Arnaud Beauville(Université Côte d’Azur)
Samir Canning(ETH)
Qingyuan Jiang(HKUST)
Chen Jiang(Fudan)
Yujiro Kawamata(Tokyo)
Yongnam Lee(IBS)
Yucheng Liu(ChongQing)
Sho Tanimoto(Nagoya)
Zhiyu Tian(Peking)
Qizheng Yin*(Peking)
Longting Wu(Sustech)
Feinuo Zhang(Fudan)
Haitao Zou(Bielefeld)
Lefschetz property and maximal variation of hyperplane sections
Arnaud Beauville(Université Côte d'Azur)
Abstract: Let X be a smooth hypersurface in P^n . Does the family of (smooth) hyperplane sections of X , up to isomorphism, have maximal dimension n ? It turns out that this is equivalent to a particular instance of the “weak Lefschetz property” (WLP), a property of graded artinian algebras which has been intensely studied by algebraists. In the talk I will first survey the known results about WLP, then explain the relation with the hyperplane sections problem. This gives a positive answer to that problem for deg(X) > n , and also for X a cubic threefold.
Cycles on moduli spaces of curves and abelian varieties
Samir Canning(ETH)
Abstract: I will explain a surprising connection between non-tautological classes on the moduli space of abelian varieties and tautological relations on the moduli space of curves. The story begins with joint work with Larson and Schmitt on the Gorenstein problem for compact type moduli and joint work with Oprea and Pandharipande on the product loci on moduli spaces of abelian varieties. I will also discuss recent progress by Iribar López toward extending the story to Noether-Lefschetz loci on the moduli space of abelian varieties.
Fano indices of canonical Fano 3-folds
Chen Jiang(Fudan)
Abstract: We show that the Q-Fano index of a canonical weak Fano 3-fold is at most 66. This upper bound is optimal and gives an affirmative answer to a conjecture of Chengxi Wang in dimension 3. During the proof, we establish a new Riemmann–Roch formula for canonical 3-folds and provide a detailed study of non-isolated singularities on canonical Fano 3-folds, concerning both their local and global properties. Our proof also involves a Kawamata–Miyaoka type inequality and geometry of foliations of rank 2 on canonical Fano 3-folds. This is a joint work with Haidong Liu.
Grassmannians of Complexes and Instantons on Blowups
Qingyuan Jiang(HKUST)
Abstract: Recent advances in derived algebraic geometry (DAG) have enabled us to extend Grothendieck’s theory of Grassmannians of sheaves to the broader framework of complexes. In my earlier work, I systematically studied the theory of Grassmannians of complexes, leading in particular to a unified formula for the derived categories of Grassmannians of two-term perfect complexes. In this talk, I will examine the representation-theoretic structure underlying this formula via Clifford algebra representations. As an application, I will discuss how this framework provides a positive answer to a question posed by Nakajima and Vafa–Witten regarding the representation-theoretic interpretation of the blowup formula for instanton moduli on surface blowups. This talk is based on joint work with Weiping Li and Yu Zhao.
On NC deformations of sheaves and NC moduli space
Yujiro Kawamata(Tokyo)
Abstract: The NC deformations of sheaves have versal families. But calculating the versal deformations is difficult. We review some cases where the versal deformations are determined and applications. I will also take Grassmann variety as an easy example, and show that even a global moduli space exists as an NC scheme.
K3 tails in the KSBA moduli of Kunev surfaces
Yongnam Lee(IBS)
Abstract: In the 1970s, Kunev constructed certain minimal surfaces of general type satisfying p_g = K^2 = 1, which gave a counterexample to the local Torelli theorem. Then Todorov, and Catanese independently, gave a description of all minimal surfaces of general type with p_g = K^2 = 1: the canonical model of such a surface is a weighted complete intersection of bidegree (6, 6) in the weighted projective space P(1, 2, 2, 3, 3). In particular, these surfaces can be viewed as degree 4 covers of the projective plane. The case in which this cover is Galois, recovers Kunev’s example, i.e. Kunev surfaces are bi-double cover of projective plane branched over two general cubics and a line.
