2024 |
09.09 |
10:00-11:30 @ MCM110 |
Paolo Cascini (Imperial College London) |
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Foliation Adjunction
We present an adjunction formula for foliations on varieties and we consider applications of the adjunction formula to the cone theorem
for rank one foliations and the study of foliation singularities. Joint work with C. Spicer.
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06.11 |
09:00-10:00 @ MCM410 |
Siqing Zhang (Institute for Advanced Study) |
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Non Abelian Hodge Theory in prime characteristic II: log-poles and G-bundles
The previous talk is about Higgs fields and flat connections on vector bundles. In the first half of this talk, we consider how the picture changes when
we add log poles. It turns out that the de Rham moduli space is, up to a twist and over an open of the Hitchin base, a Galois cover of the Dolbeault
moduli space. This happens due to a curious appearance of the Artin-Schreier map on the residues. In the second half of this talk, we consider the case
with G-bundles and no poles. We show that the analogous results on semistability and cohomology also hold. This relies on a detailed study of Ngô's regular
centralizers, and the theory of Theta-semistability. Based on joint works with Mark de Cataldo and Andres Fernandez Herrero.
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06.04 |
10:00-11:00 @ MCM410 |
Siqing Zhang (Institute for Advanced Study) |
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Non Abelian Hodge Theory in prime characteristics I: semistability and cohomology
The Non Abelian Hodge Theory for a curve in characteristic p relates the moduli of flat connections on the curve and the moduli of Higgs bundles on the
Frobenius twist of the curve. Previously, it is known that both moduli stacks sit over a Hitchin base, and that they differ by a twist of a torsor under
a Picard stack over the Hitchin base. Now we know that, by shrinking the Picard stack and its torsor simultaneously, we can make the twist respect
semistability. Consequently, the semistable Dolbeault and de Rham moduli spaces differ by a twist of a torsor under a connected group scheme over
the Hitchin base. We then deduce some results on the cohomologies of the moduli spaces, some of which entail new results in characteristic zero.
Based on joint works with Mark de Cataldo and Michael Groechenig.
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05.29 |
11:00-12:00 @ N820 |
Jian Xiao (Tsinghua University) |
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05.20/24/27 |
13:00-14:30 @ N820 |
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Numerical characterization of the hard Lefschetz classes
We give an overview on recent progress on the numerical characterization of the hard Lefschetz classes. During the lectures, the following topics
will be discussed: positivity in algebraic geometry, the extremals of geometric inequalities, Lorentzian polynomials and combinatorial poset inequalities.
Based on joint works with J. Hu and S. Shang.
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05.27 |
16:00-17:00 @ N820 |
Hao Sun (South China University of Technology) |
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Filtered Stokes G-Local Systems
The wild Riemann-Hilbert correspondence relates irregular singular connections with Stokes local systems. As a direct generalization, the correspondence
also holds for reductive groups G. Based on the wild Riemann-Hilbert correspondence, I will discuss Stokes G-local systems (or Stokes G-representations)
together with its moduli space in the viewpoint of nonabelian Hodge correspondence. This is joint work with Pengfei Huang.
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05.27 |
14:45-15:45 @ N820 |
Pengfei Huang (Max-Planck-Institut for Mathematics) |
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Filtered local systems and their moduli spaces
The celebrated Simpson¡¯s tame nonabelian Hodge correspondence shows that the correct objects from the Betti side are the so-called filtered local systems.
However, such nonabelian Hodge correspondence was only known at the level of categories until recently, due to the lack of a construction of the moduli
spaces of filtered local systems. In this talk, we will introduce filtered local systems and demonstrate the construction of their moduli spaces. This
method is applicable to general reductive groups. Based on joint work with Hao Sun.
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04.18 |
10:00-11:00 @ N818 |
Lu Qi (Princeton University) |
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Filtrations, multiplicities and singularities
Graded sequences of ideals, or filtrations on local rings, play an important role in commutative algebra, the theory of singularities and birational geometry.
