Peking University Algebraic Geometry Seminar

For a projective log canonical pair (X, B) of log general type, the set of volumes vol(X, K_X + B) satisfies the Descending Chain Condition (DCC) when the coefficients of the boundary divisor lie in a given DCC set. A natural further direction is to investigate the fine distribution of these volumes, particularly the structure of their (iterated) accumulation points.

In this talk, I will survey known results on the accumulation behavior of volumes and present recent progress on their iterated accumulation structure. Through explicit geometric constructions, we prove that even in the simplest case where the coefficient set is {0}, the local accumulation complexity of the volume set can be infinite. Our approach builds upon earlier work by Blache and Alexeev–W. Liu.

Rationality problems for conic bundles have been well studied over surfaces. In this talk, we generalize an étale cohomology diagram from the case of conic bundles to Brauer–Severi surface bundles over rational 3-folds. We use this generalization to prove a sufficient condition for a Brauer–Severi surface bundle to be not stably rational. We also give an example satisfying these sufficient conditions.

We show that a principally polarized abelian variety over a field k is, as an abelian variety, a direct summand of a product of Jacobians of curves which contain a k-point if and only if the polarization and the minimal class are both algebraic over k. This extends results of Beckmann–de Gaay Fortman and Voisin over the complex numbers to arbitrary fields, and refines an obstruction to the direct summand property over the rational numbers due to Petrov–Skorobogatov. We then give applications to the integral Tate conjecture for 1-cycles on abelian varieties over finite fields, including the case ℓ = p. Joint work with Federico Scavia.

I will present a compactification of the moduli space of maps from families of curves to certain moduli spaces M, via the example of M being the GIT moduli space of binary forms of degree 2n. One application of our results is the construction of certain moduli of surfaces and threefolds fibered in log Calabi–Yau pairs. I will then explain how to use these moduli spaces to study the boundary of certain KSBA-moduli spaces of surfaces fibered in log Calabi–Yau pairs. Based on joint work with Andrea Di Lorenzo, and joint work with Roberto Svaldi and Junyan Zhao.

Nearby cycles for D-modules along a hypersurface were introduced by Kashiwara and Malgrange via V-filtrations and by Beilinson–Bernstein via b-functions in the 1980s, providing a powerful tool in algebraic geometry and representation theory.

In this talk, I will construct (generalized) nearby cycles for regular holonomic D-modules along F, a finite union of hypersurfaces, motivated by the Beilinson–Bernstein method. Then I will give a logarithmic interpretation of Bernstein–Sato ideals of F using the log structures induced from the graph embedding of F. Finally, I will explain that the relative support of the (generalized) nearby cycles along log strata is determined by the zeroes of the Bernstein–Sato ideals along the same strata, which generalizes a classical result of Kashiwara and Malgrange.

In this talk, I will give more mathematical details of my talk on Nov 4. I will introduce the notion of limit categories for cotangent stacks of smooth stacks as an effective version of classical limits of the categories of D-modules on them. Using the notion of limit categories, I will propose a precise formulation of the Dolbeault geometric Langlands conjecture, proposed by Donagi–Pantev as the classical limit of the geometric Langlands correspondence. I will show the existence of a semiorthogonal decomposition of the limit category into quasi-BPS categories, which (when G = GL_r) categorify BPS invariants on a non-compact Calabi–Yau 3-fold playing an important role in Donaldson–Thomas theory. This semiorthogonal decomposition is interpreted as a Langlands dual to the semiorthogonal decomposition for moduli stacks of semistable Higgs bundles, obtained in our earlier work as a categorical analogue of PBW theorem in cohomological DT theory. It in particular yields a conjectural equivalence between quasi-BPS categories, which gives a categorical version of Hausel–Thaddeus mirror symmetry for Higgs bundles (for any reductive group G). Joint work with Tudor Pădurariu (arXiv:2508.19624).

In this talk, I will present joint work with Chen Jiang and Tianqi Zhang on the upper bound for anti-canonical volumes of canonical Fano 3-folds. We confirm Prokhorov's conjecture that the optimal upper bound for anti-canonical volumes of canonical Fano 3-folds should be 72. We also characterize the equality case.

We show that the eigenvalues of any polarized endomorphism acting on the ℓ-adic étale cohomology of a smooth projective variety satisfy certain parity and symmetry properties, as predicted by the standard conjectures. These properties were previously known for Frobenius endomorphisms.

Besides the hard Lefschetz theorem, a key new ingredient is a recent Weil's Riemann hypothesis-type result due to J. Xie. We also prove a "Newton over Hodge" type property for abelian varieties and Grassmannians.

We show that there exists a 2-dimensional family of smooth cubic threefolds admitting unirational parametrizations of coprime degrees. This, together with Clemens–Griffiths' work, solves the long-standing open problem whether there exists a nonrational variety with unirational parametrizations of coprime degrees. Our proof uses a new approach, called the Noether–Cremona method, for determining the rationality of quotients of hypersurfaces. Joint work with Song Yang and Zigang Zhu.

Quasi-F-splitting is a property of schemes in positive characteristic p, generalizing F-splitting. A remarkable property of quasi-F-split schemes is that they can be lifted modulo p². In fact, Achinger and Zdanowicz constructed such a lifting from the data of the splitting. In the case of F-split (i.e., ordinary) abelian varieties or K3 surfaces, the resulting lifting agrees with the canonical lifting arising from Serre–Tate theory. In this talk, I will present a deformation-theoretic viewpoint on the Achinger–Zdanowicz construction.

A classical theorem of C. Jordan asserts that finite subgroups in a general linear group over a field of characteristic zero contain normal abelian subgroups of bounded index. In general, a group G has the Jordan property if any finite subgroup of G contains a normal abelian subgroup of index at most J, where J is a constant depending only on G. J.-P. Serre proves that the Cremona group of rank 2 has the Jordan property, and he conjectures that the Cremona group of any rank has the Jordan property. The conjecture is proved by Prokhorov–Shramov and Birkar. In this talk, we give explicit bounds for the Cremona group of rank 2 in odd characteristic. Joint work with C. Shramov.

I will report on recent progress on the Mukai-type conjecture, which gives a characterization of products of projective spaces using a variant of Shokurov's complexity. Based on work with Joshua Enwright, Stefano Filipazzi, Joaquín Moraga, Roberto Svaldi, Chengxi Wang, and Kiwamu Watanabe.