Algebraic Geometry Seminar

The theory of D-modules provides very powerful tools and solved many important problems. In this talk, I will introduce a natural way to generalize the D-module theory in logarithmic geometry. I will explain log Bernstein inequality and define log holonomic D-modules on smooth log schemes. Then I will explain log constructibility by using Kato-Nakayama spaces associated to log schemes. If time allowed, I will also explain how the theory is related to the classical b-function theory as well as log Riemann-Hilbert correspondence. This is based on an ongoing project with Andreas Hohl.

In this talk, I will report recent progress on the minimal model program for foliations, applying the theory of Birkar--Zhang’s generalized pairs. Particularly, I will discuss the ACC for lc thresholds and the canonical bundle formula for foliations. Part of this talk is based on a series of joint works with Omprokash Das, Yujie Luo, and Roktim Mascharak, Fanjun Meng, and Lingyao Xie.

We show that the Chern numbers of a smooth complex projective threefold are bounded by a constant which depends only on the topological type of the threefold, provided that the cubic form of the threefold has non-zero discriminant. This is a joint work with Paolo Cascini.

In this talk, I will discuss the behavior of positivity, hyperbolicity, and Kodaira dimension under smooth morphisms of complex quasi-projective manifolds. This includes a vast generalization of a classical result: a fibration from a projective surface of non-negative Kodaira dimension to a projective line has at least three singular fibers. Furthermore, I will explain a proof of Popa's conjecture on the superadditivity of the log Kodaira dimension over bases of dimension at most three. These theorems are applications of the main technical result, namely the logarithmic base change theorem.

The Kawamata-Morrison-Totaro cone conjecture concerns the action of automorphisms on the nef effective cone in N^1(X) and pseudo-automorphisms (birational automorphisms that are isomorphisms in codimension 1) on the movable effective cone of divisors. The former classifies contractions, and the latter classifies marked SQMs (roughly, sequences of flops). It began in the 90s as a conjecture of Morrison on Calabi-Yau manifolds, inspired by intuitions from string theory, but has since extended in scope. It predicts that these actions produce only finitely many orbits of faces of the nef effective cone and finitely many chambers of the movable effective cone -- hence finitely many orbits/isomorphism classes of contractions and SQMs. This talk will motivate the conjecture, include examples, and build up to current work with Joaquin Moraga where we prove this conjecture for klt log CY fibrations of relative dimension 2.

A commutative ring is called a splinter if any finite-module ring extension splits. By the direct summand conjecture, now a theorem due to André, every regular ring is a splinter. The notion of splinter can naturally be extended to schemes. In that context, every normal scheme in characteristic zero is a splinter. In contrast, Bhatt observed in his thesis that the splinter property for proper schemes in positive characteristic imposes strong constraints on the global geometry; for instance, the structure sheaf of a proper splinter in positive characteristic has vanishing positive-degree cohomology. I will report on joint work with Johannes Krah where we describe further restrictions on the global geometry of proper splinters in positive characteristic.

The local volume of klt singularities plays an important role in the study of K-stability of Fano varieties, their moduli spaces and boundedness of singularities. Filtrations on local rings is useful in the study of this notion, as well as other aspects of singularities. In this talk I will introduce a metric on the space of filtrations on a local ring, which has some nice properties. For a klt singularity, the geometry of this geodesic metric space is related to the stable degeneration theorem, which is the fundamental problem of local volumes. This talk is based on joint work of Harold Blum and Yuchen Liu and some ongoing work.

k-Du Bois and k-rational singularities are recently introduced refinements of the classical notions of Du Bois and rational singularities, and they have been extensively studied in the local complete intersection (lci) case. Building on a well known result of Kovács that rational singularities are Du Bois, we prove that k-rational singularities are k-Du Bois. This extends previous work of Mustaţă-Popa and Friedman-Laza in the lci and the isolated singularities cases. Additionally, since Kähler differentials are typically not reflexive outside the lci case, we propose new definitions of these singularities that depend only on the cohomologies of the Du Bois complex and not on the behaviour of Kähler differentials. This is based on joint work with Wanchun Shen and Anh Duc Vo.

Motivated by studying the birational geometry of the jet and arc schemes of a scheme X, de Fernex and Docampo (2020) described the cotangent sheaf of the jets and arcs in terms of the cotangent sheaf of X. Our work (joint with Roi Docampo and Lance Edward Miller) asks a subsequent question: how may one describe the cotangent complexes? To produce an analogous version of their theorem, one must produce a derived/animated version of the jets and arcs. We define such a construction, discuss properties, and then describe the animated version of their theorem and its consequences.

The Alpha invariant of a complex Fano manifold was introduced by Tian to detect its K-stability, an algebraic condition that implies the existence of a Kähler–Einstein metric. Demailly later reinterpreted the Alpha invariant algebraically in terms of a singularity invariant called the log canonical threshold. In this talk, we will present an analog of the Alpha invariant for Fano varieties in positive characteristics, called the Frobenius-Alpha invariant. This analog is obtained by replacing “log canonical threshold” with “F-pure threshold”, a singularity invariant defined using the Frobenius map. We will review the definition of these invariants and the relations between them. The main theorem proves some interesting properties of the Frobenius-Alpha invariant; namely, we will show that its value is always at most 1/2 and make connections to a version of local volume called the F-signature. Time permitting, we will also discuss the semicontinuity properties of the Frobenius-Alpha invariant.

I will discuss how to use Laurent inversion, a technique coming from mirror symmetry which constructs toric embeddings, to study the local structure of the K-moduli space of a K-polystable toric Fano variety. More specifically, starting from a given toric Fano 3-fold X of anticanonical volume 28 and Picard rank 4, and combining a local study of its singularities with the global deformation provided by Laurent inversion, we are able to conclude that the K-moduli space is rational around X. This is joint work with Andrea Petracci.