Summer 2024

Working group (this term in person in Bonn): Kstability and moduli spaces of Fano varieties .
Friday 16:0018:00. See here for further information.
Special session of the seminar in Paris on June 17 & 18, 2024.
Monday June 17 (couloir 1525, 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.
2pm Giulia Saccà: Compactification of Lagrangian fibrations
Abstract: Lagrangian fibered HyperKähler manifolds are the natural generalization of elliptic K3 surfaces
and have been used to study and construct examples of compact HyperK\"ahler manifolds and
(possibly singular) symplectic varieties. In this talk I will talk about some compactification techniques for
quasiprojective Lagrangian fibrations, with applications to the study of Prym, Intermediate Jacobian,
Albanese, dual fibrations etc.
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 HsuehYung Lin.
Tuesday June 18 (couloir 1525, salle 104)
9:30 am: Phil Engel: Torelli for a class of elliptic surfaces
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
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 GushelMukai
fourfolds, the discriminant double covers can be described a sections of HK manifolds with an antisymplectic
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.
Summer 2023

Working group (this term in person in Bonn): Topics in hyperkähler geometry.
Friday 16:0018:00. See here for further information.
14.04: Jiexiang Huang, Xianyu Hu, Daniel Huybrechts: Eliptic K3s, moduli spaces, Brauer classes, proof of the OggShafarevich isomorphism; twisted Hodge structures
21.04: Dominique Mattei: Moduli spaces of sheaves on K3s with focus on BeauvilleMukai systems, i.e. relative compactified Jacobians of complete linear systems on K3s, including Arinkin's work on compactified Jacobians
28.04: YoonJoo 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: KuanWen Lai, Evgeny Shinder: ShafarevichTate theory for Lagrangian fibrations (following AbashevaRogov and Markman)
12.05: Yagna Dutta, Dominique Mattei, Evgeny Shinder: Prymfibrations, OG10 and its twists (following LazaSaccaVoisin, Voisin, Sacca)
19.05: Gebhard Martin, Giacomo Mezzedimi, KuanWen Lai: Formal Brauer groups, K3 and HK in char. p
26.05: YoonJoo Kim & the team: Hodge numbers of O'Grady 10, following de Cataldo, Rapagnetta and Sacca
2.06: no seminar
9.06: Nick Addington, Evgeny Shinder & the team: Derived equivalence of twisted Lagrangian fibrations, following Sawon 2004, Sawon 2008, AddingtonDonovanMeachan.
16.06: no seminar
23.06: RuiJie Yang: Hodge theory and Lagrangian fibrations, following Schnell
30.06: no seminar (Sheffield workshop)
7.07: Olivia Dumitrescu, Daniel Huybrechts, TBC, TBC: 30 minutes workinprogress talks
14.07: Celine Fietz, Moritz Hartlieb, TBC, TBC: 30 minutes workinprogress talks
End of semester celebration
Special session of the seminar in Paris on June 14 & 15.
Wednesday June 14 (couloir 1516, 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 tstructures (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 1516, salle 101)
9:30 am: Alexander Kuznetsov: Hilbert schemes of quadrics on GushelMukai varieties
Abstract: In the talk I will explain the general structure of Hilbert schemes of quadrics (of dimension 0, 1, and 2) on smooth GushelMukai 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 1525, 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 ongoing project with Stefano Filipazzi and Roberto Svaldi.
Winter 2022/23

Joint working group (BonnParis): Topics in hyperkähler geometry.
Friday 14:0016:00. 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.
14 October: This session will be replaced by the seminar talk by Chenyu Bai on Thursday 13 October:
Applications d'AbelJacobi 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
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 KontsevichTschinkel 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 HsuehYung Lin.
Ruijie Yang (16:00, ENS, salle W): The RiemannSchottky problem via singularities of theta divisors
Abstract: The RiemannSchottky 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 CasalainaMartin and provide a partial solution. To achieve this, we develop a complete theory of higher multiplier ideals for Qdivisors, using SabbahSchnell's theory of complex Hodge modules. This is joint work with Christian Schnell.
Friday
Alessio Bottini (10:00, Jussieu, salle 411, 1516): Towards a modular construction of OG10
Abstract: Moduli spaces of sheaves on holomorphic symplectic surfaces are examples of hyperKähler manifolds. Sheaves on higher dimensional hyperKähler 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 hyperKähler manifolds with beautiful properties, and they make good candidates to have wellbehaved moduli spaces. In this talk, I will go over these notions and give the first example of a nonrigid atomic stable bundle on a hyperKähler fourfold whose moduli space is birational to OG10.
Gianluca Pacienza (11:00, Jussieu, salle 411, 1516): 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 MorrisonKawamata cone conjecture for such manifolds using the analogous result (established by AmerikVerbitsky) for their universal cover.
zoom: https://unibonn.zoom.us/j/62174064883?pwd=TUZiWWFXUXpXcUJIRlJBdzBQUDFyQT09
MeetingID: 621 7406 4883
Kenncode: 111260
Winter 2021/22

Joint working group (BonnParis): Topics in hyperkähler geometry.
Friday 10:0012:00 The seminar will be organized as a hybrid event. In Bonn, the talks will be delivered/streamed in Lipschitz Saal, Endenicher Allee 60.
19 November: GushelMukai varieties I (Pietro Beri & Dominique Mattei) notes
26 November: Derived categories of GushelMukai varieties (Dmitrii Pirozhkov) notes
3 December: GushelMukai 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)
Zoom: https://unibonn.zoom.us/j/99037487887?pwd=TVZkNm9sTXZzMkJPMjNKZjdnTExrUT09
MeetingID: 990 3748 7887
Kenncode: 596770
Summer 2021

Joint working group (BonnParis): Topics in hyperkähler geometry
The notes of the talks can be downloaded here.
Friday 10:0012:00
16 April: F. Anella & D. Huybrechts: Semipositive line bundles (CampanaPeternellOguiso and Verbitsky)
23 April: A. Bottini: LooijengaLunts algebra and Verbitsky's theorem
30 April: P. Beri & O. Debarre: Betti numbers (Guan, Salamon,...)
14 May: M. Varesco & C. Voisin: KugaSatake (classical theory and work of MarkmanO'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 (GreenKimLazaRobles)
Fall 2020

Joint working group (BonnParis):