The seminar usually holds on Wednesday. For more details, please visit
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Wednesday, February 23, 9:00-10:00, Zoom link
(ID: 868 3738 6871, Code: 580336)
Yang Li (Massachusetts Institute of Technology) - Metric SYZ conjecture - Abstract
One possible interpretation of the SYZ conjecture is that for a polarized family of CY manifolds near the large complex structure limit, there is a special Lagrangian fibration on the generic region of the CY manifold.
Generic here means a set with a large percentage of the CY measure, and the percentage tends to 100% in the limit. I will discuss some recent progress on this version of the SYZ conjecture, with some emphasis on the special case of the Fermat family.
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Wednesday, March 2, 9:00-10:00, Zoom link
(ID: 873 4972 2554, Code: 728406)
Jakob Hultgren (University of Maryland) - Singular affine structures, Monge-Ampère equations and unit simplices - Abstract
Recent developments in complex geometry have highlighted the importance of real Monge-Ampère equations on singular affine manifolds, in particular for the well known SYZ conjecture concerning collapsing families of Calabi-Yau manifolds.
We show that for symmetric data, the real Monge-Ampère equation on the unit simplex admits a unique solution. This is the first general existence result for Monge-Ampère equations on a singular affine manifold.
I will outline the proof, which uses tools from optimal transport and explain a built in phenomena reminiscent of free boundary problems. Time permitting, I will discuss an application to the SYZ conjecture related to recent work by Y. Li.
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Wednesday, March 9, 16:00-17:00 (Special time), Zoom link
(ID: 846 1832 3223, Code: 449749)
Mingchen Xia (Chalmers Tekniska Högskola) - Volume of Hermitian pseudo-effective line bundles - Abstract
Given a pseudo-effective line bundle L on a compact Kähler manifold X of dimension n, it is well-known that the leading order term of dim H^0(X,L^k) is given by vol(L)k^n. When L is equipped with a singular plurisubharmonic metric \phi,
it is natural to replace H^0(X,L^k) by the subspace of L^2 integrable sections H^0(X,L^k\otimes I(k\phi)). We show that in this case, the leading order term of dim H^0(X,L^k\otimes I(k\phi)) is still of order k^n with coefficient given by a natural construction in pluripotential theory.
This is a joint work with Tamás Darvas.
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Wednesday, March 23, 9:00-10:00, Zoom link
(ID: 874 5448 7019, Code: 606066)
Chao-Ming Lin (University of California, Irvine) - The deformed Hermitian-Yang-Mills equation, the Positivstellensatz, and the Solvability - Abstract
The deformed Hermitian-Yang-Mills equation, which will be abbreviated as the dHYM equation, was discovered around the same time in the year 2000 by Mariño-Minasian-Moore-Strominger and Leung-Yau-Zaslow using different points of view.
In this talk, first, I will skim through Leung-Yau-Zaslow's approach and some known solvability results, e.g., Collins-Jacob-Yau, Chen, and Chu-Lee-Takahashi. Then, I will introduce some filtration cones, which are generalizations of the C-subsolution cone introduced by Székelyhidi (see also Guan).
Last, I will show some of my recent work on the conjecture by Collins-Jacob-Yau when the complex dimension equals four. This conjecture states that their existence theorem of the dHYM equation can be improved when the phase is close to the supercritical phase.
To be more precise, I proved that when the complex dimension equals four, if there exists a C-subsolution, then the dHYM equation is solvable.
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Wednesday, March 30, 10:00-11:00 (Special time), Zoom link
(ID: 853 2034 4180, Code: 324494)
Ved Datar (Indian Institute of Science) - Slope conditions for some complex Hessian equations - Abstract
Yau's work on the Calabi conjecture, together with the Nakai-Moizeshon criteria relates existence of smooth solutions of complex Monge-Ampere equations to the positivity of some intersection numbers. I will first review some recent works of many authors, on extending this correspondence to other complex Hessian equations.
I will then describe some ongoing work with Ramesh Mete and Jian Song on solvability of these equations if the positivity conditions fail and if one allows the solutions to be singular. I will illustrate our main ideas by focusing on the $J$ equation and the deformed Hermitian-Yang-Mills equations on surfaces.
