ERC Synergy Grant

Winter 2024

 

Working group (this term in person in Bonn): Arithmetic finiteness for very irregular varieties. 

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        Friday 16:00-18:00. See here for the program.

     

         Special session of the HyperK seminar in Bonn on January 9 & 10, 2025.

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The event will take place at the Hausdorff Institute (Poppelsdorfer Allee)

 

Thursday, January 9, 10:30 am: Frank Gounelas (Bonn): Some new results on curves on K3s

Abstract:  There are numerous questions and open conjectures about the (non-)existence of curves on K3 surfaces which vary maximally in moduli, or, respectively, do not vary at all (the isotrivial case) - a classical open problem in the area, which goes back to Deligne/Schoen/Serre is whether a very general K3 surface can be dominated by the product of two smooth curves. In this talk will summarise some recent results and in particular focus on recent work with Chen and Dutta proving that if the Picard rank is one, then there can be no generically smooth isotrivial families of curves.

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Thursday, January 9, 3 pm: Christian Lehn (Bochum): Lagrangian fibrations of hyperkähler varieties

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Thursday, January 9, 5 pm: Georg Oberdieck (Heidelberg): On invariants of moduli spaces of sheaves on Enriques surfaces

Abstract: Under a suitable primitive assumption, moduli spaces of stable sheaves on Enriques surfaces are smooth projective varieties with torsion canonical bundle, which were proven by Beckmann and Nuer-Yoshioka to be birational to Hilbert schemes of points on the Enriques or compactified Jacobians (depending on whether the dimension is even or odd). While in 
the even-dimensional case this determines their Betti numbers, not much is known in the odd-dimensional case (b_1,b_2 was computed by Sacca). In this talk I will give an overview about recent curve counting results on 
the local Enriques surfaces and explain what they predict for the moduli spaces of sheaves. In particular, we will state a conjectural relation how the perverse Hodge numbers of the compactified Jacobians are determined by their Betti numbers, and give an asymptotic formula for their Betti numbers.

 

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Friday, January 10, 9 am: Ariyan Javanpeykar (Nijmegen): The weakly special conjecture contradicts Orbifold Mordell, and hence abc

Abstract: Lang conjectured that varieties of general type over a number field have very few rational points. In 2000, guided by Lang's conjecture and in search of a converse statement, Abramovich, Colliot-Thelene, Harris, and Tschinkel formulated the "Weakly Special Conjecture": every weakly special variety over a number field has a potentially dense set of rational points. In this talk I will explain how this conjecture contradicts the abc conjecture, and more precisely Campana's "Orbifold Mordell" conjecture. Indeed, starting from an Enriques surface over Q(t) constructed by Lafon, we give the first examples of smooth projective weakly special threefolds which fiber over the projective line in Enriques surfaces with nowhere reduced, but non-divisible, fibers. The existence of these threefolds shows that the Weakly Special Conjecture contradicts the abc conjecture, but also shows that Enriques surfaces and K3 surfaces can have non-divisible but nowhere reduced degenerations, thereby answering a question raised by Campana in 2005. This is joint work with Finn Bartsch, Frederic Campana, and Olivier Wittenberg.

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Friday, January 10, 11 am: John Ottem (Oslo): Irrationality of complete intersections in projective space and Grassmannians

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Limited funds available. Please contact huybrech@math.uni-bonn.de

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Summer 2024

• Working group (this term in person in Bonn): K-stability and moduli spaces of Fano varieties . 

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Friday 16:00-18:00.  See here for further information.

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Special session of the seminar in Paris on June 17 & 18, 2024. 

Monday June 17 (couloir 15-25, salle 104)

11am: Qizheng Yin: Cohomological and motivic aspects of compactified Jacobian fibrations

 

Abstract: Beauville showed using Fourier transforms that the Chow ring/motive of an abelian variety admits a natural, multiplicative decomposition. I will explain how Beauville’s theory can be extended to certain abelian fibrations with singular fibers. One notable consequence of this extension is a proof of the P=W conjecture in nonabelian Hodge theory. In the second half of my talk I will focus on aspects beyond P=W and discuss some related open questions. Joint work in progress with Davesh Maulik and Junliang Shen.

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2pm Giulia Saccà: Compactification of Lagrangian fibrations

Abstract: Lagrangian fibered Hyper-Kähler manifolds are the natural generalization of elliptic K3 surfaces

and have been used to study and construct examples of compact Hyper-K\"ahler manifolds and
(possibly singular) symplectic varieties. In this talk I will talk about some compactification techniques for
quasi-projective Lagrangian fibrations, with applications to the study of Prym, Intermediate Jacobian,

Albanese, dual fibrations etc.

