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. This result 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.

For a general algebraic stack X, we will present combinatorial structures underlying the connected components of the stack of filtrations of X. This allows us to define analogues of the Hall algebra in different flavors, except that we do not get algebras but a more general kind of structure. Classically these Hall algebras were only defined when X parametrizes objects in an abelian category, while our construction is general. We will then discuss applications of the theory. 

In the motivic setting, we define a notion of Euler characteristic for a stack and, in the (-1)-shifted symplectic case, we give an intrinsic definition of Donaldson-Thomas invariants. The construction relies on a no-pole theorem. The invariants depend on the choice of a so-called stability measure. The space of such measures is a unipotent algebraic group that governs how invariants change under wall-crossing.

In the cohomological setting, we get an explicit form of the decomposition theorem for the map from the stack to its good moduli space, in the smooth, 0-symplectic, and (-1)-symplectic case, assuming tangent space representations at closed points are orthogonally symmetric.

This is joint work over different projects with Chenjing Bu, Ben Davison, Daniel Halpern-Leistner, Tasuki Kinjo and Tudor Pădurariu.

We say that a hyper-Kahler manifold is of Kummer type if it is birational to a quotient of an abelian variety. In this talk, I will explore the intriguing possibility that, up to deformations and birational transformations, all the hyper-Kahler manifolds are, in fact, of Kummer type. 

One case that we understand (almost) completely is the Lagrangian case. This latter part is joint work with Yoon-Joo Kim and Oliver Martin. 

Irreducible symplectic varieties are one of three building blocks of varieties with Kodaira dimension zero, which are higher-dimensional analogs of K3 surfaces. Despite their rich geometry, there have been only a limited number of approaches to construct irreducible symplectic varieties. In this talk, I will introduce a general criterion for the existence of irreducible symplectic compactifications of non-compact Lagrangian fibrations, based on the minimal model program and the geometry of general fibers. As an application, I will explain how to get a 42-dimensional irreducible symplectic variety with the second Betti number at least 24. This is a joint work with Yuchen Liu and Chenyang Xu.

In 1977 Weil identified a 2-dimensional space of rational classes of Hodge type (n,n) in the middle cohomology of every 2n-dimensional abelian variety with a suitable complex multiplication by an imaginary quadratic number field. These abelian varieties are said to be of Weil type and these Hodge classes are known as Weil classes. We prove that the Weil classes are algebraic for all abelian sixfold of Weil type of discriminant -1, for all imaginary quadratic number fields. The algebraicity of the Weil classes follows for all abelian fourfolds of Weil type (for all discriminants and all imaginary quadratic number fields), by a degeneration argument of C. Schoen. The main two ingredients of our proof are:

1) A natural equivalence, due to Orlov, between the cartesian square XxX of an abelian n-fold X and XxPic^0(X), inducing an isomorphism of cohomologies equivariant with respect to the natural action of the derived monodromy group Spin(2n,2n) of X. Orlov's equivalence maps the tensor product of two suitably chosen (secant) sheaves on X to an object E on XxPic^0(X) with a characteristic class invariant under the Mumford-Tate group of abelian varieties of Weil type.

2) When X is the Jacobian of a genus 3 curve, we prove that the object E on XxPic^0(X) is semi-regular and use the Buchweitz-Flenner semi-regularity theorem in order to deform it to deformation equivalent abelian sixfolds of Weil type.  

Cubic fourfolds have been classically studied up to birational equivalence, with an eye towards rationality problems. On the other hand, the Fano variety of lines F(X) on a cubic fourfold X is a hyperkahler manifold, and the rationality/irrationality of X is conjecturely reflected in the geometry of the Fano variety of lines.  We give examples of conjecturally irrational cubic fourfolds with birationally equivalent Fano varieties of lines. Two of our examples, which are special families in C_12, provide new examples of pairs of cubic fourfolds with equivalent Kuznetsov components. Further, we show the cubic fourfolds themselves are birational. Our examples were discovered by studying the group of birational transformations of the Fano varieties of lines of these cubic fourfolds. This is joint work with Corey Brooke and Sarah Frei, building on our previous work with Xuqiang Qin. 

The D-equivalence conjecture of Bondal and Orlov predicts that birational Calabi-Yau varieties have equivalent derived categories of coherent sheaves.  I will explain how to prove this conjecture for hyperkahler varieties of K3^[n] type (i.e. those that are deformation equivalent to Hilbert schemes of K3 surfaces).  This is joint work with Junliang Shen, Qizheng Yin, and Ruxuan Zhang.