to appear!
Nicolas Addington (University of Oregon)
Title: The Quillen-Lichtenbaum dimension of complex varieties.
Abstract: The Quillen-Lichtenbaum conjecture for smooth complex varieties states that algebraic and topological K-theory with finite coefficients become isomorphic in high degrees. We define the "Quillen-Lichtenbaum dimension" of a variety in terms of the point where this happens, show that it is surprisingly computable, and analyze many examples. It gives an obstruction to rationality, but one that turns out to be weaker than unramified cohomology and some related birational invariants defined by Colliot-Thélène and Voisin using Bloch-Ogus theory. Because it is compatible with semi-orthogonal decompositions, however, it allows us to prove some new cases of the integral Hodge conjecture using homological projective duality, and to compute the higher algebraic K-theory of the Kuznetsov components of the derived categories of some Fano varieties.
Ignacio Barros (University of Antwerp)
Title: Theta-extension and extremal divisors on moduli spaces of HK varieties.
Abstract: I will report joint work in progress with L. Flapan and R. Zuffetti where we develop tautological pull-back formulas for special cycles on orthogonal modular varieties. As a consequence, we exhibit many extremal divisors on moduli spaces of K3 surfaces and HK varieties. We also obtain equality between the effective and the NL-cone for moduli spaces of generalized Kummer varieties of low degree and dimension.
Arend Bayer (University of Edinburgh)
Title: Non-commutative abelian surfaces, Kummer-type Hyperkaehler varieties, and Weil-type abelian fourfolds
Abstract: The derived category D(A) of an abelian surface A has a six-dimensional space of deformations, but A has only a 3-dimensional space of deformations as an algebraic variety. I will explain a construction that reduces this gap, by constructing a family of categories over a four-dimensional space that we call "non-commutative abelian surfaces".This gives rise to an interpretation of every general Kummer-type Hyperkaehler variety via a moduli space. Finally, I will explain how this also leads to a different approach to and proof of O'Grady's and Markman's results concerning the relation to Weil-type abelian fourfolds. This is based on joint work with Laura Pertusi, Alex Perry and Xiaolei Zhao.
Alessio Bottini (Universität Bonn)
Title: The derived category of the variety of lines on a cubic fourfold.
Abstract: In this talk, I will discuss the conjecture saying that the derived category of the Fano variety of lines on a cubic fourfold is equivalent to the symmetric square of its Kuznetsov component. I will describe the difficulties, and outline a proof in the case where the Fano variety of lines admits a rational Lagrangian fibration. This is part of a joint work in progress with Daniel Huybrechts.
Olivier Debarre (Université Paris Cité)
Title: On a conjecture of Kazhdan and Polishchuk.
Abstract: I will discuss a conjecture on rank 2 vector bundles on curves that Kazhdan made at his plenary talk at ICM 2022.
Laure Flapan (Michigan State University)
Title: Cones of Noether-Lefschetz divisors and moduli of hyperkähler manifolds.
Abstract: We describe how to compute cones of Noether-Lefshetz divisors on orthogonal modular varieties with a particular view towards moduli spaces of polarized K3 surfaces and hyperkähler manifolds. We then describe some geometric applications of these cone computations for these moduli spaces. This is joint work with I. Barros, P. Beri, and B. Williams.
Salvatore Floccari (Universität Bielefeld)
Title: The hyperKummer construction
Abstract: Any manifold K of Kum^3-type has a naturally associated manifold Y_K of K3^[3]-type, defined as crepant resolution of the quotient K/G by the action of a finite group G=(Z/2)^5. I will report on joint work with Lie Fu in which we explore this hyperKummer construction guided by the analogy with the classical Kummer construction in dimension 2. We obtain a birational characterization of hyperKummer varieties of K3^[3]-type as well as relations between K and Y_K at the level of motives and derived categories. As I will explain, our study leads to the possibility of constructing very general projective varieties of Kum^3-type as rational covers of suitable moduli spaces of sheaves on certain K3 surfaces. If time permits, I will also discuss a somewhat surprising relation with hyperKähler varieties of OG6-type.
Robert Laterveer (Université de Strasbourg)
Title: On the Chow ring of double EPW sextics.
Abstract: Chow rings of HK varieties are expected to have particular properties. Notably, a famous conjecture made by Beauville and Voisin predicts that the subring (of the Chow ring of a HK variety) generated by divisors and Chern classes should inject into cohomology. Building on the work of many other people, I proved that this conjecture is true for all double EPW sextics. I will also report on a recent joint work with M. Bolognesi, concerning double EPW sextics and their covering involution. We relate the action of the covering involution on zero-cycles to Voisin’s rational orbit filtration, in accordance with general conjectures.
