Cécile Gachet |
Our guest seminar takes place on Thursdays, starting at 14:15, in Room IA1/135. The talks last about one hour and are followed by questions. Here is the line-up:
I will report on work in progress with Matteo Costantini and Daniel Greb providing an intrinsic numerical characterization of totally geodesic ball quotients inside the moduli space of principally polarized abelian varieties over the complex numbers, obtained in terms of an Arakelov (in)equality associated with the underlying variation of Hodge structure. Our result extends work of Möller, Viehweg, and Zuo by removing some positivity conditions imposed in their statement. Our approach involves first showing that the period map associated with a family of Abelian varieties factors through certain MMP operations, and then generalizing the results of Möller, Viehweg, and Zuo to a singular setting.
Together with Prof. Greb (Universität Duisburg-Essen) and Prof. Christian Lehn (RUB), we organize a reading group on the preprint Baily--Borel compactifications of period images and the b-semiampleness conjecture by Bakker--Filipazzi--Mauri--Tsimerman. Here is a tentative schedule. The room is WSC-S-U-3.03 in Essen.
Thursdays from 14:15 to 15:45 in Room IA1/81.
The geometry of irreducible holomorphic symplectic (IHS, sometimes referred to as hyperkähler) manifolds can be studied through the numerical properties of algebraic classes with respect to a non-degenerate quadratic form on the second cohomology group. In this context, a famous conjecture (SYZ) predicts that the existence of Lagrangian fibrations is detected by the presence of certain isotropic classes. While the conjecture holds in all known examples, it remains open in general. Recently, singular analogues of IHS manifolds have been proposed, providing a new framework to test the conjecture in a singular setting. In this talk, I will focus on Nikulin orbifolds, which are among the simplest singular examples, and present recent work classifying possible fibrations in this deformation class, from which the SYZ conjecture follows in this specific case.
We discuss basic properties of currents on a complex manifold such as positivity, Lelong numbers, Siu's theorem, and relation to positivity properties from algebraic geometry.
The U-hat theorem of Bérczi, Doran, Hawes and Kirwan gives conditions for when a linear action of a complex graded unipotent group admits a geometric quotient. It is one of the key results non-reductive geometric invariant theory is built on. We give a stacky re-interpretation of this theorem in terms of Θ-strata, as introduced by Halpern-Leistner, of algebraic stacks. As a corollary we generalize the U-hat theorem to not necessarily linear actions of graded unipotent groups over a Noetherian base scheme.
For a convex polygon bounded by n lines there exists a unique curve -the so called adjoint- of degree n-3 passing through all intersection points of the lines except for those points, which are vertices of the polygon. Motivated by possible applications in finite element methods they were introduced by Wachspress in 1975 in his work on generalized barycentric coordinates. The definition of the adjoint extends to polygons in the complex projective plane, which are bounded by n lines in general position. The adjoint map, which maps an n-gon to its adjoint curve, happens to be dominant and generically finite only in the case of quartics and their associated heptagons. After establishing 864 by numerical certification as a lower bound, Kohn et al. (2021) conjectured this to be the precise number of heptagons associated to a generic plane quartic. In this talk, I will present joint work with Daniele Agostini, Daniel Plaumann and Rainer Sinn, which proves the conjecture: Employing intersection theory and the Scorza correspondence for quartics we show that 864 is indeed an upper bound. Furthermore we present a new proof for the lower bound revealed to us by a careful study of the Klein quartic.
This talk presents a unified framework for finiteness results concerning arithmetic points on algebraic curves, exploring the analogy between number fields and function fields. The number field setting, joint work with F. Janbazi, generalizes and extends classical results of Birch–Merriman, Siegel, and Faltings. We prove that the set of Galois-conjugate points on a smooth projective curve with good reduction outside a fixed finite set of places is finite, when considered up to the action of the automorphism group of a proper integral model. Motivated by this, we consider the function field analogue, involving a smooth and proper family of curves over an affine curve defined over a finite field. In this setting, we show that for a fixed degree, there are only finitely many étale relative divisors over the base, up to the action of the family's automorphism group (and including the Frobenius in the isotrivial case). Together, these results illustrate both the parallels and distinctions between the two arithmetic settings, contributing to a broader unifying perspective on finiteness.
(joint work with Víctor Gonzàlez-Alonso) Barth and Peters showed that a generic complex Enriques surface has exactly 527 isomorphism classes of elliptic fibrations. We show that every Enriques surface has precisely 527 isomorphism classes of elliptic fibrations when counted with multiplicity. Their reducible singular fibers and the multiplicities can be calculated explicitly. The statements holds over any algebraically closed field of characteristic not two. To explain these results, we construct a moduli space of complex elliptic Enriques surfaces and study the ramification behavior of the forgetful map to the moduli space of unpolarized Enriques surfaces. Curiously, the ramification indices of a similar map computes the hyperbolic volume of the rational polyhedral fundamental domain appearing in the Morrison-Kawamata cone conjecture.
