researchseminars.org

Ryushi GOTO (Osaka University)
2021-07-12 01:00 - 01:50 UTC | https://stable.researchseminars.org/talk/2021PRCSG/1/
Title: Scalar curvature and moment map in Generalized Kähler geometry (https://stable.researchseminars.org/talk/2021PRCSG/1/)
by Ryushi GOTO (Osaka University) as part of 2021 Pacific Rim Complex & Symplectic Geometry Conference

Abstract
We introduce a notion of scalar curvature of a twisted generalized Kähler manifold in terms of pure spinors formalism. A moment map framework on an arbitrary compact twisted generalized Kähler manifold is provided and then it turns out that a moment map is given by the scalar curvature under the certain condition, which is a generalization of the result of the scalar curvature as a moment map in the ordinary Kähler geometry, due to Fujiki and Donaldson. A noncommutative compact Lie group G does not have any Kähler structure. However, we show that a compact Lie group has a family of generalized Kähler structures twisted by the Cartan 3-form, which is constructed by the action of the real Pin group of the double of Cartan subalgebra. Then we show that an arbitrary compact Lie group admits generalized Kähler structures with constant scalar curvature. In particular, generalized Kähler structures with constant scalar curvature on the standard Hopf surface are explicitly given.

Byunghee AN (Kyungpook National University)
2021-07-12 02:10 - 03:00 | https://stable.researchseminars.org/talk/2021PRCSG/2/
Title: Augmentations and ruling polynomials for Legendrian graphs (https://stable.researchseminars.org/talk/2021PRCSG/2/)
by Byunghee AN (Kyungpook National University) as part of 2021 Pacific Rim Complex & Symplectic Geometry Conference

Abstract
In this talk, we will show the equivalence between two Legendrian isotopy invariants of Legendrian graphs: (i) augmentation number via point-counting over a finite field for the augmentation variety of the associated Chekanov-Eliashberg DGA, and (ii) the ruling polynomial via combinatorics of the decompositions of the associated front projections. This is a joint work with Youngjin Bae(Incheon National University) and Tao Su(Yau Mathematical Sciences Center, Tsinghua University).

Youngjin Bae (Incheon National University)
2021-07-13 00:30 - 01:20 | https://stable.researchseminars.org/talk/2021PRCSG/3/
Title: Seeds many Lagrangian fillings for Legendrian links (https://stable.researchseminars.org/talk/2021PRCSG/3/)
by Youngjin Bae (Incheon National University) as part of 2021 Pacific Rim Complex & Symplectic Geometry Conference

Abstract
I will introduce Legendrian links of finite and affine Dynkin diagrams, and then argue that there are at least as many Lagrangian fillings as seeds in the corresponding cluster structure. The main ingredients are N-graphs developed by Casals-Zaslow, and cluster structures by Fomin-Zelevinsky. This is a joint work with Byung Hee An and Eunjeong Lee.

Huai-Liang CHANG (Hong Kong University of Science and Technology)
2021-07-13 01:35 - 02:25 | https://stable.researchseminars.org/talk/2021PRCSG/4/
Title: Structure of high genus Gromov Witten invariants (https://stable.researchseminars.org/talk/2021PRCSG/4/)
by Huai-Liang CHANG (Hong Kong University of Science and Technology) as part of 2021 Pacific Rim Complex & Symplectic Geometry Conference

Abstract
Gromov Witten invariants Fg encodes the numbers of genus g curves in Calabi Yau threefolds and play an important role in enumerative geometry. In 1993, Bershadsky, Cecotti, Ooguri, Vafa exhibited a hidden ``Feynman structure” governing all Fg’s at once, using path integral methods. The counterpart in mathematics has been missing for many years. After a decades of search, in 2018, a mathematical approach: Mixed Spin P field (MSP) moduli, is finally developed to provide the wanted ``Feynman structure”, for quintic CY 3fold. Instead of enumerating curves in the quintic 3fold, MSP enumerate curves in a large N dimensional singular space with quintic-3-fold boundary. The “P fields” and “cosections” are used to formulate counting in the singular space via a Landau Ginzburg type construction. In this talk, I shall focus on geometric ideas behind the MSP moduli. The results follows from a decade of joint works with Jun Li, Shuai Guo, Young Hoon Kiem, Weiping Li, Melissa C.C. Liu, Jie Zhou, and Yang Zhou.

