Title: Optimal bounds for local volumes of threefold singularities.
Abstract: We establish an optimal upper bound for local volumes of Gorenstein canonical non-hypersurface threefold singularities. Specifically, we show that a klt threefold singularity with local volume at least 9 is either a hypersurface singularity or a quotient singularity. As applications, we obtain new restrictions on the singularities of members in K-moduli spaces of Fano threefolds, and we establish a sharp inequality between local volumes and minimal log discrepancies for threefold singularities.
Title: D-Geometric Hilbert and Quot DG-Schemes (derived Hilbert scheme of solutions to nonlinear PDE).
Abstract: We report on recent series of joint works with Jacob Kryczka and Shing-Tung Yau on construction of derived moduli spaces of solutions to nonlinear PDE. We construct a parameterizing space of ideal sheaves of involutive and formally integrable non-linear partial differential equations in the algebraic-geometric setting. We elaborate on the construction of a D-geometric analog of Grothendieck's Quot (resp. Hilbert) functor and prove that its is represented by a D-scheme which is suitably of finite type. A natural derived enhancement of the so-called D-Quot (resp. D-Hilbert) moduli functor is constructed and its representability by a differentially graded D-manifold with corresponding finiteness properties is studied. If time permits, we further elaborate on how this technology leads to construction of derived Donaldson-Uhlenbeck-Yau (DUY) correspondences.
Abstract: For Fano varieties, significant progress has been made recently in the study of K-stability, while the understanding of the weaker but more algebraic concept of (-K)-slope stability remains intricate. For instance, a conjecture attributed to Iskovskikh states that the tangent bundle of a Picard rank one Fano manifold is slope stable. Peternell-Wisniewski and Hwang proved this conjecture up to dimension five in 1998, but Kanemitsu later disproved it in 2021. To address this gap in understanding, we present a method that aims to characterize the geometry associated with the maximal destabilizing sheaf of the tangent sheaf of a Fano variety. This approach utilizes modern advancements in the foliated minimal model program. In dimension two, our approach leads to a complete classification of (-K)-slope unstable weak del Pezzo surfaces with canonical singularities. As by-products, we provide the first conceptual proof that P1 ×P1 and F1 are the only (-K)-slope unstable
nonsingular del Pezzo surfaces, recovering a classical result of Fahlaoui in 1989. This is the joint work with Ching-Jui Lai.
Abstract: In this talk I will present a construction of the Chow quotient of the complete flag variety of C^4 by the action of a maximal torus in its automorphism group, and its birational properties. We will also discuss how to extend this construction to other rational homogeneous spaces.
Title: The Fano of lines and the Kuznetsov component of cubic fourfolds.
Abstract: A smooth cubic fourfold gives rise to two kinds of hyperkähler fourfolds: one is classical --the variety of lines on the cubic; and the other is "non-commutative" --arising from the symmetric square of the Kuznetsov component. Galkin conjectured that these two objects should be derived equivalent. In this talk, I’ll explain a proof of this conjecture, which uses matrix factorizations and a wall-crossing derived equivalence for a particular 12-dimensional flop. This is joint work with Ed Segal.
Abstract: My talk is about group actions on projective varieties. Every algebraic group G over the complex numbers is an extension of an abelian variety A by a linear algebraic group L (Chevalley's theorem). The main result is that if G acts algebraically on a projective
variety X, then the cohomology of X is "free" over the cohomology of A. The precise statement involves Hopf algebras and comodules. This is joint work with Mark de Cataldo and Yoonjoo Kim.
Abstract: In this talk I will discuss the K-stability of weighted Fano hypersurfaces of dimension n>=3. The 3-fold case has been established by [Kim-Okada-Won], [Sano-Tasin], [Campo-Okada]. In a joint work with Kento Fujita, Taro Sano, and Luca Tasin we studied the n-fold case with n>3 producing bounds for delta invariants of weighted Fano n-fold hypersurfaces embedded in certain weighted projective spaces.
Title: On the boundedness of birational automorphisms.
Abstract: We formulate boundedness questions for birational automorphisms and discuss to which extent the Cremona groups exhibit unbounded behavior. As an example, we show that over a field of characteristic zero, the weak factorization centers occurring in the Cremona transformations of P^N do not belong to a birationally bounded family whenever N = 4, even after cancellations up to birational equivalence of their MRC bases. (Based on joint work with Evgeny Shinder.)