Proving unobstructedness of a deformation problem is usually a difficult problem, especially when the obstruction space is not zero. In the nineties, Ran and Tian developed a new technique called T1-lifting to prove unobstructedness of a deformation problem under mild conditions; this is namely a key ingredient of the proof of the celebrated Bogomolov–Tian–Todorov (BTT) theorem that says that deformations of compact Calabi–Yau manifolds are unobstructed. In the talk, we consider some ruled surfaces glued along disjoint sections with a shift; these varieties are singular so BTT does not apply; we would nevertheless prove that their deformations are unobstructed. In order to do that, we will use logarithmic geometry and more precisely that, from the point of view of logarithmic geometry, these varieties are smooth. We will use logarithmic replacements to the ingredients of the proof of BTT to prove the unobstructedness of logarithmic deformations and then, we will link logarithmic deformations to classical deformations to derive unobstructedness for usual flat deformations.
An important result of Bogomolov from 1978 asserts that Chern numbers of semistable vector bundles on projective surfaces satisfy a certain inequality. In this talk we report on recent joint work with Mihai Pavel and Julius Ross. We draw a parallel between the Bogomolov inequality, the Hodge Index Theorem and Lübke's inequality for Hermite-Einstein vector bundles and propose generalizations in the case of higher dimensional manifolds. We prove that the expected generalizations hold in a number of cases and show how one can apply them to boundedness questions of semistable coherent sheaves.
In this talk, I will explain a moduli-theoretic way to prove projectivity of moduli spaces of vector bundles on (orbifold) curves and
quiver representations by producing sections of determinantal line bundles. These moduli spaces have geometric invariant theory constructions, but they can also be constructed abstractly using existence criteria for stacks - the latter does not yield projectivity that one usually gets from GIT. Our approach to projectivity involves characterising semistability via Hom-vanishing conditions, and using dualities and elementary modifications to separate points, and moreover gives effective bounds. I will ignore all the details on the stacky-construction, and will instead focus on producing sections. I plan to highlight the similarities and differences between the case of quiver representations and vector bundles on curves. This is based on two joint papers with Pieter Belmans, Chiara Damiolini, Hans Franzen, Svetlana Makarova, Lisanne Taams and Tuomas Tajakka.
Mondays from 14:15 to 15:45 in Room IA1/75.
Hyperbolic curves are curves in the real projective plane with the maximal number of nested ovals. They arise in several different ways and have been studied extensively, along with their generalizations to any dimension. In this talk, I will give an overview, focussing on examples, determinantal representations, computational questions and open problems. (Partly based on joint work with Mario Kummer, Simone Naldi, Bernd Sturmfels, Cynthia Vinzant).
In 1913, De Franchis proved that the number of surjective holomorphic maps from X to Y is finite when X and Y are compact Riemann surfaces and Y has genus at least 2. This result was extended to higher dimensions by Noguchi for certain hyperbolic varieties, and Campana established an analogous statement for hyperbolic orbifold curves. In this talk, we will introduce various notions related to hyperbolicity and orbifolds in order to understand certain finiteness properties of holomorphic maps between hyperbolic varieties or between hyperbolic orbifold pairs, thus generalizing the De Franchis theorem.
In deformation theory, it is customary to seek to construct for a given structure a family that accounts, at least locally, for all its deformations and in the most "economical" way possible. Kuranishi's theorem, for example, guarantees the existence of such a family for any compact complex manifold. The work of Girbau, Haefliger, Nicolau and Sundararaman reproduces Kuranishi's arguments and constructs such families for holomorphic and transversely holomorphic foliations, and a particular type of deformation for holomorphic foliations. All the above structures are examples of a structure introduced by Kodaira and Spencer: multifoliate structures. In this talk, I shall present what these structures are, the deformation theory associated with them and, finally, I shall introduce the notion of Calabi-Yau foliations and some remarkable properties of these foliations.
Enriques manifolds are an higher-dimensional analogue of Enriques surfaces. While Enriques surfaces are all obtained as a quotient of a fixed-point-free involution on a K3 surface, in higher dimension the situation is more intricate: there are examples of Enriques manifolds obtained as quotients of irreducible symplectic manifolds by fixed-point-free automorphisms of order 2, 3 or 4 and, moreover, for some deformation types of irreducible symplectic manifolds it is not known if finite order fixed-point-free automorphisms exist. We use the Looijenga–Lunts–Verbitsky algebra to describe the action of a finite order automorphism on the total cohomology of a manifold of OG10 type. As an application, we prove that no Enriques manifolds arise as quotients of manifolds of OG10 type. This answers to a question recently raised by Pacienza and Sarti.
