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.