Abstract: We study the moduli space of higher rank marginally stable pairs (E,s), where E is a torsion-free coherent sheaf of rank r on a smooth projective surface and s = (s_1, …, s_r) is a collection of r sections of E. Fixing the Chern character of E, the moduli space is realised as a subscheme of an appropriate Quot-scheme that parametrises quotient sheaves with the corresponding Hilbert polynomial. We establish a precise link between these moduli spaces and the stable minimal models determined by E and its sections, together with the (relative) log canonical model of the base surface. Using the birational geometry of such minimal models, we analyse in detail the components of the Hilbert-Chow morphism from the moduli space to the Hilbert scheme of effective Cartier divisors on the surface.
Title: Automorphism groups of toroidal horospherical varieties.
Abstract: We present our recent work on the structure of the connected part of the automorphism group of a smooth, complete, toroidal horospherical variety by generalizing the notion of Demazure roots using the toric bundle structure. In particular, we provide a criterion for the reductivity of $\mathrm{Aut}^0(X)$ in terms of an analogous notion of Demazure roots for such toric bundles, i.e., projective toric bundles over rational homogeneous spaces. As an application, we prove the K-unstability of certain $\mathbb{P}^1$-bundles over rational homogeneous spaces. This is joint work with Lorenzo Barban and Minseong Kwon.
Abstract: Secant varieties are classical objects in algebraic geometry. Given a smooth projective variety inside a projective space, its secant variety is by definition the closure of the union of secant lines. It is almost always singular and sits inside the same projective space by its construction. In this talk, we will discuss the singularities of secant varieties. In particular, we will study the Du Bois complex of secant varieties and will also discuss its local cohomology. The results are obtained in various collaborations with Q. Chen, B. Dirks, S. Olano and L. Song.
Title: Contractions and flops via moduli of non-pure sheaves.
Abstract: When introducing the notion of moduli problems, perhaps the most trivial example one may consider is the "functor of points" of a variety X, whose "classifying space" (moduli space) is X itself. Its very close relative is the stack of ideal sheaves of length 1 subschemes: when X is normal and of dimension at least 2, this stack is the product of X with BG_m. Despite the apparent triviality, modifying this stack gives rise to interesting birational phenomena, about which I will speak. I will describe a certain enlargement U of the stack of ideal sheaves that yields, via taking the good moduli space, a birational contraction of X, and an open substack of U that yields a surgery diagram via non-GIT wall-crossing. If time permits, I will explain how the interpretation of the surgery as a fine moduli of sheaves helps prove instances Kawamata’s DK-hypothesis in this setting. This is joint work with Andres Fernandez Herrero.
Title: Singularities of Foliations and Good Moduli Spaces of Algebraic Stacks.
Abstract: Drawing on the theory of Minimal Model Program singularities for foliations, we define relative canonical and log canonical singularities for algebraic stacks with finite generic stabilisers. We show that if a point has log-canonical singularities, its stabiliser group is a finite extension of an algebraic torus, thus, etale locally, the good moduli space exists. If the singularity is canonical, we further show that the locus of stable points is non-empty.
Title: Equidimensional morphisms onto splinters are pure.
Abstract: Pure ring maps arise naturally in algebraic geometry. For example, inclusions of rings of invariants by linearly reductive groups, split maps, and faithfully flat maps are all pure. It is useful to know when a map is pure because pure maps satisfy effective descent properties and many classes of singularities are preserved under the operation of taking pure subrings. For example, a theorem of Boutot (for rings of finite type over a field) and myself (in general) says that for Noetherian Q-algebras, pure subrings of rings with (at worst) rational singularities have (at worst) rational singularities. Such results are often called Boutot-type theorems. In this talk, I will discuss a new class of pure ring maps which arise geometrically from families of varieties of the same dimension. In fact, this yields a characterization of the splinter property: A Noetherian ring R is a splinter (when R is a domain, this means that all module-finite extensions R -> S are split) if and only if every locally equidimensional surjective morphism Spec(S) -> Spec(R) is pure. Since not all pure maps are locally equidimensional and Boutot-type theorems fail for some classes of singularities like F-rationality, this raises the question: Are there "weak" Boutot-type theorems for pure ring maps that are also equidimensional? I will discuss my affirmative solution to this question for F-rationality.
Abstract: We will show that there are infinitely many non-Q-Gorenstein examples with all but finitely many irrational but algebraic (F-)jumping numbers, confirming the suspicion of many experts.
Our main tool is an application of Schwede and Tucker's result (2012), which shows that the multiplier and test ideals are equal in a specific cone situation. We will pose 2 questions in the end.