Spring 2025
Seminar time and location:
Odd week Wednesday and even week Thursday (1st week is the week of Feb 17th),
15:00-16:00, Jingchunyuan 77201
Information can also be found on Professor Qizheng Yin's webpage or the official webpage.
February 19
Speaker: Omprokash Das (Tata Institute)
Title: Transcendental base-point freeness and minimal models for projective varieties
Abstract: 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, which says that if X is a projective manifold and \alpha is a (1,1) Bott-Chern cohomology class on X such that \alpha-K_X is nef and big (in the analytic sense), then there is a projective morphism f:X \to Y to a normal compact Kahler variety Y and a Kahler form \omega_Y such that \alpha=f^*\omega_Y.
February 27
Speaker: 李家俊(Jia-Choon Lee) (BICMR)
Title: Deligne-Simpson Problem via Relative Spectral Correspondence
Abstract: 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.
March 13
Speaker: 周正一 (Zhengyi Zhou) (Chinese Academy of Sciences)
Title: Kähler compactification of Cn and Reeb dynamics
Abstract: I will explain a formula relating the minimal discrepency of Fano cone singularities and Reeb dynamics on the Sasaki link as well as Floer theoric 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 discrepency of Fano cone singularities motivated from this formula as well as compactifications of affine varieties beyond Cn. This is based on joint works with Chi Li.
March 27
Speaker: Quentin Posva (Heinrich Heine Universität)
Title: Foliations and singularities of the MMP in positive characteristic
Abstract: 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).
April 2
Speaker: 中村勇哉 (Yusuke Nakamura) (Nagoya University)
Title: A counterexample to the PIA conjecture
Abstract: In this talk, I will give a counterexample to the PIA (precise inversion of adjunction) conjecture for MLD's (minimal log discrepancy). 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 MLD (minimal log discrepancy), the invariant of the singularity. 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.
April 8 (Special time, Colloquium talk)
Speaker: Frédéric Bruno Campana (Institut Élie Cartan de Lorraine)
Title: About a conjecture of Mihnea Popa.
Abstract. M. Popa conjectured that if $f:X\to Y$ is a projective and submersive map beween complex quasi-projective manifolds, then $\overline{\kappa}(X)=\kappa(X_y)+\overline{\kappa}(Y)$, where $\overline{\kappa}$ is the Logarithmic Kodaira dimension.
We prove this, assuming that the fibres $X_y$ have good minimal models.
April 10
Cancelled because of Beijing Algebraic Geometry Day in Chinese Academy of Science
April 16
Speaker: Nikolaos Tsakanikas (EPFL)
Title: Singular Enriques varieties
Abstract: 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.
April 24
Speaker: 崔星乐 (Sung Rak Choi) (Yonsei University)
Title: A valuative approach to the -K-MMP
Abstract: We study the geometry of the triples which consist of a usual pair and a pseudoeffective 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. This is based on the joint work with S.Jang, D.Kim, and D.Lee.
May 8
Speaker: 钟一鸣 (Yiming Zhong) (BICMR)
Title: On moduli spaces of sextic curves with simple singularities
Abstract: 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. This is based on joint work with Chenglong Yu and Zhiwei Zheng.
May 9
Beijing Algebraic Geometry Day in Peking University
Speakers:
Laurent Manivel (CNRS, Paul Sabatier University)
Title: A family of Fano manifolds obtained as linear sections of the spinor tenfold
Abstract: 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, that 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
E_8, 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)
孟晟 (Sheng Meng) (ECNU)
Title: On dynamical Iitaka fibration
Abstract: I will introduce the dynamical Iitaka fibration and its several recent applications. Based on several joint works.
申屠钧超 (Junchao Shentu) (USTC)
Title: Higher dimensional geometric Shafarevich program
Abstract: 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.
许福临 (Fulin Xu) (Tsinghua Unviersity)
Title: Morrison-Kawamata cone conjecture for log Calabi-Yau pairs of lower dimension
Abstract: The Morrison-Kawamata cone conjecture predicts a mysterious finiteness property for log Calabi-Yau pairs. In particular, this conjecture implies finiteness of fiber space structure for any fixed log Calabi-Yau pair. In this talk, I will explain some results on the cone conjecture with some positivity conditions on the boundary divisor, and finiteness of fiber space structure for log Calabi-Yau pairs of dimension 3.
赵禹 (Yu Zhao) (Beijing Institute of Technology)
Title: Instantons from blow ups and free fermions.
Abstract: the semi orthogonal decomposition of the cohomological theory of grassmannian of two term complexes is studied by a series paper of 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 the instanton spaces 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.
May 14
Speaker: 郝峰(Feng Hao) (Shandong Unviersity)
Title: Topological Circle Bundle Structures on Complex Smooth Projective Varieties
Abstract: In this talk, I will present some properties of complex smooth projective varieties with topological $S^1$-actions. Then I will discuss some results on the nonexistence of topological $S^1$-bundle structure on complex smooth projective varieties of general type.
May 22
Speaker: Erik Paemurru (Universität des Saarlandes)
Title: Local Inequalities for $cA_k$ Singularities
Abstract: I will talk about a generalisation of an intersection-theoretic local inequality of Fulton–Lazarsfeld to weighted blowups. As a consequence, we obtain the $4/(k+1) n^2$-inequality for isolated $cA_k$ singularities, an analogue of the $4n^2$-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 $cA_1$ and ordinary $cA_2$ points. This is a joint work with Igor Krylov and Takuzo Okada.
May 28
Cancelled because of a scheduled conference
June 5
Speaker: Niklas Maximilian Müller (Universität Duisburg-Essen)
Title: Inequalities of Miyaoka-type and Uniformisation for Varieties of Intermediate Kodaira Dimension
Abstract: Abstract: 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 the 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 I will present a characterisation of those varieties achieving the equality. This talk is partially based on joint work with Masataka Iwai and Shin-ichi Matsumura.