type) and Fourier transforms from a topological point of view.
27.1.2020 Thomas Kahle, OvGU Magdeburg
Central limit theorems for permutation statistics
The number of inversions or descents of a random permutation in a large symmetric group is asymptotically normally distributed. We discuss extensions of this principle to arbitrary families of finite Coxeter groups of increasing rank. As a prerequisite we find uniform formulas for the means and variances in terms of Coxeter group data. The main gadget for central limit theorems is the Lindeberg—Feller theorem for triangular arrays. Transferring the Lindeberg condition to the combinatorial setting, one finds that the validity of a central limit theorem depends on the growth of the dihedral subgroups in the sequence.
3.2.2020 Marco Hien, Universität Augsburg
Stokes data for unramified confluent hypergeometric equations
Hypergeometric functions are a classical example of special functions, studied by Gauß, Klein, Riemann and others. We consider the confluent version arising from the non-confluent ones by Fourier-Laplace transform. In particular, we are interested in the Stokes phenomenon of their irregular/confluent singularity. We explain how to determine the Stokes invariants in a purely topological way by understanding the rigid local system of the non-confluent equations. We obtain a new version of some results by Duval-Mitschi, which we can generalize to include further interesting classes.




Vergangene Semester:

Datum Sprecher Titel und Zusammenfassung
09.06.2016 Damian Brotbek, Strasbourg On the hyperbolicity of general hypersurfaces

A complex manifold is said to be (Brody) hyperbolic if it doesn't contain any entire curves. Kobayashi conjectured in 1970 that a general hypersurface of sufficiently high degree in projective space is hyperbolic. This statement was only settled recently by a work of Siu.
In this talk, after a gentle introduction to hyperbolic manifolds, we outline a new proof of this conjecture of Kobayashi.
11.08.2016 Diletta Martinelli, Imperial College What are minimal models? How many are there?

Algebraic geometry studies the structure of algebraic varieties, solutions of a system of polynomial equations in an affine or projective space. The final goal of the subject is to achieve a complete classification of algebraic varieties up to some kind of equivalence relations. I will explain in the talk why one of the most natural choice is the classification up to birational equivalence (two varieties are birational if they are isomorphic up to some subvarieties of smaller dimension).
Then the first step of the classification is to find a representative inside the birational equivalence class that is in some sense simpler than the others, we call this variety a minimal model. The Minimal Model Program (MMP) is an algorithm that establishes a series of steps to find the minimal model.
A very natural question is whether the minimal model is unique, and if not how many minimal models does a variety admit and how are they related. After describing the general ideas of the MMP I will focus on these last questions and explain that for a special class of varieties the number of minimal model is finite and that in some cases it is possible to bound this number using topological information. These results are part of my PhD thesis and of a recent joint work with Stefan Schreieder and Luca Tasin.
24.10.2016 Christian Rose, TU Chemnitz The first Betti number and the Kato class on compact Riemannian manifolds
The starting point of this talk will be the dependence of the first Betti number of a compact Riemannian manifold on the so-called Kato-condition on the negative part of the Ricci curvature. This kind of condition can be controlled as soon as one knows something about the short time behavior of the heat semigroup. We show that L^p-type curvature conditions lead to upper bounds on the heat kernel. Furthermore, we discuss that in fact the Kato-condition on the negative part of the curvature is enough to obtain such bounds.
03.11.2016 Fabrizio Catanese, Bayreuth (im CMC) Configurations of lines and interesting algebraic surfaces.
Many important questions in the theory of surfaces and in algebraic geometry have been solved thanks to explicit constructions of algebraic surfaces as abelian coverings branched over special configurations of lines. After recalling the classical configurations (Pappus, Desargues, Fano, Hesse) and some new ones, I shall describe simple equations for such surfaces, as the Fermat, and Hirzebruch-Kummer coverings. As the configuration of lines becomes special some interesting geometry shows up, as in the case of the six lines of a complete quadrangle, related to the Del Pezzo surface of degree 5 and its icosahedral symmetry. After mentioning many important such examples and applications, by several authors, I shall concentrate on a recent simple series of such surfaces, studied in my joint work with Ingrid Bauer and Michael Dettweiler, discussing new results and quite general open questions, concerning rigid manifolds, and projective classifying spaces.