At the beginning of the 20th century, it was known that any compact connected, simply connected Riemann surface is biholomorphic to the projective line. Subsequently, several characterizations of projective spaces were established. For instance, Siu and Yau stated that projective spaces are the only Kähler manifolds with positive holomorphic bisectional curvature, and Mori proved that they are the only projective manifolds that have an ample tangent bundle. In a different direction, projective spaces are the only Kähler-Einstein manifolds with a positive constant satisfying the equality in the Miyaoka-Yau inequality. This result originating from uniformization theory was generalized in the singular setting by Greb, Kebekus, Peternell and Druel, Guenancia, Păun. More precisely, they characterize singular quotients of Pn by finite groups acting freely in codimension 1. The aim of this talk is to discuss a generalization of Greb-Kebekus-Peternell's result in order to characterize quotients of Pn by any group action.
For any lattice polarised elliptic K3 surface, van Geemen's Brauer twist construction associates to any order 2 element in its Brauer group another elliptic K3 surface, where the original K3 surface can be recovered by taking the relative Jacobian fibration. We will give explicit geometric constructions of some of the Brauer twists of a very general K3 surface that admits a van Geemen-Sarti involution, as well as their birational models. We also observe the same construction works to geometrically realise the Brauer twists of K3 surfaces in some families of higher Picard ranks. This is an ongoing work with Adrian Clingher and Andreas Malmendier.
Invited speakers
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Titles of the talks
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Abstracts
Given a family of proper algebraic varieties arranged as the fibres of a smooth variety over a smooth base, the decomposition theorem captures how the singular cohomologies of these varieties in the fibres vary as they become more and more singular. In this setup, lot of symmetries, e.g. Hard Leschetz, Poincaré duality, that are enjoyed by smooth projective varieties, manifest themselves fibrewise. Cohomologies of smooth projective varieties also enjoy a very symmetric diamond shaped decomposition in subvector spaces (known as the Hodge decomposition). This kind of symmetry, albeit mysterious for general families, shows up very elegantly for a certain degenerate family of Abelian varieties; the Lagrangian fibrations of hyperkähler manifolds. Hodge module is a powerful tool for proving these in a rigorous, yet relatively lazy way. Furthermore, in some examples these abelian varieties generically arise as (intermediate) Jacobian of curves and cubic threefolds. Hodge module theoretic techniques also allow us to consider the relative sheaf of (intermediate) Jacobian, a gadget that helps us construct new Lagrangian fibrations from the old one.
The plan for the three talks will roughly be as follows. 1) Crash course on decomposition theorem via various examples after de Cataldo-Migliorini. 2) Lagrangian fibration of hyperkähler manifolds and symmetries after Matsuhita and Schnell 3) Intermediate Jacobians in family after D-Mattei-Shinder.
In the theory of linear complex differential equations, an essential distinction is that between two types of singular points: While regular singularities have been well understood for a long time, the classification of irregular ones is much more recent. A key ingredient in the latter is the Stokes phenomenon, which was originally discovered by Stokes while performing computations in optics. It allows us to adapt the concepts of monodromy, local systems and perverse sheaves to the irregular case.
In this lecture series, we will learn about topological perspectives on systems with possibly irregular singularities that have been developed
in the last 50 years, and we will use them to explain some explicit results on Fourier transforms, an integral transform that is ubiquitious
in mathematics and physics. In the first lecture, we survey some basics about D-modules, and we are going to see in particular how to classify them via so-called Riemann-Hilbert correspondences. In the second lecture, we will learn about the Stokes phenomenon in the context of irregular singularities, and in particular different ways of representing it geometrically. In the final lecture, the Fourier transform will come into play, and we will investigate the question of how the Stokes data behave under this transform.
To any perverse sheaf on an abelian variety one may attach a linear algebraic group by applying Tannaka duality to the tensor category generated by its convolution powers. The arising groups play a fundamental role in the geometry and arithmetic of irregular varieties. We will give a self-contained introduction to the topic starting from generic vanishing and the geometry of Gauss maps, and then discuss some recent applications in two directions: (1) Singularities of theta divisors and the moduli of abelian varieties, and (2) big monodromy results in arithmetic geometry.
Meet-and-greet session
On Wednesday, March 5th, we will have three 20-minute talks by participants.
Prospective timetable
The room is ID 03/653 (in the building ID, on the floor 03 = "negative 3", in the room numbered 653). Building ID can be found on the map of the campus here .
| Time | Wednesday 05.03 | Thursday 06.03 | Friday 07.03 |
| 10 - 11 | Krämer 2 | Hohl 3 | |
| 11:30 - 12:30 | Dutta 2 | Dutta 3 | |
| 13 - 14 | Dutta 1 | ||
| 14:30 - 15:30 | Hohl 1 | Hohl 2 | |
| 16 - 17 | Krämer 1 | Krämer 3 | |
| 17:30 - 18:30 | Meet-and-greet |
Organisation