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Wednesday, April 6, 9:00-10:00, Zoom link
(ID: 835 9969 2836, Code: 475209)
Charles Ouyang (University of Massachusetts, Amherst) - Higgs bundles and SYZ geometry - Abstract
Special Lagrangian 3-torus fibrations over a 3 dimensional base
play an important role in mirror symmetry and the SYZ conjecture. In this
talk, we discuss the construction via Higgs bundles of an infinite family
of semi-flat Calabi-Yau metrics on special Lagrangian torus bundles over
an open ball in R^3 with a Y-vertex deleted. This is joint work with S.
Heller and F. Pedit.
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Wednesday, April 13, 9:00-10:00, Zoom link
(ID: 882 9753 6698, Code: 089974)
Qiang Tu (Hubei University) - The Dirichlet problem for mixed Hessian equations on Hermitian manifolds - Abstract
In this talk we consider the Dirichlet problem for a class of Hessian type equation with its structure as a combination of elementary symmetric functions on Hermitian manifolds. This kind of equations includes some of the most partial differential equations in complex geometry and analysis.
Under some conditions with the initial data on manifolds and admissible subsolutions, we derive a priori estimates for this complex mixed Hessian equation and solvability of the corresponding Dirichlet problem.
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Wednesday, April 20, 9:00-10:00, Zoom link
(ID: 883 5927 4553, Code: 488369)
Jingchen Hu (Chinese Academy of Sciences) - Regularity of Geodesics in the Space of Kahler Potentials - Abstract
In Kahler geometry, the geodesics in the space of Kahler potentials, which is governed by a homogenous complex Monge-Ampere equation, is an import tool. And the research on the regularity of geodesics has been very intense, and it shows that the optimal regularity of a general geodesic is $C^{1,1}$.
However, to freely use geometric tools and as a degenerate PDE problem, it's interesting to study if geodesics have higher regularity, locally or in some special cases. In this talk, we will briefly introduce the concept of the geodesics in the space of Kahler potentials, and present some former results on it,
then we discuss the problem that to what extent can we extend the regularity beyond $C^{1,1}$.
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Wednesday, April 27, 9:00-10:00, Zoom link
(ID: 892 4928 0436, Code: 266443)
Sungmin Yoo (Institute for Basic Science) - K-stability of Gorenstein Fano group compactifications with rank two - Abstract
In this talk, We give a classification of Gorenstein Fano equivariant compactifications of complex semisimple Lie groups with rank two, and determine which of them are equivariant K-stable and admit (singular) Kähler-Einstein metrics.
As a consequence, we obtain several explicit examples of K-stable Fano varieties admitting (singular) Kahler-Einstein metrics. We also compute the greatest Ricci lower bounds, equivalently the delta invariants for K-unstable varieties.
This gives us three new examples on which each solution of the Kähler-Ricci flow is of type II. This is based on the joint work with Jae-Hyouk Lee and Kyeong-Dong Park.
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Wednesday, May 4, 9:00-10:00, Zoom link
(ID: 870 3571 8641, Code: 431261)
Tristan Collins (Massachusetts Institute of Technology) - Complete Calabi-Yau metrics on the complement of two divisors - Abstract
In 1990 Tian-Yau proved the fundamental result that if Y is a Fano manifold and D is a smooth anti-canonical divisor, the complement X=Y\D admits a complete Calabi-Yau metric. A long standing problem has been to understand the existence of Calabi-Yau metrics when D is singular.
I will discuss the resolution of this problem when D=D_1+D_2 has two components and simple normal crossings. I will also explain a general picture which suggests the case of general SNC divisors should be inductive on the number of components. This is joint work with Y. Li.
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Wednesday, May 11, 9:00-10:00, Zoom link
(ID: 836 5873 7813, Code: 129822)
Haidong Liu (Sun Yat-sen University) - Some recent progress on Serrano's conjecture - Abstract
The positivity (nefness, strict nefness, ampleness) of divisors plays an important role in birational geometry. It is well-known that strictly nef divisors are not necessarily ample. However, Serrano's conjecture predicts that a small deformation of a strictly nef divisor in the direction of the canonical divisor is ample.
In this talk, I will survey some recent progress on this conjecture. Part of my works are joint with Roberto Svaldi, Part of them are joint with Shin-ichi Matsumura.