10.11.2016 Bernd Sturmfels, UC Berkeley and MPI Leipzig
time: 11.30 h, room: 2/B102
Nearest Point on Toric Varieties
We determine the Euclidean distance degree of a projective toric variety. This is the intrinsic algebraic complexity of a ubiquitous optimization problem, namely to compute the point on a real toric variety that is closest to a given data point. Our results generalize the formula due to Matsui and Takeuchi for the degree of the A-discriminant in terms of Euler obstructions. This is joint work with Martin Helmer.
17.11.2016 Giovanni Mongardi, Bologna Chow rings of holomorphic symplectic manifolds: known structure, conjectures and speculations In this talk I will present some results on holomorphic symplectic manifolds and their Chow rings, in particular i will outline possible approaches to solve a series of conjectures made by Voisin.
17.11.2016 Joe Kileel, UC Berkeley Using Algebraic Geometry for Computer Vision
In computer vision, 3D reconstruction is a fundamental task: starting from photographs of a world scene, taken by cameras with unknown positions and orientations, how can we best create a 3D model of that world scene? Algorithms that do this built Street View (Google) and are instrumental in autonomous robotics. In 2004, David Nister (Tesla) used Grobner bases to build a solver for robust reconstruction given just two photographs. This is a key routine in much larger-scale reconstructions today. In this talk, I will discuss reconstruction given three photographs, where efforts to replicate Nister have so far proven elusive. My approach relies on applied algebraic geometry. In particular, I shall introduce an algebraic variety whose points are 3x3x3 tensors in correspondence with configurations of three calibrated cameras. Special linear sections of this variety recover camera configurations from image data. The main result is the determination of the algebraic degree of minimal problems for this recovery. These comprise interesting enumerative geometry problems; the solution is by way of homotopy continuation calculations.
16.01.2017 Vladimir Shikhman, Chemnitz On the local representation of piecewise smooth equations as a Lipschitz manifold
We study systems of equations given by piecewise differentiable functions The focus is on the representability of their solution set locally as a Lipschitz manifold. For that, nonsmooth versions of inverse function theorems are applied. It turns out that their applicability depends on the choice of a particular basis. To overcome this obstacle we introduce a strong full-rank assumption (SFRA) in terms of Clarkeʼs generalized Jacobians. The SFRA claims the existence of a basis in which Clarkeʼs inverse function theorem can be applied. Aiming at a characterization of SFRA, we consider also a full-rank assumption (FRA). The FRA insures the full rank of all matrices from the Clarkeʼs generalized Jacobian. The talk is devoted to the conjectured equivalence of SFRA and FRA.
19.01.2017 Stefan Müller-Stach, Mainz (im CMC) Von Dedekind zur modernen Arithmetik: Rekursionstheorie und Perioden
Dedekind (1831-1916) hat neben seinen vielen anderen einflussreichen Werken die Rekursionstheorie begründet. Von dort kann man eine Verbindung zur modernen Logik und zur Arithmetischen Geometrie und insbesondere zu berechenbaren reellen Zahlen und Perioden herstellen. Diese historische Reise versuchen wir im Vortrag darzustellen.
30.01.2017 Paolo Stellari, Milano A derived category approach to some moduli spaces on cubic threefolds and fourfolds.
We exploit the homological properties and the geometric meaning of Kuznetsov's semiorthogonal decomposition of the derived categories of cubic fourfolds (and threefolds) to study the (birational) geometry of some interesting moduli spaces on such varieties. We will start working out the very instructive example of the moduli space of stable aCM bundles of a given rank on a cubic threefold. Then we will discuss our main result concerning the case of generalized twisted cubics on cubic fourfolds not containing a plane. We will show that we can recover the picture by Lehn-Lehn-Sorger-van Straten in terms of moduli spaces of (weakly) stable sheaves/complexes. This is joint work with M. Lahoz, M. Lehn, and E. Macri'.