Emanuele Macrì (Université Paris-Saclay)
Title: Deformation of t-structures with applications to hyperkaehler geometry, II.
Abstract: This the second of two lectures which illustrate a new technique that allows to deform t-structures and stability conditions at various levels of generality. The same circle of ideas leads to the proof of a conjecture by Belmans-Okawa-Ricolfi about deformations of semiorthogonal decompositions. Additional and more geometric applications of our deformation result for stability conditions include:
- A proof of a conjecture by Kuznetsov and Shinder relating the Fano variety of lines and the Fano variety of conics on special Fano 3-folds;
- The construction of stability conditions for very general abelian varieties and HK manifolds of Hilb^n(K3)-type in any dimension;
- The connectedness of the stability manifold for generic projective K3 surfaces;
- A proof that the very general member of a locally complete family of HK manifolds of Hilb^n(K3)-type is a moduli space of stable objects on a noncommutative K3 surface. This is joint work (in progress) with C. Li, A. Perry, P. Stellari and X. Zhao.
Alina Marian (ICTP)
Title: The geometry of Quot schemes of zero-dimensional quotients on a curve.
Abstract: I will give an overview of the cohomology and derived category of a basic moduli space: the Quot scheme of finite-length quotient sheaves of a vector bundle on a smooth projective curve. In particular, I will describe a canonical Nakajima basis for its cohomology, and a parallel semi-orthogonal decomposition of its derived category. The decomposition allows one to calculate the cohomology of interesting tautological vector bundles over the Quot scheme in compact formulas. These formulas are strikingly similar to well-known calculations of the cohomology of tautological vector bundles over Hilbert schemes of points on a surface. The talk is based on joint work with Andrei Negut.
Eyal Markman (University of Massachussets)
Title: Algebraic Weil classes on abelian 2n-f olds from secant sheaves on abelian n-folds.
Abstract: Let X be an n-dimensional abelian variety and let A be the 2n-dimensional abelian variety A=X x Pic^0(X). Automorphisms of the cohomology of X induced by autoequivalences of the derived category of X and by monodromy operators are both elements of the derived monodromy group DMon(X)=Spin(2n,2n). The cohomology ring H(X) is the spin representation of DMon(X), the even and odd cohomologies are the half spin representations. The vector space V:=H^1(A) is the vector representation of DMon(X) and it comes with a symmetric bilinear pairing. The Grassmannian IGr_+ of even maximal isotropic subspaces of V naturally embeds in the projectivization of the half-spin representation H^{ev}(X). A coherent sheaf E on X is a K-secant sheaf, if its chern character ch(E) belongs to a line secant to IGr_+ intersecting it in two complex conjugate points, each defined over the imaginary quadratic number field K. Orlov defined an equivalence F:D^b(XxX) -> D^b(A), which conjugates the diagonal action of DMon(X) on the tensor square H(XxX) of the spin representation to the natural action of DMon(X) on the exterior algebra H(A) of the vector representation. We show that two K-secant sheaves E' and E'' on X, with linearly independent ch(E') and ch(E'') in the same K-secant, determine a complex multiplication e:K->End_Q(A) on A by K. Furthermore, F maps the outer tensor product of E' and the dual of E" to an object E on A, whose characteristic class ch(E)exp[-c_1(E)/rk(E)] remains of Hodge type, under all deformations of the pair (A,e). The complex multiplication e determines a 2-dimensional subspace of Hodge classes in the middle cohomology of A known as Weil classes. When X is the Jacobian of a genus 3 curve, we show that there is a choice of K-secant sheaves E' and E" for which E deforms with (A,e) as a twisted sheaf in an open subset of the 9-dimensional moduli space of abelian sixfolds of Weil-type. We conclude the algebraicity of the Hodge Weil classes for all abelian sixfolds of Weil type of discriminant -1, for all imaginary quadratic number fields K.
Davesh Maulik (MIT)
Title: Algebraic cycles and Hitchin systems.
Abstract: (joint with Junliang Shen and Qizheng Yin) In earlier work, we gave a partial generalization of the Beauville decomposition for abelian schemes to compactified Jacobian fibrations with integral fibers). In this talk, I will discuss how to extend these ideas to the Hitchin system. In particular, I will explain how to use certain representation-theoretic ideas to handle non-reduced fibers that appear.