Hisashi KASUYA (Osaka University)
2021-07-13 02:40 - 03:30 | https://stable.researchseminars.org/talk/2021PRCSG/5/
Title: Non-invariant deformations of left-invariant complex structures on compact Lie groups (https://stable.researchseminars.org/talk/2021PRCSG/5/)
by Hisashi KASUYA (Osaka University) as part of 2021 Pacific Rim Complex & Symplectic Geometry Conference

Abstract
It is known that every compact Lie group of even dimension admits left-invariant complex structures. We study deformations of left-invariant complex structures on simply connected semisimple compact Lie groups which are non-invariant. We compute cohomology of vector bundles for such non-invariant complex structures and see the difference between invariant complex structures and non-invariant complex structures. This talk is a joint work with Hiroaki Ishida (Kagoshima).

Hyungryul Baik (Korea Advanced Institute of Science and Technology)
2021-07-14 01:00 - 01:50 | https://stable.researchseminars.org/talk/2021PRCSG/6/
Title: Limits of canonical metrics in low-dimensions (https://stable.researchseminars.org/talk/2021PRCSG/6/)
by Hyungryul Baik (Korea Advanced Institute of Science and Technology) as part of 2021 Pacific Rim Complex & Symplectic Geometry Conference

Abstract
For a tower of finite normal covers of graphs or surfaces, one can consider a sequence of metrics on the base given by pull-back of canonical metrc of the covers. We show that such a sequence has a limit and it depends only on the cover approximated by the tower up to scaling. The case of compact Riemann surface where the tower approximates the universal cover is due to Kazhdan. In this talk, we will mostly focus on the surface case and explain how the L^2-theory can be applied. This talk is based on a joint work with Farbod Shokrieh and Chenxi Wu.

Ziyu ZHANG (ShanghaiTech University)
2021-07-14 02:10 - 03:00 | https://stable.researchseminars.org/talk/2021PRCSG/7/
Title: Degenerations of Hilbert schemes of points on K3 surfaces (https://stable.researchseminars.org/talk/2021PRCSG/7/)
by Ziyu ZHANG (ShanghaiTech University) as part of 2021 Pacific Rim Complex & Symplectic Geometry Conference

Abstract
It is a widely open problem to understand the degenerations of higher dimensional hyperkähler manifolds. The simplest case would be the degenerations of Hilbert schemes of points on K3 surfaces. Given a simple degeneration family of K3 surfaces, there are two existing constructions of the degenerations of the Hilbert schemes of its fibers in the literature, due to Nagai and Gulbrandsen-Halle-Hulek respectively. I will compare the two constructions with an emphasis on the geometry of the latter. Based on joint work with M.G.Gulbrandsen, L.H.Halle and K.Hulek.

Ngoc Cuong Nguyen (Korea Advanced Institute of Science and Technology)
2021-07-15 00:30 - 01:20 | https://stable.researchseminars.org/talk/2021PRCSG/8/
Title: Continuous solutions to Monge-Amp ere equations on Hermitian manifolds for measures dominated by capacity (https://stable.researchseminars.org/talk/2021PRCSG/8/)
by Ngoc Cuong Nguyen (Korea Advanced Institute of Science and Technology) as part of 2021 Pacific Rim Complex & Symplectic Geometry Conference

Abstract
We prove the existence of a continuous quasi-plurisubharmonic solution to the Monge-Amp ere equation on a compact Hermitian manifold for a very general measre on the right hand side. We admit measures dominated by capacity in a certain manner, in particular, moderate measures studied by Dinh-Nguyen-Sibony. As a consequence, we give a characterization of measures admitting Holder continuous quasi-plurisubharmonic potential, inspired by the work of Dinh-Nguyen. This is joint work with S lawomir Ko lodziej.