02.02.2017 Simon Brandhorst, Leibniz Universität Hannover Minimal Salem numbers on supersingular K3 surfaces
The entropy of a surface automorphism is either zero or the logarithm of a Salem number, that is an algebraic integer $lambda>1$ which is conjugate to $1/lambda$ and all whose other conjugates lie on the unit circle. In the case of a complex K3 surface McMullen gave a strategy to decide whether a given Salem number arises in this way. To do this he combined methods from linear programming, number fields, lattice theory and the Torelli theorems. In this talk we extend these methods to automorphisms of supersingular K3 surfaces using the crystalline Torelli theorems and apply them in the case of characteristic $5$. This is joint work with Víctor González-Alonso.
23.03.2017 Ziyu Zhang, Leibniz Universität Hannover Holomorphic symplectic manifolds among Bridgeland moduli spaces
We consider moduli spaces of semistable complexes on a projective K3 surface with respect to generic Bridgeland stability conditions. Similar to the sheaf case, the smooth ones among them are holomorphic symplectic manifolds, and the 10-dimensional singular ones admit symplectic resolutions. I will explain why these examples of holomorphic symplectic manifolds are all deformation equivalent to the known ones. By generalizing the prominent work of Bayer and Macri, we can also study the birational geometry of the 10-dimensional singular moduli spaces via wall-crossing on the stability manifold. This is a joint work with C.Meachan.
03.04.2017 Carsten Liese, Leibniz Universität Hannover The KSBA compactification of the moduli space of degree 2 K3 pairs: a toroidal interpretation
Work of Gross, Hacking, Keel and Siebert shows that the Gross-Siebert reconstruction algorithm provides a partial toroidal compactification of the moduli space of polarized K3 surfaces for any genus. The construction comes with a family $mathfrak{X}to bar{mathbb{P}}^g$ over a partial toroidal compactification $bar{mathbb{P}}^g$ of a subset $mathbb{P}^g$ of the Kollár-Shepherd-Barron moduli space of stable K3 pairs $M_{SP}$. A conjecture of Keel says that $mathfrak{X}to bar{mathbb{P}}^g$ extends to a compactification of ${mathbb{P}^g}$ and in particular, all surfaces in the boundary of $M_{SP} appear as fibres of $mathfrak{X}to bar{mathbb{P}}^g$. In the genus $2$ case, $M_{SP}$ is known by work of Laza.
In this talk, I check the prediction of Keel's conjecture and show that all degenerate $K3$ surfaces in the boundary of $M_{SP}$ appear as fibres of $mathfrak{X}to mathbb{P}^g$. I do not assume familiarity with any part of the subject.
06.04.2017 Maxim Smirnov, Augsburg On quantum cohomology of isotropic Grassmannians
Dubrovin’s conjecture (ICM 1998) predicts an intriguing relation between the quantum cohomology ring of a smooth projective variety X and its derived category of coherent sheaves. I will explain some aspects of this story taking symplectic isotropic Grassmannians IG(m,2n) as the main example and stress the importance of the big quantum cohomology in the formulation of the conjecture. If time permits I will exhibit a relation between the quantum cohomology of IG(m,2n) and unfoldings of isolated hypersurface singularities, and its counterpart for the derived category of coherent sheaves on IG(m,2n). The talk is based on joint works, some finished and some still in progress, with A. J. Cruz Morales, S. Galkin, A. Mellit, N.Perrin, and A. Kuznetsov.
08.06.2017 Emanuel Scheidegger, Freiburg On the hemisphere partition function
Starting with a variation of GIT quotients with potential we will review how to associate a function on the K-theory of the associated derived categories, called the hemisphere partition function, which conjecturally defines a Bridgeland stability condition. We will discuss various (conjectural) properties of this function and relate it to known functions and differential equations.