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4pm Zhi Jiang: Severi type inequalities and equalities

Abstract: Barja，Pardini，and Stoppino introduced continuous rank functions on varieties of maximal Albanese dimension around 2015 and they applied these functions to improve Severi type inequalities and characterized partially varieties on the Severi line. Together with G. Pareschi, we extended the construction of continuous rank functions to get cohomological rank functions on Abelian varieties and get precise local polynomial expressions of these functions.
In this talk, we explain that the precise expression can be used to understand the structures of irregular surfaces with small birational invariants or on another Severi line. This talk is based on a joint work in progress with Hsueh-Yung Lin.

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Tuesday June 18 (couloir 15-25, salle 104)

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9:30 am: Phil Engel: Torelli for a class of elliptic surfaces

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Abstract: ​I will discuss a proof that the period mapping is dominant for elliptic surfaces over an elliptic curve with 12 nodal fibers, but that its degree is larger than 1. This is joint work with Francois Greer and Abigail Ward. 

 

11am: Giovanni Mongardi: General type surfaces of K3 type and HK manifolds

 

Abstract: Some Fano fourfolds of K3 type have natural conic fibrations, whose discriminant 

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locus has a double cover of general type. The antiinvariant cohomology of this surfaces carries a natural Hodge

 

structure, corresponding to the K3 structure of the Fano fourfold. In a work in collaboration with Bernardara,

 

Fatighenti, G. Kapustka, M. Kapustka,  Manivel and Tanturri, we prove that in the case of Gushel-Mukai

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fourfolds, the discriminant double covers can be described a sections of HK manifolds with an anti-symplectic

 

involution. 

 

2pm: Susanna Zimmermann: Algebraic groups acting on fibrations

 

Abstract: What algebraic groups act birationally and faithfully on a projective space? In dimension 2 and 3, 

every such group is contained in a maximal such group (with respect to inclusion). I will explain why this

no longer holds in dimension 5 and higher. This is a collaboration with A. Fanelli and E. Floris. 

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Summer 2023

• Working group (this term in person in Bonn): Topics in hyperkähler geometry. 

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Friday 16:00-18:00.  See here for further information.

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14.04: Jiexiang Huang, Xianyu Hu, Daniel Huybrechts: Eliptic K3s, moduli spaces, Brauer classes, proof of the Ogg-Shafarevich isomorphism; twisted Hodge structures

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21.04: Dominique Mattei: Moduli spaces of sheaves on K3s with focus on Beauville-Mukai systems, i.e. relative compactified Jacobians of complete linear systems on K3s, including Arinkin's work on compactified Jacobians

 

28.04: Yoon-Joo Kim: Overview of Lagrangian fibrations on HK varieties (Matsushita's conjecture, examples), versions of duality of Lagrangian fibrations (following Sawon, Kim), what can be these fibrations twisted by (following Sawon)

 

5.05: Kuan-Wen Lai, Evgeny Shinder: Shafarevich-Tate theory for Lagrangian fibrations (following Abasheva-Rogov and Markman) 

 

12.05: Yagna Dutta, Dominique Mattei, Evgeny Shinder: Prym-fibrations, OG10 and its twists (following Laza-Sacca-Voisin, Voisin, Sacca)

 

19.05: Gebhard Martin, Giacomo Mezzedimi, Kuan-Wen Lai: Formal Brauer groups, K3 and HK in char. p

 

26.05: Yoon-Joo Kim & the team: Hodge numbers of O'Grady 10, following de Cataldo, Rapagnetta and Sacca 

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2.06: no seminar 

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9.06: Nick Addington, Evgeny Shinder & the team: Derived equivalence of twisted Lagrangian fibrations, following Sawon 2004, Sawon 2008, Addington-Donovan-Meachan. 

 

16.06: no seminar

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23.06: Rui-Jie Yang: Hodge theory and Lagrangian fibrations, following Schnell

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30.06: no seminar (Sheffield workshop)

 

7.07: Olivia Dumitrescu, Daniel Huybrechts, TBC, TBC: 30 minutes work-in-progress talks

 

14.07: Celine Fietz, Moritz Hartlieb, TBC, TBC: 30 minutes work-in-progress talks

 

End of semester celebration

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Special session of the seminar in Paris on June 14 & 15. 