In this talk, I will introduce the saturation of a filtration and discuss several results about multiplicities of filtrations. The first is a theorem generalizing
Rees' theorem on Hilbert-Samuel multiplicities of ideals. Then I will talk about convexity of multiplicities along geodesics, with applications in valuation theory
and K-stability. I will also introduce a metric on the space of filtrations as an analogue of the Darvas metric in complex geometry, and prove the semiconuity of log
canonical threshold functions in the spirit of Koll¨¢r-Demailly. This talk is partially based on joint work with Harold Blum and Yuchen Liu.
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04.17 |
10:00-11:30 @ MCM110 |
Konstantin Shramov (Steklov Mathematical Institute and NRU Higher School of Economics) |
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Del Pezzo surfaces of degree 4
A del Pezzo surface is a smooth projective surface with ample anticanonical divisor. Over an algebraically closed field, any surface like this is rational. However,
without this assumption del Pezzo surfaces exhibit very interesting birational properties. I will survey some old and new results about birational geometry of
del Pezzo surfaces over arbitrary fields, mostly focusing on del Pezzo surfaces of degree 4. These can be realized as intersections of two quadrics. It is known
that minimal del Pezzo surfaces of degree 4 over a perfect field are never rational. I will provide a detailed description of birational models for such surfaces,
and will discuss the properties of their birational automorphism groups. The talk is based on a joint work in progress with A.Trepalin.
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2023 |
11.15 |
10:00-11:30 @ MCM410 |
Andreas Höring (Université Côte d'Azur) |
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Projective manifolds with pseudoeffective cotangent bundles II
Symmetric holomorphic forms on projective manifolds have been studied from various angles in the last ten years : Brotbek-Darondeau and independently Song-Yan Xie
have shown that complete intersections of sufficiently high degree and codimension have an ample cotangent bundle, so many symmetric forms. Vice versa Campana and
Paun show that if the cotangent bundle is big, then \(X\) is of general type. In this series of lectures I will work under the assumption that the cotangent bundle is
pseudoeffective. In general this positivity assumption is too weak to have any useful implications, but we will see that for the cotangent bundle it leads to
surprisingly many restrictions on the geometry of the manifold. In particular we expect a nonvanishing property that is analogous to the famous nonvanishing conjecture
for the canonical bundle. These talks are based on joint papers with Thomas Peternell and Junyan Cao.
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11.13 |
14:00-15:30 @ MCM410 |
Andreas Höring (Université Côte d'Azur) |
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Projective manifolds with pseudoeffective cotangent bundles I
Symmetric holomorphic forms on projective manifolds have been studied from various angles in the last ten years : Brotbek-Darondeau and independently Song-Yan Xie
have shown that complete intersections of sufficiently high degree and codimension have an ample cotangent bundle, so many symmetric forms. Vice versa Campana and
Paun show that if the cotangent bundle is big, then \(X\) is of general type. In this series of lectures I will work under the assumption that the cotangent bundle is
pseudoeffective. In general this positivity assumption is too weak to have any useful implications, but we will see that for the cotangent bundle it leads to
surprisingly many restrictions on the geometry of the manifold. In particular we expect a nonvanishing property that is analogous to the famous nonvanishing conjecture
for the canonical bundle. These talks are based on joint papers with Thomas Peternell and Junyan Cao.
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09.26 |
14:00-15:00 @ N913 |
Yizheng Yin (Peking University) |
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Abelian fibrations, Perverse = Chern, and multiplicativity
The perverse filtration captures interesting homological information of algebraic maps. Recent studies of the Hitchin and Beauville-Mukai integrable systems suggest two
common features of the perverse filtration for abelian fibrations:
1) the perverse filtration is multiplicative with respect to the cup product;
2) the perversity of tautological classes is governed by the Chern degree.
