Algebraic Geometry Day

2021.04.16-04.16
Beijing, China & Zoom

Organizers:

Baohua Fu (MCM)   Yifei Chen (AMSS)

 

Speakes:

Kuan-Wen Lai (University of Massachusetts Amherst)

Yu-Shen Lin (Boston University)

Yun Gao（Shanghai Jiao Tong University）

Changzheng Li（Sun Yat-sen University）

 

Schedule:

 

April 16
Time	Place	Speakers	Titles
9:00–10:00	Zoom: 4663562952 Pswd: mcm1234	Kuan-Wen Lai	On the irrationality of moduli spaces of K3 surfaces
10:15–11:15		Yu-Shen Lin	Special Lagrangian Fibrations in Log Calabi-Yau Surfaces and Mirror Symmetry
13:30-14:30	MCM410	Yun Gao	Dimension estimate ,orthogonal maps and CR maps
14:45-15:45		Changzheng Li	On quantum cohomology and quantum K-theory of flag varieties

 

Speaker:  Kuan-Wen Lai (University of Massachusetts Amherst)

Title: On the irrationality of moduli spaces of K3 surfaces

Abstract: As for moduli spaces of curves, the moduli space of polarized K3 surfaces of genus g is of general type and thus is irrational for g sufficiently large. In this work, we estimate how the irrationality grows with g in terms of the measure introduced by Moh and Heinzer. We proved that the growth is bounded by a polynomial in g of degree 15 and, for three sets of infinitely many genera, the bounds can be refined to polynomials of degree 10. These results are built upon the modularity of the generating series of these moduli spaces in certain ambient spaces, and also built upon the existence of Hodge theoretically associated cubic fourfolds, Gushel–Mukai fourfolds, and hyperk？hler fourfolds. This is a collaboration with Daniele Agostini and Ignacio Barros (arXiv:2011.11025).

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Speaker: Yu-Shen Lin (Boston University)

Title: Special Lagrangian Fibrations in Log Calabi-Yau Surfaces and Mirror Symmetry

Abstract: Strominger-Yau-Zaslow conjecture predicts that the Calabi-Yau manifolds admit special Lagrangian fibrations and the mirror can be constructed via the dual torus fibration. The conjecture has been the guiding principle for mirror symmetry while the original conjecture has little progress. In this talk, I will prove that the SYZ fibration exists in certain log Calabi-Yau surfaces and their mirrors indeed admit the dual torus fibration under suitable mirror maps. The result is an interplay between geometric analysis and complex algebraic geometry. The talk is based on joint works with T. Collins and A. Jacob. 

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Speaker:  Prof. Yun Gao（Shanghai Jiao Tong University）

Title: Dimension estimate ,orthogonal maps and CR maps

Abstract: In this talk, we will introduce a dimension formula for local holomorphic mappings which is inspired from Green hyperplane restriction Theorem. This formula gives a dimension estimate for the linear spans of the images of linear subspaces. As an application, we use this formula to study the holomorphic mappings between real hyperquadrics, which is a very classical topic in Several Complex Variables. Combining the dimension formula and a notion of orthogonality, we will present a new coordinate-free approach to the study of holomorphic mappings. Using these method, we are able generalize many well-known rigidity theorems for the holomorphic mappings between the hyperquadrics, especially the degenerated cases. In addition, we will talk about Huang-Ji-Yin Conjecture on the gap phenomenon for the complex unit balls and extend it to all generalized balls. This is a joint work with Sui-Chung Ng.

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Speaker:  Prof. Changzheng Li（Sun Yat-sen University）

Title: On quantum cohomology and quantum K-theory of flag varieties

Abstract: In this talk, we will review some facts on the symmetry on quantum cohomology of flag varieties G/P. In particular, we will introduce a cyclic group symmetry on the quantum cohomology of complex Grassmannian Gr(k, n), due to a formula by Belkale. We will show its generalization to the quantum K-theory QK(Gr(k,n)).