Recently, Kerr and Laza classified K3 tails which comes from the isolated quasi-homogeneous hypersurface singularities. In the pure tail case, these are exactly the 14 Dolgachev singularities (exceptional unimodal singularities), the 6 quadrilateral singularities, and 2 trimodal singularities.
Motivated by Kerr-Laza’s classification and recent work by Gallardo, Pearlstein, Schaffler and Zhang on the construction of specific boundary divisors in KSBA moduli space of I-surfaces which come from the exceptional unimodal singularities, we study specific boundary divisors in KSBA moduli of Kunev surfaces which come from K3 tails corresponding KSBA stable limits. In this talk, we focus on two type of singularities which are double coverer of Z_{12} and W_{15}singularities. Both are codimension 2 weighted K3 surfaces (No. 42 and 13 in Iano-Fletcher’s list). This work is in progress with Luca Schaffler.
Constructing stable Hilbert bundles via Diophantine approximation
Yucheng Liu(Chongqing)
Abstract: In this talk, I will present a construction of stable Hilbert bundles on any compact Riemann surfaces with positive genera. The main constructive step is via investigating the arithmetic property of the upper half plane in Bridgeland’s definition of stability conditions and its homological parts. We will also briefly discuss its relation to noncommutative geometry and possible applications on quantum field theory. This is based on work joint with Heng Du, Qiangyuan Jiang and work joint with Biao Ma.
Homological stability and weak approximation
Sho Tanimoto(Nagoya)
Abstract: Motivated by a topological proof of Manin’s conjecture over global function fields, Ellenberg and Venkatesh envisioned homological stability for the space of sections of Fano fibrations. In this talk we discuss this property in the context of weak approximation and establish such a stability for certain Fano fibrations. This is joint work with Yuri Tschinkel.
Poincaré polynomials of moduli spaces of 1-dimensional sheaves on the projective plane
Longting Wu(Sustech)
Abstract: The geometry of moduli spaces of one-dimensional sheaves on the projective plane has attracted a lot of study recently. In this talk, I will give a new calculation of the Betti numbers of the moduli spaces of one-dimensional sheaves on the projective plane using Gromov-Witten invariants of local P^2 and local curves. The new calculation is based on the refined sheaves/GW correspondence established by Bousseau and all genus local/relative correspondence given by Bousseau-Fan-Guo-Wu. It can be used to prove the divisibility property of Poincaré polynomials of moduli spaces of one-dimensional sheaves on projective plane conjectured by Choi-van Garrel-Katz-Takahashi, and can also be used to determine the leading Betti numbers. Some conjectures concerning the higher range Betti numbers will be proposed if time permits. This is based on a joint work with Shuai Guo and Miguel Moreira.
Moduli of one-dimensional sheaves: stabilization phenomena and pathologies
Feinuo Zhang(Fudan)
Abstract: Moduli spaces of sheaves on surfaces play an important role in algebraic geometry, yet many natural questions about them remain open. Motivated by stabilization phenomena for moduli spaces of torsion free sheaves, we explore the behavior of the moduli space of one-dimensional sheaves supported on sufficiently positive divisors on a surface. In this talk, I will report recent progress on the Betti and Picard numbers of such moduli spaces, along with unexpected examples of reducible moduli spaces that arise for certain surfaces of general type. This talk is based on joint work arXiv:2406.10004 with Weite Pi, Junliang Shen, and Fei Si, and arXiv:2503.06153 with Fei Si.
Pointed Shafarevich conjecture for primitive symplectic varieties
Haitao Zou(Bielefeld)
Abstract: The Shafarevich conjecture predicts that families of varieties over number fields with “good reduction” outside finitely many places are severely restricted—in classical cases like curves or polarized abelian varieties, only finitely many isomorphism classes occur. In this talk, I will present a geometric Shafarevich conjecture for primitive symplectic varieties in characteristic zero. This version fixes a base point (a fiber over a marked point on a curve) and asks for finiteness of families of primitive symplectic varieties (PSVs) whose fiber at the marked point is isomorphic to a given PSV. I will sketch new results toward this conjecture under natural hypotheses. This is joint work with Lie Fu, Zhiyuan Li, and Teppei Takamatsu.