In this talk, I will explain a unified approach to bothstatements for a natural class of abelian fibrations, namely fibrations incompactified Jacobians.
I will discuss several ingredients from coherent derived categories, K-theory/Chow theory, and constructible categories, as well as some applications of our result.
Joint work with Davesh Maulik and Junliang Shen.
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08.02 |
14:00-15:00 @ N913 |
Mingchen Xia (Institut de mathématiques de Jussieu - Paris Rive Gauche) |
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Characterizations of I-good singularities
We continue the discussion of the last talk. I will explain the properties of the \(d_S\)-pseudometric and prove various characterizations of I-good singularities.
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08.01 |
14:00-15:00 @ N913 |
Mingchen Xia (Institut de mathématiques de Jussieu - Paris Rive Gauche) |
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Ross-Witt Nyström correspondence
In this talk, we will discuss a geometric application of the theory of I-good singularities. I will prove the bijection between geodesic rays and
test curves and the bijection between Berman-Boucksom-Jonsson maximal geodesic rays and I-model test curves.
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07.24 |
10:00-11:00 @ N913 |
Ya Deng (Université de Lorraine) |
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Green-Griffiths-Lang conjecture for algebraic varieties with big reductive fundamental groups
The Green-Griffiths-Lang (GGL) conjecture asserts that any entire curve in a complex projective variety of general type cannot be Zariski dense. This conjecture fascinates
many complex geometers, in part due to its arithmetic analogy with the Bombieri-Lang conjecture for rational points in algebraic varieties over number fields. In this talk
I will report a recent work with Cadorel and Yamanoi, focusing on the proof of the generalized GGL conjecture for quasi-projective varieties whose topological fundamental
groups possess a big and reductive representation into a complex general linear group.
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07.11 |
10:45-11:45 @ N913 |
Lei Wang (Huazhong University of Science and Technology) |
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On the rigidity of proper holomorphic self-mappings of the Hua domains
Hua domain, named after Chinese mathematician Loo-Keng Hua, is defined as a domain in \(\mathbb{C}^n\) fibered over an irreducible bounded symmetric domain
\(\Omega \subset \mathbb{C}^d\) with the fiber over \(z\in \Omega\) being a \((n-d)\)-dimensional generalized complex ellipsoid \(\Sigma(z)\). In 2015, Tu-Wang
obtained the rigidity result that proper holomorphic mappings between two equidimensional Hua domains are biholomorphisms when the sets consisting of boundary
points of Hua domains which are not strongly pseudoconvex have complex codimension at least $2$. In this article, we find a counter-example to show that the
rigidity result is not true for Hua domains without this condition and obtain the rigidity of proper holomorphic self-mappings of the Hua domains in this case.
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07.11 |
09:30-10:30 @ N913 |
Sheng Rao (Wuhan University) |
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Projective, Moishezon and Kahler loci of family II
This talk mainly concerns the projective, Moishezon and Kahler loci of a family. We first review our recent results on deformation limit and invariance of plurigenera
of Moishezon manifolds, based on several joint works with I-Hsun Tsai, Yi Li and Runze Zhang. Then we talk about an in-process joint work with Mu-Lin Li and Mengjiao Wang,
on various loci of a family and their applications. Among them are a Chow-type Lemma and a reverse side of one theorem of Zhiwei Wang on the modification of some special
complex structure.
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06.01 |
14:00-15:00 @ N913 |
Guodu Chen (Westlake University) |
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On multiplicities of fibres of Fano fibrations
Let \(f:X\rightarrow Z\) be a Mori fibre space. McKernan conjectured that the singularities of \(Z\) are bounded in terms of the singularities of \(X\). Shokurov independently proposed a
more general conjecture in the setting of pairs. In this talk, we may survey some recent progress on Shokurov¡¯s conjecture on the singularities of the base. Then
we reduce various conjectures in birational geometry, including Shokurov¡¯s conjecture on the singularities of the base and boundedness conjecture for rationally
connected Calabi-Yau varieties, to a conjecture on multiplicities of fibers of Fano fibrations over curves. This is a joint work with Chuyu Zhou.
