Schedule and Abstracts

Zhiyu Tian

Title: Integral Tate conjecture, local-global principle, the space of one-cycles

Abstract: Motivated by some arithmetic questions, we study the space of one-cycles using geometric methods. In a joint work with Kollár, we obtain a new understanding of algebraic equivalence between one-cycles via deformation of stable maps. This leads to subsequent work where we gain some  understanding of the first two homotopy groups of the space of one-cycles, and verify the integral Tate conjecture for one-cycles and local-global principle for zero-cycles for some classes of varieties.

In this minicourse, I will explain the above story, and if time permits, explain how this work is related to conjectures on homological stability of space of curves of Manin-Tschinkel-Batyrev-Peyre-Cohen-Jones-Segal type.

Theodosis Alexandrou

Title: Torsion in Griffiths Groups

Abstract: The Griffiths group Griff^i(X) of a smooth complex projective variety X is the group of nullhomologous codimension-i cycles on X modulo algebraic equivalence. Recently Schreieder gave the first examples of smooth complex projective varieties X for which the Griffiths group has infinite torsion. In his examples the infinitely many torsion classes are of order 2. In this talk we show that for any integer n\geq 2, there is a smooth complex projective 5-fold X whose third Griffiths group contains infinitely many torsion elements of order n.

Giuseppe Ancona

Title: Lefschetz standard conjecture for some lagrangian fibrations

Abstract: The Lefschetz standard conjecture predicts the existence of some specific algebraic classes on the square of an algebraic variety, namely the inverse of the Lefschetz operator should be induced by an algebraic correspondence. We will show this conjecture for the hyper-kähler varieties constructed by Laza-Saccà-Voisin. This will be a special case of a general criterion which will tell that hyper-kähler varieties admitting sufficiently "nice" lagrangian fibrations satisfy this conjecture. 

I will start by recalling the conjecture and the known results.Then I will study how the conjecture behaves under fibration and explain why several problems appear. Finally I  show that one gets indeed a good behaviour in the setting of lagrangian fibrations. This is a joint work with Mattia Cavicchi, Robert Laterveer and Giulia Saccà. 

Jean-Louis Colliot-Thélène

Title: Quadric surface fibrations over the real projective line (joint work with Alena Pirutka)

Abstract: We consider the question whether a real threefold fibred into quadric surfaces over the real projective line is stably rational (over the reals) if the topological space of real points is connected. We give a counterexample. When all geometric fibres are irreducible, the question is open. We investigate a family of such fibrations for which the intermediate jacobian technique is not available. For these, we produce two independent methods which in many cases enable one to prove decomposition of the diagonal.

Salvatore Floccari

Title: The Hodge conjecture for sixfolds of generalized Kummer type

Abstract: The Hodge conjecture is a central problem in complex algebraic geometry. It is notoriously difficult to attack and we still lack general evidence towards its validity.

In my talk I will present a proof of the Hodge conjecture for all six-dimensional hyper-Kähler varieties of generalized Kummer type, i.e. those arising as deformations of Beauville's generalized Kummer varieties built from abelian surfaces. The result presented yields the first complete families of projective hyper-Kähler varieties of dimension larger than two for which the Hodge conjecture is verified. A key ingredient for the proof is the construction of a K3 surface naturally associated to a sixfold of generalized Kummer type.

Federico Scavia

Title: Varieties over Q¯ with infinite Chow groups modulo almost all primes

Abstract: Let E be the Fermat cubic curve over Q¯. In 2002, Schoen proved that the group CH^2(E^3)/ℓ is infinite for all primes ℓ≡1(mod3). We show that CH^2(E^3)/ℓ is infinite for all prime numbers ℓ>5. This gives the first example of a smooth projective variety X over Q¯ such that CH^2(X)/ℓ is infinite for all but at most finitely many primes ℓ. A key tool is a recent theorem of Farb--Kisin--Wolfson, whose proof uses the prismatic cohomology of Bhatt--Scholze. 

Charles Vial

Title: Around the de Rham-Betti conjecture

Abstract: For smooth projective varieties defined over the rational numbers, the de Rham-Betti conjecture is the analogue of the Hodge conjecture, except that one replaces the Hodge filtration on the algebraic de Rham cohomology with its Q-vector space structure. It is a special case of the Grothendieck Period Conjecture. I will report on joint work with Tobias Kreutz and Mingmin Shen, where we explore the validity of (stronger forms of) the de Rham-Betti conjecture for abelian varieties and hyper-Kähler varieties.

