Tag - Algebraic geometry

Igor Klep: Factorization of non-commutative polynomials and Nullstellensätze for the free algebra

10th November, 2022

The singularity set of a non-commutative polynomial f=f(x1, . . . ,xd) is the graded set Z(f)=(Zn(f))n, where Zn(f)={ X ∈ Mnd : det f(X) = 0 }. Two main results will be presented. First, irreducible factors of f are shown to be in a natural bijective correspondence with irreducible components of Zn(f) for every sufficiently large n. In particular, f is irreducible if and only if Zn(f) is eventually irreducible. Second, we give Nullstellensätze for non-commutative polynomials. For instance, given two non-commutative polynomials f1, f2, we have Z(f1) ⊆ Z(f2) if and only if each irreducible factor of f1 is (up to stable associativity) an irreducible factor of f2. Along the way an algorithm for factorization of non-commutative polynomials will be presented.

Grigory Garkusha: Homological Algebra for Enriched Grothendieck Categories

8th November, 2022

Enriched Grothendieck categories naturally occur in algebraic geometry, where associated abelian categories rarely have projectives but have plenty of information encoded by enriched category theory. In this talk general properties of derived categories for Grothendieck categories of enriched functors and various recollements of such categories will be presented. Applications are given for Voevodsky's triangulated categories of motives.

Carolina Araujo: Higher Fano manifolds

13th October, 2022

Fano manifolds are complex projective manifolds having positive first Chern class. The positivity condition on the first Chern class has far reaching geometric and arithmetic implications. For instance, Fano manifolds are covered by rational curves, and families of Fano manifolds over one dimensional bases always admit holomorphic sections. In recent years, there has been great effort towards defining suitable higher analogues of the Fano condition. Higher Fano manifolds are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional rational varieties, and families of higher Fano manifolds over higher dimensional bases should admit meromorphic sections (modulo Brauer obstruction). In this talk, I will discuss a possible notion of higher Fano manifolds in terms of positivity of higher Chern characters, and describe special geometric features of these manifolds.

David Barnes: Sheaf models for rational stable equivariant homotopy theory

28th June, 2022

Sheaves sit at an interface of algebra and geometry. Equivariant sheaves offer even more structure, allowing for different group actions at different stalks. We are interested in the case where both the base space and group of equivariance are profinite (that is, compact, Hausdorff and totally disconnected). This combination provides many useful consequences, such as a good notion of equivariant presheaves and an explicit construction of infinite products. 

The 2019 PhD thesis of Sugrue used equivariant sheaves to give an algebraic model for rational G-equivariant stable homotopy theory, where G is profinite. In this talk I will explain the model and related results, such as the equivalence between equivariant sheaves and rational Mackey functors (for profinite G).

Damien Calaque: Vertex models and En-algebras

7th June, 2022

I will explain and state a conjecture of Kontsevich, that relates vertex models from statistical mechanics to En-algebras (i.e., algebras for the n-dimensional little disks operad). I will also give the main ingredients of the proof of Kontsevich's conjecture, that involve discretized versions of the little disks operad. This is a work in progress with Damien Lejay.

Yujiro Kawamata: Deformations over non-commutative base

8th May, 2022

If one allows the base of the deformations to be non-commutative, then there are more deformations than usual deformations. The deformations over commutative base can sometimes be regarded as the first order approximation of more general higher order deformations. Though the formal theories of deformations are parallel and the extension to the non-commutative case is simple, some new phenomena and invariants appear. I will explain these by some examples.

Giulia Saccà: Moduli spaces on K3 categories are Irreducible Symplectic Varieties

7th May, 2022

Recent developments by Druel, Greb-Guenancia-Kebekus, Horing-Peternell have led to the formulation of a decomposition theorem for singular (klt) projective varieties with numerical trivial canonical class. Irreducible symplectic varieties are one the building blocks provided by this theorem, and the singular analogue of irreducible hyper-Kahler manifolds. In this talk I will show that moduli spaces of Bridgeland stable objects on the Kuznetsov component of a cubic fourfold with respect to a generic stability condition are always projective irreducible symplectic varieties. I will rely on the recent work of Bayer-Lahoz-Macri-Neuer-Perry-Stellari, which, ending a long series of results by several authors, proved the analogue statement in the smooth case.

Lena Ji: The Noether–Lefschetz theorem in arbitrary characteristic

7th May, 2022

The classical Noether–Lefschetz theorem over the complex numbers states that the restriction map on divisor class groups from P3 to a very general surface of degree at least 4 is an isomorphism. In this talk, we will show a Noether–Lefschetz result for varieties over fields of arbitrary characteristic. The proof uses the relative Jacobian of a curve fibration, and it also works for singular varieties (for Weil divisors).

Ziquan Zhuang: Boundedness of singularities and minimal log discrepancies of Kollár components

7th May, 2022

Several years ago, Chi Li introduced the local volume of a klt singularity in his work on K-stability. The local-global analogy between klt singularities and Fano varieties, together with recent study in K-stability lead to the conjecture that klt singularities whose local volumes are bounded away from zero are bounded up to special degeneration. In this talk, I will discuss some recent work on this conjecture through the minimal log discrepancies of Kollár components.

Claire Voisin: A topological characterization of hyper-Kähler fourfolds of Hilb2(K3) type

7th May, 2022

There are two known deformations types of hyper-Kähler (HK) fourfolds, namely Hilb2(K3) (Beauville, Fujiki) and the generalized Kummer variety K2(A) (Beauville). It is however still unknown whether there are other topological types or deformation types of HK fourfolds. Some strong restrictions on the Betti numbers of HK fourfolds are known by work of Beauville, S. Salamon, Verbitsky and Guan. In this talk, I will sketch the proof of the following:

Theorem. A hyper-Kähler fourfold X is a deformation of Hilb2(K3) if and only if it has two integral degree 2 cohomology classes satisfying the conditions l4=0, m4=0, l2m2=2. In particular, a HK fourfold which is homeomorphic to Hilb2(K3) is a deformation of Hilb2(K3).