Wednesday June 14 (couloir 15-16, salle 413)

2pm: Paolo Stellari: Deformations of stability conditions with applications to Hilbert schemes of points and very general abelian varieties
 
Abstract: The construction of stability conditions on the bounded derived category of coherent sheaves on smooth projective varieties is
notoriously a difficult problem, especially when the canonical bundle is trivial. In this talk, I will illustrate a new and very effective  technique based on deformations. A key ingredient is a general result about deformations of bounded t-structures (and with some additional and mild assumptions). Two remarkable applications are the  construction of stability conditions for very general abelian varieties in any dimension and for some irreducible holomorphic symplectic manifolds, again in all possible dimensions. This is joint  work with C. Li, E. Macri' and X. Zhao.

4pm: Martí Lahoz: Cohomological rank functions on abelian surfaces via Bridgeland stability

Abstract: In the context of polarized abelian varieties, Zhi Jiang and Giuseppe Pareschi have introduced the cohomological rank functions associated to a (complex of) coherent sheaves. These functions are closely related to the continuous rank functions introduced previously by Miguel Angel Barja, and studied together with Rita Pardini and Lidia Stoppino in the context of irregular varieties. I will present joint work with Andrés Rojas that show that, in the case of abelian surfaces, Bridgeland stability provides an alternative description of the cohomological rank functions. This helps to understand their general structure, and allows to compute geometrically meaningful examples. I will illustrate the potential of this reinterpretation by presenting new results on syzygies of abelian surfaces proven by Andrés Rojas.

Thursday June 15 (couloir 15-16, salle 101)

9:30 am: Alexander Kuznetsov: Hilbert schemes of quadrics on Gushel-Mukai varieties

Abstract: In the talk I will explain the general structure of Hilbert schemes of quadrics (of dimension 0, 1, and 2) on smooth Gushel-Mukai varieties (of dimension 2 \le n \le 6), and some explicit examples of these schemes. This is a joint work in progress with Olivier Debarre.

11 am: Justin Sawon: Lagrangian fibrations in four dimensions

Abstract: We consider Lagrangian fibrations by abelian surfaces over the complex projective plane, with total space holomorphic symplectic manifolds and orbifolds. There are examples whose fibres are (1,d) polarized for d=1 to 4. We recall some classification results of Markushevich and Kamenova in the principally polarized case, and a new classification result in the (1,2) polarized case (joint work with Xuqiang Qin). We also describe restrictions on the polarization; indeed, `most' polarizations are not possible.

Thursday June 15 (couloir 15-25, salle 502) Séminaire de Géométrie algébrique

2pm: Mirko Mauri: Remarks on the topology of hyperkähler varieties

Abstract: The Nagai conjecture and the SYZ conjecture concern respectively the geometry of degenerations and fibrations of hyperkähler varieties. In this talk I will explore some topological consequences of these conjectures. This is an account on a joint project with Daniel Huybrechts and an on-going project with Stefano Filipazzi and Roberto Svaldi.

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Winter 2022/23

• Joint working group (Bonn-Paris): Topics in hyperkähler geometry. 

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Friday 14:00-16:00. The seminar will be organized as a hybrid event. Typically we will have two speakers with talks of 50 min each. In Bonn, the talks will be delivered/streamed in Lipschitz Saal, Endenicher Allee 60.

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14 October: This session will be replaced by the seminar talk by Chenyu Bai on Thursday 13 October:

Applications d'Abel--Jacobi des familles lagrangiennes, 14:00 in the Séminaire de géométrie algébrique

21 October: LLV algebra acting in Chow: Talks by G. Oberdieck and Y. Kim (Bonn)

28 October: No seminar (school holidays in France)

04 November: No seminar (school holidays in France)

11 November: This session will be replaced by the seminar talk by Pietro Beri on Thursday 10 November

Kodaira dimension of some moduli spaces of polarized hyperkähler manifolds 14:00 in the Séminaire de géométrie algébrique

18 November: Cones and birational maps. Talks by Francesco Denisi & Yajnaseni Dutta/Dominique Mattei

​25 November: Filtration of Chow groups: Talks by Charles Vial & Zhiyuan Li

02 December: André motives of hyperkähler varieties: Talks by Salvatore Floccari & Andrey Soldatenkov

​09 December: Giulia Saccà: cancelled

 

 

​12 & 13 January: Special Paris addition:

 

 Thursday

 E. Shinder (14:00, ENS, salle W): Motivic invariants of birational maps

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Abstract: I explain how exceptional divisors of birational maps can be assembled into invariants taking values in the Grothendieck ring of varieties and in the Kontsevich-Tschinkel ring. Using these invariants we prove new results about the structure of the groups of birational isomorphisms; in particular, we prove that various Cremona groups are not generated by regularizable elements. This is joint work with Hsueh-Yung Lin.