Mirko Mauri (Université Paris Cité)
Title: The P=W paradigm for compact hyperkähler manifolds
Abstract: The P=W paradigm for compact hyperkähler manifolds suggests surprising relations between degenerations of these manifolds and Lagrangian fibrations on top of them. I will exemplify this principle in two instances: 1. the perverse-Hodge octahedron, i.e., a 3D enhancement of the classical Hodge diamond; 2. the ubiquity of tori as fiber of Lagrangian fibrations, vanishing cycles of type III degenerations, and now also as deeper strata of type II degenerations. This talk is based on a joint project with D. Huybrechts and an ongoing project with P. Engel.
Giovanni Mongardi (Università di Bologna)
Title: Regenerations and applications.
Abstract: Chen-Gounelas-Liedtke recently introduced a powerful regeneration technique, a process opposite to specialization, to prove existence results for rational curves on projective K3 surfaces. We show that, for projective irreducible holomorphic symplectic manifolds, an analogous regeneration principle holds and provides a very flexible tool to prove existence of uniruled divisors, significantly improving known results. This is joint work with G. Pacienza.
Georg Oberdieck (KTH)
Title: On invariants of moduli 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 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.
Laura Pertusi (Università di Milano)
Title: Cubic threefolds and noncommutative curves.
Abstract: The bounded derived category of a cubic threefold X admits a semiorthogonal decomposition formed by two exceptional line bundles and their orthogonal complement, which we denote by Ku(X). Although stability conditions are known to exist on Ku(X), the geometric structure of the associated moduli spaces of semistable objects is rather mysterious. In this talk, I will present structure results on moduli spaces and Abel-Jacobi maps, proving some interesting analogies with moduli spaces on curves, and applications to the construction of Lagrangian subvarieties in hyperkahler manifolds. This is a joint work with Chunyi Li, Yinbang Lin and Xiaolei Zhao.
Antonio Rapagnetta (Università Tor Vergata)
Title: Singular moduli spaces of sheaves on K3 surfaces
Abstract: In this talk I will present some results, obtained in collaboration with A. Perego and C. Onorati, on the geometry of the moduli spaces of sheaves with non-primitive Mukai vector on K3 surfaces.
Giulia Saccà (Columbia University)
Title: Lefschetz Standard conjecture for Lagrangian fibrations.
Abstract: If X is a smooth projective variety of dimension n and L is the class of an ample line bundle on X, the Hard Lefschetz Theorem says that for every k=0,...,n, the cup product map L^{n-k}: H^k(X) \to H^{2n-k}(X) is an isomorphism. The Lefschetz Standard conjecture predicts that the inverse of this isomorphism is induced by an algebraic cycle on X \times X. In this talk I will present some recent results proving the Lefschetz Standard conjecture for certain Lagrangian fibered hyper-K\"ahler manifolds. This is joint work with G. Ancona, M. Cavicchi, and R. Laterveer.
Paolo Stellari (Università di Milano)
Title: Deformation of t-structures with applications to hyperkaehler geometry, I
Abstract: This the first of two lectures which illustrate a new technique that allows to deform t-structures and stability conditions at various levels of generality. The same circle of ideas leads to the proof of a conjecture by Belmans-Okawa-Ricolfi about deformations of semiorthogonal decompositions. Additional and more geometric applications of our deformation result for stability conditions include:
- A proof of a conjecture by Kuznetsov and Shinder relating the Fano variety of lines and the Fano variety of conics on special Fano 3-folds;
- The construction of stability conditions for very general abelian varieties and HK manifolds of Hilb^n(K3)-type in any dimension;
- The connectedness of the stability manifold for generic projective K3 surfaces;
- A proof that the very general member of a locally complete family of HK manifolds of Hilb^n(K3)-type is a moduli space of stable objects on a noncommutative K3 surface. This is joint work (in progress) with C. Li, E. Macrì, A. Perry and X. Zhao.
Claire Voisin (IMJ-PRG)
Title: On a question of Borel and Haefliger.
Abstract: Borel and Haefliger asked in 1961 whether cycle classes on smooth projective complex varieties are smoothable, that is, belong to the subgroup generated by classes of smooth subvarieties. This was disproved for cycles of dimension greater than or equal their codimension. I will explain the proof of smoothability of cycles of dimension smaller than their codimension (joint work with Kollár).
Qizheng Yin (Peking University)
Title: D-equivalence conjecture for K3^[n]
Abstract: I will explain how to use Markman's hyperholomorphic bundles to show that birational hyper-Kähler varieties of K3^[n] type are derived equivalent. This is joint work with Davesh Maulik, Junliang Shen, and Ruxuan Zhang.