Peng WU (Fudan University)
2021-07-15 01:35 - 02:25 | https://stable.researchseminars.org/talk/2021PRCSG/9/
Title: Complex structures on Einstein four-manifolds of positive scalar curvature (https://stable.researchseminars.org/talk/2021PRCSG/9/)
by Peng WU (Fudan University) as part of 2021 Pacific Rim Complex & Symplectic Geometry Conference

Abstract
The question that when a four-manifold with a complex structure admits a compatible Einstein metric of positive scalar curvature has been answered by Tian, LeBrun, respectively. Tian classified Kahler-Einstein four-manifolds with positive scalar curvature, LeBrun classified Hermitian, Einstein four-manifolds of positive scalar curvature. In this talk we consider the inverse problem, that is, when a simply connected four-manifold with an Einstein metric of positive scalar curvature admits a compatible complex structure. We will show that if the determinant of the self-dual Weyl curvature is positive then the manifold admits a compatible complex structure.

Ryosuke TAKAHASHI (Kyushu University)
2021-07-16 01:00 - 01:50 | https://stable.researchseminars.org/talk/2021PRCSG/10/
Title: Some geometric flow approaches for deformed Hermitian-Yang-Mills equation (https://stable.researchseminars.org/talk/2021PRCSG/10/)
by Ryosuke TAKAHASHI (Kyushu University) as part of 2021 Pacific Rim Complex & Symplectic Geometry Conference

Abstract
On SYZ mirror symmetry, a deformed Hermitian-Yang-Mills (dHYM) metric is a fiber metric on a holomorphic line bundle, which is the mirror object to a special Lagrangian section of the dual torus fibration. As a parabolic analogue, Jacob-Yau introduced the Line Bundle Mean Curvature Flow (LBMCF) as the mirror of the Lagrangian Mean Curvature Flow. In this talk, we explore some geometric flow approaches for dHYM metrics: (A) On K\”ahler surfaces, it is known that the existence of dHYM metrics is equivalent to a K\”ahler condition for a certain cohomology class. We relax this condition and study how the LBMCF blows up. (B) Recently, Collins-Yau discovered a new variational characterization for dHYM metrics. Motivated by this, we introduce a new geometric flow which is designed to deform a given metric to a dHYM one. Then we show that this new flow potentially has more global existence and convergence properties than the LBMCF.

Kyeongsu CHOI (Korea Institute for Advanced Study)
2021-07-16 02:10 - 03:00 | https://stable.researchseminars.org/talk/2021PRCSG/11/
Title: Translators of the Gauss curvature flow (https://stable.researchseminars.org/talk/2021PRCSG/11/)
by Kyeongsu CHOI (Korea Institute for Advanced Study) as part of 2021 Pacific Rim Complex & Symplectic Geometry Conference

Abstract
We begin by reviewing the blow-up analysis for the minimal surfaces at isolated singularities, and will quickly discuss about some related recent developments in the singularity analysis for the mean curvature flow. Then, we will classify the translating surfaces under the flows by sub-affine-critical powers of the Gauss curvature, which is a Liouville theorem for a class of Monge-Ampere equations. We will put an emphasis on the divergence free property of the linearized operator of the Monge-Ampere equation. This is a joint work with Beomjun Choi and Soojung Kim.

Genki HOSONO (Tohoku University)
2021-07-15 02:40 - 03:30 | https://stable.researchseminars.org/talk/2021PRCSG/12/
Title: On Berndtsson-Lempert's proof of optimal $L^2$ extension theorem and extension from non-reduced varieties (https://stable.researchseminars.org/talk/2021PRCSG/12/)
by Genki HOSONO (Tohoku University) as part of 2021 Pacific Rim Complex & Symplectic Geometry Conference

Abstract
I'd like to talk about the proof of an optimal version of the Ohsawa-Takegoshi $L^2$ extension theorem and its application to an extension theorem from non-reduced varieties.