29.06.2017 Ferran Dachs-Cadefau, Halle Computing jumping numbers in higher dimensions
Multiplier ideals and jumping numbers are invariants that encode relevant information about the structure of the ideal to which they are associated. A first part of this talk will be devoted to introduce some basics about multiplier ideals in the case of 2-dimensional local rings. We will also present the relations with other invariants. In the second part, we present a formula to compute the multiplicity of jumping numbers of an m-primary ideal in a 2-dimensional local ring with rational singularities. This formula leads to a simple way to detect whether a given rational number is a jumping number. Another consequence of the formula is that it allows us to give an explicit rational expression for the Poincar é series of the multiplier ideals introduced by Galindo and Monserrat in 2010. This Poincar é series encodes in a unified way the jumping numbers and its corresponding multiplicities. In the third part of the talk, we will introduce some results for the multiplier ideals in the higher-dimensional case. For this, we introduce the notion of π-antieffective divisors, a generalization of antinef divisors to higher dimensions. Using these divisors, we present a way to find a small subset of the ‘classical’ candidate jumping numbers of an ideal, containing all the jumping numbers. Moreover, many of these numbers are automatically jumping numbers, and in many other cases, it can be easily checked. The second part of this talk is a joint work with Maria Alberich Carramiñana, Josep Àlvarez Montaner and Víctor González Alonso, while the third is a joint work with Hans Baumers.
06.07.2017 Andrés Reyes, Bogotá Topology and the phase transition of the Ising model
After briefly reviewing the Shale-Stinespring theorem, I will discuss the relevance of the choice of complex structure for a topological characterization of quantum phase transitions in systems that are described in terms of quasi-free states of the CAR (i.e. fermionic) algebra. Making use of the transfer matrix method, I will then show how this interpretation can be carried over to the case of the thermodynamic phase transition of the two dimensional classical Ising model.
12.07.2017 Martin Ulirsch, MPI Leipzig Tropical and logarithmic moduli theory
The moduli space of tropical curves (and its variants) are some of the most-studied objects in tropical geometry. So far this moduli space has only been considered as an essentially set-theoretic coarse moduli space (sometimes with additional structure). As a consequence of this restriction, the tropical forgetful map does not define a universal curve (at least in the positive genus case). The classical work of Deligne-Knudsen-Mumford has resolved a similar issue for the algebraic moduli space of curves by considering the fine moduli stacks instead of the coarse moduli spaces.
In this talk I am going to give an introduction to these fascinating moduli spaces and report on joint work with Renzo Cavalieri, Melody Chan, and Jonathan Wise, where we propose the notion of a moduli stack of tropical curves as a geometric stack over the category of rational polyhedral cones. Using this $2$-categorical framework one can give a natural interpretation of the forgetful morphism as a universal curve. Our theory naturally arises as a combinatorial shadow of the F. Kato's moduli stack of logarithmically smooth curves, which we can now think of as a hybrid of both the moduli stack of stable algebraic curves and our tropical moduli stack. The process of tropicalization connects these two worlds via a natural smooth and surjective tropicalization morphism.
13.07.2017 Patrick Graf, Bayreuth Finite quotients of complex tori
Let $X$ be a compact Kähler threefold with canonical singularities and vanishing first Chern class. I will show that if the second orbifold Chern class of $X$ intersects some Kähler form trivially, then $X$ admits a quasi-étale (i.e. étale in codimension one) cover by a complex torus. This result generalizes a theorem of Shepherd-Barron and Wilson for projective varieties. It should be seen as complementing the structure theory of Kähler threefolds (aka Minimal Model Program). Part of the talk is devoted to explaining the notion of second orbifold Chern class for complex spaces with canonical singularities, since this topic has not been treated in the literature up to now. If time permits, I will also discuss possible generalizations to klt singularities and to higher dimensions. Joint with Tim Kirschner (Essen).
