Peking University Algebraic Geometry Seminar

The minimal model program for generalized pairs has become one of the fundamental tools for classifying higher dimensional algebraic varieties since its inception due to Birkar and Zhang. In this talk I will introduce an analytic version of generalized pairs, namely a triplet (X, B, T), where X is an analytic variety, B a boundary divisor and T is a bi-degree (1,1) current. The current T is the analog of a b-divisor which appears in the generalized pairs of Birkar and Zhang. We will then see that many expected results of MMP, e.g. the results parallel to BCHM, still hold in this generality. Finally, as an application of this kind of MMP we will show that the transcendental base-point free theorem holds for projective varieties: if X is a projective manifold and α is a (1,1) Bott–Chern cohomology class on X such that α − K_X is nef and big (in the analytic sense), then there is a projective morphism f : X → Y to a normal compact Kähler variety Y and a Kähler form ω_Y such that α = f^*ω_Y.

The multiplicative Deligne–Simpson problem (DSP) asks the following question: given an n-tuple of conjugacy classes of matrices, can we choose n matrices from these classes such that their product is the identity and they have no common invariant subspace? Another way to formulate the DSP is to ask for a criterion for the existence of irreducible local systems on the punctured sphere with prescribed monodromy data. Many works have been done in this direction using various methods by Simpson, Katz, Kostov, Crawley-Boevey, and Shaw. In this talk, I will present an alternative approach to the DSP by establishing a relative spectral correspondence for parabolic Higgs bundles. This is joint work with Sukjoo Lee.

I will explain a formula relating the minimal discrepancy of Fano cone singularities and Reeb dynamics on the Sasaki link as well as Floer-theoretic invariants. Such a result can be used to obtain the uniqueness of Kähler compactification of Cⁿ provided the added divisor is smooth. Time permitting, I will also explain a sharp upper bound of minimal discrepancy of Fano cone singularities motivated from this formula as well as compactifications of affine varieties beyond Cⁿ. Based on joint works with Chi Li.

Given a foliation on a variety in positive characteristic, one can define an associated infinitesimal quotient of the variety. This construction is the source of many surprising examples: while it does not change the topology of the variety, it may alter drastically its singularities and its cohomology. In this talk, I will present an efficient way of understanding the MMP singularities of such infinitesimal quotients. Then I will apply this method to exhibit pathological MMP singularities (such as non-Cohen–Macaulay terminal singularities, or locally stable families with non-S₂ special fibers).

In this talk, I will give a counterexample to the PIA (precise inversion of adjunction) conjecture for minimal log discrepancies. The usual inversion of adjunction is a type of claim "the information of the singularity of a pair (X, D) can be recovered from the information of the singularity of D". The precise version (PIA conjecture) states that this is correct at the level of the minimal log discrepancy. The PIA conjecture is known to be true in dimension 3. In this talk, I will give a counterexample in dimension 5. This talk is based on joint work with Kohsuke Shibata.

M. Popa conjectured that if f : X → Y is a projective and submersive map between complex quasi-projective manifolds, then κ̄(X) = κ(X_y) + κ̄(Y), where κ̄ is the logarithmic Kodaira dimension. We prove this, assuming that the fibres X_y have good minimal models.

In this talk, which is based on joint work with Denisi, Ortiz and Xie, I will introduce the class of primitive Enriques varieties. I will discuss the basic properties of these objects, showing in particular that the smooth ones are Enriques manifolds, and I will also present some examples of (singular) primitive Enriques varieties. Finally, I will sketch the proof of the following termination statement: if X is an Enriques manifold and B is an R-divisor on X such that the pair (X, B) is log canonical, then any (K_X + B)-MMP terminates.

We study the geometry of triples consisting of a usual pair and a pseudo-effective divisor. We prove that there exists a quasi-monomial valuation which computes the log canonical threshold of the triple if the triple is potentially klt. As a by-product, we show that in such a case we can run the −K-MMP. Based on joint work with S. Jang, D. Kim, and D. Lee.

In this talk, I will describe moduli spaces for sextic curves with fixed types of simple singularities. I will explain that such moduli spaces admit algebraic open embeddings into arithmetic quotients of type IV domains. I will also describe the identifications of GIT compactifications with the Looijenga compactifications. Furthermore, I will discuss the Picard lattices and the relations of orbifold structures on two sides of the period maps. Based on joint work with Chenglong Yu and Zhiwei Zheng.

Many nice Fano manifolds and K3 surfaces can be obtained as linear sections of homogeneous spaces. I will study low-codimensional sections of the spinor tenfold, which admit non-trivial moduli starting from codimension four. The corresponding family exhibits an extremely rich geometry, connected with the exceptional complex Lie algebra of type E₈, the theory of graded Lie algebras, as well as the classical Kummer quartic surfaces in three-dimensional projective space. Joint work with Yingqi Liu (AMSS).

I will introduce the dynamical Iitaka fibration and its several recent applications. Based on several joint works.

I will introduce the higher-dimensional version of the geometric Shafarevich program, which technically depends on the construction of compact moduli of varieties (the KSBA moduli and Birkar's compact moduli of stable minimal models). If time permits, I will report some works in progress.

The Morrison–Kawamata cone conjecture predicts a mysterious finiteness property for log Calabi–Yau pairs. In particular, this conjecture implies finiteness of fiber-space structures for any fixed log Calabi–Yau pair. In this talk, I will explain some results on the cone conjecture under positivity conditions on the boundary divisor, and finiteness of fiber-space structures for log Calabi–Yau pairs of dimension 3.

The semi-orthogonal decomposition of the cohomological theory of the Grassmannian of two-term complexes is studied in a series of papers by Jiang. In this talk, we will reinterpret it as a representation of the Clifford algebra. As an application, we will explain a relation between the basic representation of the affine Lie algebra and the moduli space of instantons on the blow-up of a point in a surface. It verifies predictions of Li–Qin and Feigin–Gukov. Based on joint work with Qingyuan Jiang and Wei-Ping Li.

In this talk, I will present some properties of complex smooth projective varieties with topological S¹-actions. Then I will discuss some results on the non-existence of topological S¹-bundle structures on complex smooth projective varieties of general type.

I will talk about a generalisation of an intersection-theoretic local inequality of Fulton–Lazarsfeld to weighted blow-ups. As a consequence, we obtain the 4/(k+1) n²-inequality for isolated cA_k singularities, an analogue of the 4n²-inequality for smooth points. We use this to prove birational rigidity of many families of Fano 3-fold weighted complete intersections with terminal quotient singularities and isolated cA_k singularities, including sextic double solids with cA₁ and ordinary cA₂ points. Joint work with Igor Krylov and Takuzo Okada.

Let X be a minimal complex projective variety. Over the past years, many similar inequalities between the Chern classes of X have been obtained. Moreover, it is known precisely which varieties X can achieve equality. However, so far all results in this direction have focused on the case where the numerical dimension of X is either very small or very large. In this talk, I will present analogous inequalities for varieties of intermediate Kodaira dimension and a characterisation of those varieties achieving equality. Partially based on joint work with Masataka Iwai and Shin-ichi Matsumura.