 

Rui-jie Yang (16:00, ENS, salle W): The Riemann-Schottky problem via singularities of theta divisors

Abstract: The Riemann-Schottky problem is the problem of determining which principally polarized abelian varieties (PPAV) arise as Jacobians of curves. Riemann showed that the theta divisor on the Jacobian of a hyperelliptic curve has singularity of codimension three. A hundred years later, Debarre conjectured that any irreducible PPAV with such property must come from hyperelliptic curves. In this talk, I will discuss a refinement of this conjecture by Casalaina-Martin and provide a partial solution. To achieve this, we develop a complete theory of higher multiplier ideals for Q-divisors, using Sabbah-Schnell's theory of complex Hodge modules. This is joint work with Christian Schnell.

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Friday

Alessio Bottini (10:00, Jussieu, salle 411, 15-16): Towards a modular construction of OG10

 

Abstract: Moduli spaces of sheaves on holomorphic symplectic surfaces are examples of hyper-Kähler manifolds. Sheaves on higher dimensional hyper-Kähler manifolds have proven much more difficult to study, although it is believed that their moduli spaces could lead to new examples. Recently, the introduction of modular and atomic sheaves has lead to developments in this subject. They are special classes of sheaves on hyper-Kähler manifolds with beautiful properties, and they make good candidates to have well-behaved moduli spaces. In this talk, I will go over these notions and give the first example of a non-rigid atomic stable bundle on a hyper-Kähler fourfold whose moduli space is birational to OG10.

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Gianluca Pacienza (11:00, Jussieu, salle 411, 15-16): On the cone conjecture for Enriques manifolds. 

Abstract: Enriques manifolds are non simply connected manifolds whose universal cover is irreducible holomorphic symplectic, and as such they are natural generalizations of Enriques surfaces. In this talk I will report on a joint work in progress with Alessandra Sarti in which  we study the Morrison-Kawamata cone conjecture for such manifolds using the analogous result (established by Amerik-Verbitsky) for their universal cover.

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zoom: https://uni-bonn.zoom.us/j/62174064883?pwd=TUZiWWFXUXpXcUJIRlJBdzBQUDFyQT09

Meeting-ID: 621 7406 4883

Kenncode: 111260

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Winter 2021/22

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• Joint working group (Bonn-Paris): Topics in hyperkähler geometry. 

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Friday 10:00-12:00 The seminar will be organized as a hybrid event. In Bonn, the talks will be delivered/streamed in Lipschitz Saal, Endenicher Allee 60.

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19 November: Gushel-Mukai varieties I (Pietro Beri & Dominique Mattei) notes

26 November: Derived categories of Gushel-Mukai varieties (Dmitrii Pirozhkov) notes

3 December: Gushel-Mukai varieties II  (Pietro Beri & Olivier Debarre) notes

10 December: Characteristic foliations (Fabrizio Anella & Daniel Huybrechts) notes

17 December: Dual fibrations (Thorsten Beckmann & Daniel Huybrechts) notes

14 January: Vector bundles on hyperkähler manifolds (Alessio Bottini & Emanuele Macrì)

21 January: Integral Hodge conjecture (Ignacio Barros & Claire Voisin)

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Zoom: https://uni-bonn.zoom.us/j/99037487887?pwd=TVZkNm9sTXZzMkJPMjNKZjdnTExrUT09

Meeting-ID: 990 3748 7887

Kenncode: 596770

Summer 2021

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• Joint working group (Bonn-Paris): Topics in hyperkähler geometry

 

The notes of the talks can be downloaded here.

 

Friday 10:00-12:00

16 April: F. Anella & D. Huybrechts: Semipositive line bundles (Campana-Peternell-Oguiso and Verbitsky)

23 April: A. Bottini: Looijenga-Lunts algebra and Verbitsky's theorem

30 April: P. Beri & O. Debarre: Betti numbers (Guan, Salamon,...)

14 May: M. Varesco & C. Voisin: Kuga-Satake (classical theory and work of Markman-O'Grady)

21 May: T. Beckmann: Derived categories and derived monodromy (work of Taelman)

4 June: D. Huybrechts & M. Mauri: Lagrangian fibrations (Matsushita, Shen & Yin) 

11 June: G. Oberdieck & J. Song: Representation theory and cohomology (Green-Kim-Laza-Robles)

      

 

• SAG (Bonn)

• SAGA (Orsay)

• SGA (Paris)