Pascal Fong

Title: Unbounded algebraic subgroups of groups of birational transformations

Abstract: From the work of Enriques and Fano, it is known that every connected algebraic subgroup of the Cremona groups of ranks 2 and 3 is contained in a maximal connected algebraic subgroup. This result simplifies the study of connected algebraic subgroups by focusing on the maximal ones. However, for some varieties X, recent results have shown the existence of connected algebraic subgroups that are not contained in any maximal.

Andrea Gallese

Title: Monodromy groups and exceptional Tate classes

Abstract: The Tate conjecture predicts that all Tate classes (in the ℓ-adic étale cohomology) of an abelian variety A/k arise from algebraic cycles. The field of definition of these Tate classes over k is an interesting arithmetic invariant of A, and it is independent of the prime ℓ. In this work, we consider the Fermat Jacobian J_m/Q, the Jacobian of the hyperelliptic curve defined by y^2 = x^m + 1. We compute a finite set of generators for the algebra of Tate classes of A = J_m and determine their field of definition. Since there is no general algorithm for performing this computation, we develop an ad-hoc method. This example is particularly interesting because not all Tate classes of J_m are generated by divisors.

Jan Lange

Title: On the rationality problem for low degree hypersurfaces

Abstract: We present our recent result that a very general hypersurface of degree d \geq 4 and dimension N \leq (d+1)2^{d-4} does not admit a decomposition of the diagonal. Hence such a very general hypersurface is neither stably nor retract rational nor A^1-connected. This joint work with Stefan Schreieder improves earlier results by Schreieder and Moe.

Luigi Martinelli

Title: Towards a categorical resolution of some singular moduli spaces on K3 surfaces

Abstract: Singular moduli spaces of sheaves on a K3 surface rarely admit a crepant resolution. When such a resolution exists, it provides remarkable examples of hyper-Kähler manifolds. When it does not exist, one may still look for a categorical crepant resolution, that would make the previous sporadic examples part of a more general phenomenon, though noncommutative. We present, for certain singular moduli spaces, the preliminary step needed to find such a categorical resolution.

Erik Nikolov

Title: A semi-orthogonal sequence for the Hilbert scheme of three points

Abstract: The Hilbert scheme of n points of a quasi-projective variety X over a field is the fine moduli space parametrizing zero-dimensional subschemes of X of length n. In contrast to the cases where X is a curve, a surface or where n equals two, much less is known about the Hilbert scheme of three points in arbitrary dimension, especially from the derived category point of view.

The main result establishes a collection of semi-orthogonal fully faithful Fourier-Mukai transforms from the derived category of X to the one of its Hilbert scheme of three points. Their construction uses the locus of planar subschemes of X, for which an explicit normal bundle description is found. A conjecture on where to find the correct "complementary" subcategories in the derived category of the Hilbert scheme of three points is presented.

Matthias Paulsen

Title: The degree of algebraic cycles on hypersurfaces

Abstract: Let X be a very general hypersurface of dimension 3 and degree d at least 6. Griffiths and Harris conjectured in 1985 that the degree of every 1-cycle on X is divisible by d. Substantial progress on this conjecture was made by Kollár in 1991 via degeneration arguments. However, the conjecture of Griffiths and Harris remained open in any degree d. Building on Kollár's method, we prove this conjecture (and its higher-dimensional analogues) for infinitely many degrees d.

Morena Porzio

Title: On the stable birationality of Hilbert schemes of points on surfaces

Abstract: The aim of this work is to understand for which pairs of integers (n,n') the variety Hilb^n_X  is stably birational to Hilb^n'_X, when X is a surface with H^1(X, O_X)=0. We then restrict our attention to geometrically rational surfaces, proving that there are only finitely many stable birational classes among the Hilb^n_X 's. As a corollary, we deduce the rationality of the motivic zeta function \zeta(X, t) in K_0(Var/k)/([A^1_k])[[t]] when char(k) = 0.

Francesca Rizzo

Title: On the geometry of singular EPW sextics

Abstract: EPW cubes form a locally complete family of smooth projective hyper-Kähler varieties of dimension 6, constructed by Iliev--Kapustka--Kapustka--Ranestad. Their construction and behavior share a lot of similarities with the double EPW sextics constructed by O'Grady. Adapting the methods of O'Grady, we construct a projective smooth small resolution of singular EPW cubes.