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Wednesday, May 18, 9:00-10:00, Zoom link
(ID: 814 7792 7989, Code: 709381)
Deping Ye (Memorial University of Newfoundland) - The $L_p$ Brunn-Minkowski theory for $C$-coconvex sets - Abstract
The classical Minkowski problem was posed by Minkowski in
1897 and has great influence on convex geometry and partial differential
equations. It asks whether a given nonzero finite Borel measure on the
unit sphere can be the surface area measure of some convex body. Based
on the $L_p$ addition of convex bodies, Lutwak derived the $L_p$ surface
area measure and initiated the study of the $L_p$ Minkowski problem in
1993. Note that solving the $L_p$ Minkowski problem requires to find
(weak) solutions to some Monge-Ampere equations.
In this talk, we will discuss the $L_p$ Brunn-Minkowski theory for
$C$-coconvex sets. The $C$-coconvex sets have found important
applications in many fields, such as algebraic geometry, singularity
theory, etc. We will talk about the $L_p$ addition of $C$-coconvex
sets, the $L_p$ Brunn-Minkowski and Minkowski inequalities, and a
variational formula which derives the $L_p$ surface area measure for
$C$-coconvex sets. The related Minkowski problem will be presented and
its solution will be provided.
We also discuss the case for $p=0$ which leads to the log-Minkowski
problem for $C$-coconvex sets. In particular, we will discuss the
log-Brunn-Minkowski and log-Minkowski inequalities for $C$-coconvex
sets. These inequalities solve an open problem regarding the uniqueness
of the solutions to the log-Minkowski problem for $C$-coconvex sets
raised by Schneider.
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Wednesday, May 25, 9:00-10:00, Zoom link
(ID: 826 1081 2154, Code: 134494)
Connor Mooney (University of California, Irvine) - Some regularity questions for the special Lagrangian equation - Abstract
The special Lagrangian equation is a fully nonlinear elliptic PDE whose solutions are potentials for volume-minimizing Lagrangian graphs. There exist continuous viscosity solutions to the Dirichlet
problem for this equation, but many basic regularity questions (such as whether these solutions have bounded gradient) remain open. In this talk
we will discuss some of these questions. We will use them to motivate the more general question of whether homogeneous functions with nowhere
vanishing Hessian determinant can change sign when the degree of homogeneity is between zero and one, and we will answer this question in the negative.
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Wednesday, June 1, 9:00-10:00, Zoom link
(ID: 891 4853 3378, Code: 079126)
Kyle Broder (The Australian National University & Beijing International Center for Mathematical Research) - Recent developments concerning the Schwarz lemma, with applications to the Wu-Yau theorem - Abstract
The Schwarz lemma is an essential technique in complex geometry. We will discuss some recent developments concerning the Schwarz lemma, using the one-parameter family of connections introduced by Gauduchon. We will also discuss applications to the Wu-Yau theorem in the Hermitian category.
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Wednesday, June 8, 9:00-10:00, Zoom link
(ID: 874 7040 1536, Code: 146008)
Nicholas McCleerey (University of Michigan) - Lelong Numbers of $m$-Subharmonic Functions Along Submanifolds - Abstract
We study the possible singularities of an $m$-subharmonic function $\varphi$ along a complex submanifold $V$ of a compact K\"ahler manifold, finding a maximal rate of growth for $\varphi$ which depends only on $m$ and $k$, the codimension of $V$. When $k < m$, we show that $\varphi$ has at worst log poles along $V$,
and that the strength of these poles is moreover constant along $V$. This can be thought of as an analogue of Siu's theorem. This is joint work with Jianchun Chu.
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Wednesday, June 15, 9:00-10:00, Zoom link
(ID: 839 3427 3174, Code: 324536)
Beomjun Choi (Pohang University of Science and Technology) - Liouville theorem for surfaces translating by sub-affine-critical powers of Gauss curvature - Abstract
We classify the translators to the flows by sub-affine-critical powers of Gauss curvature in $\mathbb{R}^3$. If $\alpha$ denotes the power, this is a Liouville theorem for degenerate Monge-Ampere equations $\det D^2u = (1+|Du|^2)^{2-\frac{1}{2\alpha}}$ for $0<\alpha<1/4$.
For the affine-critical-case $\det D^2u =1$, the classical result by Jorgens, Calabi and Pogorelov shows the level curves of given solution are homothetic ellipses. In our case, the level curves converge asymptotically to a round circle or a curve with $k$-fold symmetry for some $k>2$.
More precisely, these curves are closed shrinking curves to the $\frac {\alpha}{1-\alpha}$-curve shortening flow that were previously classified by B. Andrews in 2003. This is a joint work with K. Choi and S. Kim.