01.03.2018 Carolin Peternell, Leibniz Universität Hannover Birational Models for Moduli of Quartic Rational Curves
Ch. Lehn, M. Lehn, Ch. Sorger and D. van Straten constructed a family of irreducible holomorphic symplectic manifolds via a contraction of the compactified moduli space $M_3(Y)$ of rational curves of degree 3 on a smooth cubic fourfold Y. This suggests that also the moduli space $M_4(Y)$ could be connected to a family of holomorphic symplectic manifolds. In order to understand $M_4(Y)$, we study the Hilbert scheme of curves on $P^4$ with Hilbert polynomial 4n+1, a moduli space of Kronecker modules, a moduli space of semi-stable sheaves and their relations.
30.04.2018 Mathew Dawes, Riemann Fellow,
Leibniz Universität Hannover
Note that this talk will exceptionally
take place in W037 from 11-12!
Modular forms and the birational geometry of modular varieties
Orthogonal modular varieties are arithmetic quotients of a Hermitian symmetric domain of type IV by a subgroup of O(2,n). They often arise in connection with moduli; for example, the moduli of K3 surfaces or the moduli of irreducible symplectic manifolds. Many aspects of the geometry of orthogonal modular varieties can be understood by studying modular forms for O(2,n); in particular, their Kodaira dimension. This problem is particularly interesting in small dimensions as, in addition to encountering questions on modular forms, one also needs a detailed understanding of singularities in modular varieties. I shall discuss my work on this problem for modular varieties associated with the moduli of irreducible symplectic manifolds of generalised Kummer type. If time permits, I shall discuss current work to generalise these results and explain how my methods shed some light on automorphisms and certain families of lattice polarised varieties.
24.05.2018 Milena Wrobel, MPI MiS Leipzig On Cox rings of T-varieties
An important invariant of an algebraic variety is its Cox ring. We consider structural properties of these rings and give for varieties with torus action an explicit description via generators and relations. Based on this description we look at the question of iterability of Cox rings and give classification results for smooth Fano varieties.
28.05.2018 Thomas Jahn, TU Chemnitz
Note that this talk will
take place in 39-733 from 11-12!
Uniqueness of circumcenters in generalized Minkowski spaces
We study an extension of a familiar concept of Euclidean plane geometry: circumballs. After an introduction to our setting of generalized Minkowski spaces, a characterization of uniqueness of circumcenters in terms of the boundary structure of balls will be presented. Finally, we discuss an extension involving the dimension of the set of circumcenters. This is joint work with Bernardo González Merino and Christian Richter.
31.05.2018 Emre Sertöz, MPI MiS Leipzig Computing periods of hypersurfaces
The periods of a smooth complex projective variety X are complex numbers, typically expressed as integrals, which give an explicit representation of the Hodge structure on the cohomology of X. Although they provide great insight, periods are often very hard to compute. In the past 20 years, an algorithm for computing the periods existed only for plane curves. We will give a different algorithm which can compute the periods of any smooth projective hypersurface and can do so with much higher precision. As an application, we will demonstrate how to reliably guess the Picard rank of quartic K3 surfaces and the Hodge rank of cubic fourfolds from their periods.
07.06.2018 Thomas Krämer, Humboldt-Universität zu Berlin A categorification of Kashiwara's index formula
On any commutative group variety, the category of holonomic D-modules has a natural Tannakian description with respect to the convolution product. While for affine varieties this has been studied by Gabber, Loeser, Sabbah, Katz, Dettweiler and others from many perspectives, much less is known in the case of abelian varieties. The talk will discuss a microlocal categorification of Kashiwara's index formula that gives a dictionary between characteristic cycles of holonomic D-modules and Weyl group orbits of weights for the corresponding reductive Tannakian Galois groups. While most previous results required the characteristic cycles to be reduced, we will explain how to get rid of this assumption and what this means for the geometry and singularities of subvarieties in abelian varieties.
11.06.2018 Caterina Cejp, Leibniz Universität Hannover
Note that this talk will
exceptionally take place in 39-633 from 15.30-16.30!
Die Methode der Borel Invarianten für das invariante Hilbertschema
14.06.2018 Ronan Terpereau, Université de Bourgogne, Dijon Automorphism groups of P1-bundles over rational surfaces.
In this talk I will explain 1) how to construct a very simple moduli space (=a projective space) for the non-decomposable P1-bundles with no "jumping fibres" over Hirzebruch surfaces, and 2) how to deduce the classification of all the P1-bundles over rational surfaces whose automorphism group G is maximal in the sense that every G-equivariant birational map to another P1-bundle is necessarily a G-isomorphism. (This is a joint work with Jérémy Blanc and Andrea Fanelli.)