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04.18 |
15:30-16:30 @ MCM410 |
Yalong Cao (RIKEN) |
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From curve counting on Calabi-Yau 4-folds to quasimaps for quivers with potentials
I will start by reviewing an old joint work with Davesh Maulik and Yukinobu Toda on relating Gromov-Witten, Gopakumar-Vafa and stable pair invariants on compact Calabi-Yau
\(4\)-folds. For non-compact CY\(4\) like local curves, similar invariants can be studied via the perspective of quasimaps to quivers with potentials. In a joint work in progress
with Gufang Zhao, we define a virtual count for such quasimaps and prove a gluing formula. Computations of examples will also be discussed.
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04.04 |
16:30-17:30 @ Tencent Meeting: 377-9935-3697 |
Qingyuan Jiang (University of Edinburgh) |
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Borel-Weil-Bott theorem for Derived Flag Schemes
The framework of derived algebraic geometry (DAG), developed by Toen-Vezzosi, Lurie and many others, allows us to extend Grothendieck's theory of flag schemes of sheaves
to the cases of complexes. In the first half of the talk, we will define and study flag schemes of complexes using DAG.
The Borel-Weil-Bott theorem provides a fundamental link between algebraic geometry and representation theory. In the second half of the talk, we will establish a derived
generalization of the Borel-Weil-Bott theorem for flag schemes of \([-1,0]\)-perfect complexes.
Based on the second part of the paper arXiv:2212.10488.
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04.03 |
16:30-17:30 @ Tencent Meeting: 377-9935-3697 |
Qingyuan Jiang (University of Edinburgh) |
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Non-abelian Derived Categories and Derived Schur and Weyl Functors
We'll start with reviewing Quillen and Lurie's theory of non-abelian derived categories (also known as the "animation" theory). Following that, we will apply this theory
to generalize the classical Schur and Weyl functors of \(GL_n\)-representation theory to the cases of connective complexes, and investigate their fundamental properties.
If time allows, we will also talk about how to generalize them to non-connective cases using Goodwillie's theory of calculus of functors.
Based on the first part of the paper arXiv:2212.10488.
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03.09 |
15:00-16:00 @ N933 |
Sichen Li (East China University of Science and Technology) |
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Density of orbits and arithmetic degrees of automorphisms of projective threefolds
The Zariski dense orbit conjecture of the arithmetic dynamic is to study the existence of a rational point \(P\) with a Zariski dense orbit of a
dominant rational self-map \(f\) of a projective variety \(X\). When there exist such a rational point \(P\), Prof. Kawaguchi and Prof. Silverman conjecture
defined the arithmetic degree at \(P\) and claimed that its arithmetic degree is equal to the first dynamical degree of \(f\), i.e., Kawaguchi-Silverman
conjecture of the arithmetic dynamic. In this talk, I will explain my recent progress towards on the Zariski dense orbit conjecture and
Kawaguchi-Silverman conjecture on automorphisms of projective threefolds from the viewpoint of the explicit birational geometry.
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02.28 |
9:30-11:30 @ MCM410 |
Qifeng Li (Shandong University) |
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Lines on holomorphic contact manifolds and a generalization of \((2,3,5)\)-distributions to higher dimensions
Since the celebrated work by Cartan, distributions with small growth vector \((2,3,5)\) have been studied extensively. In the holomorphic setting, there
is a natural correspondence between holomorphic \((2,3,5)\)-distributions and nondegenerate lines on holomorphic contact manifolds of dimension \(5\).
We generalize this correspondence to higher dimensions by studying nondegenerate lines on holomorphic contact manifolds and the corresponding
class of distributions of small growth vector \((2m, 3m, 3m+2)\) for any positive integer \(m\). This is a joint work with Jun-Muk Hwang.
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