# Seminars & Events for Algebraic Geometry Seminar

##### Hodge theory and derived categories of cubic fourfolds

##### Generic K3 categories and Hodge theory

##### TBA - Nori

##### TBA - Huh

##### The topology of proper toric maps

**Please note special day and time. ** I will discuss some of the topology of the fibers of proper toric maps and a combinatorial invariant that comes out of this picture. Joint with Luca Migliorini and Mircea Mustata.

##### Chow rings and modified diagonals

Beauville and Voisin proved that decomposable cycles (intersections of divisors) on a projective K3 surface span a 1-dimensional subspace of the (infinite-dimensional) group of 0-cycles modulo rational equivalence. I will address the following question: what is the rank of the group of decomposable 0-cycles of a smooth projective variety? Beauville and Voisin also proved a refinement of the result mentioned above, namely a decomposition (modulo rational equivalence) of the small diagonal in the cube of a K3. Motivated by this result we will discuss modified diagonals and their relation with conjectures of Beauville and Voisin on the Chow ring of hyperkaehler varieties.

##### Two counterexamples arising from infinite sequences of flops

I will explain how infinite sequences of flops give rise to some interesting phenomena: first, an infinite set of smooth projective varieties that have equivalent derived categories but are not isomorphic; second, a pseudoeffective divisor for which the asymptotic multiplicity along a certain subvariety is infinite, in the relative setting.

##### The construction problem for Hodge numbers

What are the possible Hodge numbers of a smooth complex projective variety? We construct enough varieties to show that many of the Hodge numbers can take all possible values satisfying the constraints given by Hodge theory. For example, there are varieties such that a Hodge number hp,0 is big and the intermediate Hodge numbers hi,p−i are small.

##### Positive cones of higher (co)dimensional numerical cycle classes

##### The structure of instability in moduli theory

In many examples of moduli stacks which come equipped with a notion of stable points, one tests stability by considering "iso-trivial one parameter degenerations" of a point in the stack. To such a degeneration one can often associate a real number which measures "how destabilizing" it is, and in these situations one can ask the question of whether there is a "maximal destabilizing" or "canonically destabilizing" degeneration of a given unstable point. I will discuss a framework for formulating and discussing this question which generalizes several commonly studied examples: geometric invariant theory, the moduli of bundles on a smooth curve, the moduli of Bridgeland-semistable complexes on a smooth projective variety, the moduli of K-semistable varieties.

##### Extending differential forms and the Lipman-Zariski conjecture

The Lipman-Zariski conjecture states that if the tangent sheaf of a complex variety is locally free then the variety is smooth. In joint work with Patrick Graf we prove that this holds whenever an extension theorem for differential 1-forms holds, in particular if the variety in question has log canonical singularities.

##### Singular moduli spaces and Nakajima quiver varieties

The aim of this talk is to study a class of singularities of moduli spaces of sheaves on K3 surfaces by means of Nakajima quiver varieties. The singularities in question arise from the choice of a non generic polarization, with respect to which we consider stability, and admit natural symplectic resolutions corresponding to choices of general polarizations. By establishing the stability of the Lazarsfeld-Mukai bundle for some class of rank zero sheaves on a K3 surface, we show that these moduli spaces are, locally around a singular point, isomorphic to a quiver variety in the sense of Nakajima and that, via this isomorphism, the natural symplectic resolutions correspond to variations of GIT quotients of the quiver varieties. This is joint work with E. Arbarello.

##### Mirror symmetry & Looijenga's conjecture

A cusp singularity is an isolated surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. In the 1980's Looijenga conjectured that a cusp singularity is smoothable if and only if the minimal resolution of the dual cusp is the anticanonical divisor of some rational surface. This conjecture can be related to the existence of certain integral affine-linear structures on a sphere. Existence of such integral-affine structures follows from constructions originally discovered in symplectic geometry.

##### Beauville's splitting principle for Chow rings of projective hyperkaehler manifolds

Being the natural generalization of K3 surfaces, hyperkaehler varieties, also known as irreducible holomorphic symplectic varieties, are one of the building blocks of smooth projective varieties with trivial canonical bundle. One of the guiding conjectures in the study of algebraic cycles of such varieties is Beauville's splitting principle. Concerning the weak form of the splitting principle, I want to report some progress on the closely related Beauville-Voisin conjecture. As a continuation of the recent work of Vial, I will formulate a motivic version of Ruan's hyperkaehler crepant resolution conjecture and explain a work in progress for the Hilbert schemes of K3 surfaces.

##### Elliptic genera of Pfaffian-Grassmannian double mirrors

For an odd integer n>3 the data of generic n-dimensional subspace of the space of skew bilinear forms on an n-dimensional vector space define two different Calabi-Yau varieties of dimension n−4. Specifically, one is a complete intersection of n hyperplanes in the Grassmannian G(2,n) and the other is a complete intersection of n(n−3)/2hyperplanes in the Pfaffian variety of degenerate skew forms. In n=7 case, these have been investigated by Rodland and were (heuristically) found to have the same mirror family. As a result, these Calabi-Yau varieties share many common features. For example, it has been verified that they are derived equivalent even though they are not birational to each other. In the ongoing project, joint with Anatoly Libgober, we are trying to verify that elliptic genera of these Calabi-Yau varieties coincide.

##### TBA - Deligne

##### Zarhin's trick and geometric boundedness results for K3 surfaces

**Please note special time. **Tate's conjecture for divisors on algebraic varieties can be rephrased as a finiteness statement for certain families of polarized varieties with unbounded degrees. In the case of abelian varieties, the geometric part of these finiteness statements is contained in Zarhin's trick. We will discuss such geometric boundedness statements for K3 surfaces over arbitrary fields and holomorphic symplectic varieties, with application to direct proofs of the Tate conjecture for K3 surfaces that do not involve the Kuga-Satake correspondence.

##### Universal Chow group of zero-cycles on cubic hypersurfaces

**Please note special day and time. ** We discuss the universal triviality of the CH0-group of cubic hypersurfaces, or equivalently the existence of a Chow-theoretic decomposition of their diagonal. The motivation is the study of stable irrationality for these varieties. Our main result is that this decomposition exists if and only if it exists on the cohomological level. As an application, we find that a cubic threefold has universally trivial CH0 group if and only if the minimal class θ4/4! of its intermediate jacobian is the class of a 1-cycle (only twice this class is known to be algebraic).