21.06.2018 Bernd Schober, Leibniz Universität Hannover Resolution of Surface singularities
Nowadays, there are several quite accessible accounts for Hironaka's theorem of resolution of singularities for algebraic varieties over a field of characteristic zero. In contrast to this, there exist only results in small dimensions for the case of positive or mixed characteristic. Based on a result by Hironaka for excellent hypersurfaces of dimension two Cossart, Jannsen and Saito (CJS) gave a proof for resolution of singularities of two dimensional excellent schemes via blowing ups in regular centers. More precisely, they introduced a canonical strategy for the choice of the centers and showed by contradiction that the constructed sequence of blowing ups can not be infinite. In order to avoid technical details as good as possible, I will consider only the case of hypersurfaces (i.e., varieties defined by the vanishing of a single polynomial) in my talk. First, I will give an introduction to the problem of resolution of singularities and the strategy of CJS. After that I will explain how polyhedra can be used to obtain an invariant that captures the strict improvement of the singularity along the CJS process. Therefore, the invariant provides the basis for a direct proof of the result by CJS. The constructions involve Hironaka’s characteristic polyhedron which is a certain minimal projection of the Newton polyhedron. Hence, the ideas for the very technical proof of the improvement, can be explained by drawing rather simple pictures. This is joint work with Vincent Cossart.
26.06.2018 Henri Guenancia, Université Paul Sabatier
Note that this talk will exceptionally take
place in W065 from 13:45 -14:45!
Bochner principle on singular varieties
The well-known Bochner principle states that on any compact Kähler manifold with trivial first Chern class, any global holomorphic tensor is parallel with respect to any Ricci flat Kähler metric (provided by Yau's theorem), thus establishing a 1-1 correspondence between holomorphic tensors and invariant vectors under the so-called holonomy representation. I will explain a singular version of this statement for projective varieties with klt singularities and trivial first Chern class. This is joint work with Daniel Greb and Stefan Kebekus.
28.06.2018 Tim Kirschner, Universität Duisburg-Essen Deformations of hyperkähler twistor spaces
I present some novel results concerning the deformation theory of twistor spaces of hyperkähler type. First and foremost, I show that the (local) deformations of such twistor spaces are unobstructed—a result which is new even in the K3 surface case. Time permitting, I touch upon the (global) structure of the moduli of hyperkähler twistor spaces.
05.07.2018 Anna-Laura Sattelberger, Universität Augsburg Topological Computation of Stokes Data of Weighted Projective Lines
The quantum connection of (weighted) projective lines P(a, b) defines a system of linear differential equations, which have an irregular singularity at ∞. According to a conjecture of B. Dubrovin, the Stokes matrices can be obtained by the Gram matrix of an exceptional collection of the bounded derived category of coherent sheaves on P(a, b). By mirror symmetry, the quantum connection of P(a, b) is closely related to the Fourier transform of the Gauß–Manin connection of a Landau–Ginzburg model (X, f) of P(a, b). We apply the results of A. D’Agnolo, G. Morando, M. Hien and C. Sabbah from 2017 to the perverse sheaf Rf∗C[1] in order to obtain the Stokes matrices in a purely topological way and compare them to the Gram matrix that we obtain by Dubrovin’s conjecture.
03.09.2018 Makiko Mase, Tokyo Metropolitan University Polytope/Lattice dualities among families of K3 surfaces associated to strange duality of singularities
As a generalisation of Arnold’s strange duality, Ebeling and Takahashi found a strange duality of invertible polynomials. We consider the case of bimodal singularities, which is studied by Ebeling and Ploog, and study the families of K3 surfaces associated to them. Our question is whether or not the strange duality of bimodal singularities can be explained in terms of the families. As a conclusion, we can prove that the families are polytope dual, and that with some exceptions, they are lattice dual shown by combining toric geometry and lattice theory.
24.01.2019 Konstantin Jakob,
Universität Duisburg-Essen
Rigid Local Systems and the Geometric Langlands Correspondence
Classically, rigid local systems arise as solution sheaves of certain regular singular differential equations in the complex domain. They have been studied in depth by N. Katz who devised a way of constructing these local systems from systems of rank one by means of a convolution operation. D. Arinkin has generalized this to allow for the construction of arbitrary rigid connections (with possibly irregular singularities). However, this construction has several disadvantages, for example it only works for GL_n-local systems. In this talk I will report on work in progress (joint with Z. Yun) on another approach to construct rigid local systems for more general reductive groups via the geometric Langlands correspondence based on ideas of Heinloth, Ngô & Yun.
29.01.2019 Robert Laterveer,
IRMA, Strasbourg
Algebraic cycles and Verra fourfolds
A Verra fourfold is a smooth projective complex variety defined as double cover of P^2x P^2 branched along a divisor of bidegree (2,2). These varieties are similar to cubic fourfolds in several ways (Hodge theory, relation to hyperkaehler fourfolds, derived categories). Inspired by these multiple analogies, in this talk I will consider the Chow ring of a Verra fourfold. Among other things, I will show that the multiplicative structure of this Chow ring has a curious K3-like property.
31.01.2019 Vladimir Lazić,
Universität des Saarlandes
On Generalized Abundance
11.04.2019 Martin Schwald,
Universität Duisburg-Essen
Fibrations of irreducible symplectic varieties
A recently proven generalization of the Beauville-Bogomolov decomposition theorem motivates the study of the geometry of a new singular analogue of irreducible symplectic manifolds. Their fibrations behave arguably more similar to the smooth case than the fibrations of older definitions used by Matsushita and Namikawa.
25.04.2019 Valeria Bertini,
University of Strasbourg
Rational Curves on Irriducible Holomorphic Symplectic Varieties of OG10-type
Thanks to some recent works due to F. Charles, C. Lehn, G. Mongardi and G. Pacienza, we know that, in order to show the existence of rational curves on an irreducible holomorphic symplectic (IHS) variety of fixed deformation type, it is enough to do it for special points of their moduli space, thanks to the study of the monodromy group Mon^2 of the variety. In this talk I will start introducing the problem of finding rational curves on IHS varieties and presenting some motivation behind it; then I will describe the strategy of the authors above, focusing on a completely solved case: the Hilbert scheme of points of a general K3 surface. Finally I will present my contribution to the OG10-case, giving an example of uniruled divisor on a OG10-variety and describing the monodromy invariants of this divisor.
09.05.2019 Davide Cesare Veniani,
Johannes Gutenberg Uni-
versität Mainz
The two most algebraic Enriques surfaces
In 2007 Ohashi observed that any K3 surface covers a finite number $n$ of Enriques surfaces up to isomorphism. Building on his work, we gave an exact formula for this number $n$ in terms of lattice theory. Together with Shimada, we computed $n$ for all singular K3 surfaces of discriminant less than or equal to 36. We discovered that the K3 surface of discriminant 7 - the smallest one for which n > 0 - covers exactly two Enriques surfaces: the two most algebraic ones. We investigate their automorphism groups and give explicit equations. If time permits, I will also report on an ongoing project with Brandhorst and Sonel on the classification of K3 surfaces which do not cover any Enriques surface.
16.05.2019 Carsten Liese,
Leibniz Universität,
Cusp models and Morifans
23.06.2019 Nidhi Kaihnsa,
Computing Convex hull of Trajectories
I will talk about the convex hulls of trajectories of polynomial dynamical systems. Such trajectories also include real algebraic curves. The boundary of the resulting convex bodies are stratified into families of faces. I will discuss the numerical algorithms we developed for identifying these patches. This work is also a step towards computing the attainable region of a trajectory. This is a joint work with Daniel Ciripoi, Andreas Loehne, and Bernd Sturmfels.