Workshop on algebraic and complex geometry in Jinju

Invited Speakers

• Ye-won Luke Cho (Gyeongsang National University)

• In-Kyun Kim (KIAS) 

• Yeongrak Kim (Pusan National University)

• Kang-Hyurk Lee (Gyeongsang National University)

• Hoseob Seo (IBS-CCG) 

Program 

• May 2 (Thursday) 15:00~16:30 Registration & Discussion, 16:30~17:30 Ye-won Luke Cho, 18:00~ Dinner 

• May 3 (Friday) 10:30~11:30 Kang-Hyurk Lee, 12:00~14:00 Lunch, 14:00~15:00 In-Kyun Kim, 15:20~16:20 Yeongrak Kim, 16:40~17:40 Hoseob Seo, 18:00~ Dinner 

• May 4 (Saturday) 10:00~12:00 Discussion (Kyeong-Dong Park)

Titles and Abstracts 

• Ye-won Luke Cho (Gyeongsang National University) 

Title: A brief introduction to singular Kähler-Einstein metrics 

Abstract: I shall summarize the construction of singular Kähler-Einstein metrics on a compact klt pair with negative or trivial first Chern class (Eyssidieux-Guedj-Zeriahi 2009). Not only does the notion of SKE metrics generalize canonical metrics introduced by Aubin and Yau, but it also found some applications in the uniformization theory of complex varieties with log terminal singularities. If time permits, I will also present my recent work with Y.-J. Choi on the regularity of SKE potentials and some geometric consequences. 

• In-Kyun Kim (KIAS) 

Title: K-stability and anti-canonical polar cylinder of Fano varieties  

Abstract: A cylinder in a normal projective variety is a Zariski open subset that is isomorphic to a product of an affine line and an affine variety. There exists a connection between rationality and cylindricity for a normal variety. Additionally, a normal projective space has an anti-canonical polar cylinder if and only if its affine cone admits a nontrivial unipotent group action. Therefore, proving the existence of an anti-canonical polar cylinder is crucial. Using the alpha-invariant suggested by Tian, we can prove the existence of an anti-canonical polar cylinder for Fano varieties. Consequently, there is a connection between K-stability and the existence of an anti-canonical polar cylinder for Fano varieties. In this talk, we study a relationship between K-stability and the existence of an anti-canonical polar cylinder for Fano varieties.

• Yeongrak Kim (Pusan National University) 

Title: Koszul flattenings and tensor ranks of determinant and permanent tensors 

Abstract: The rank of a tensor T is the minimum number of decomposable tensors whose sum equals to T so that it extends the notion of the matrix rank. Understanding the rank of a given tensor has great theoretical and practical applications, however, the rank of a tensor of high order is very hard to determine in most cases. For instance, Strassen's algorithm for matrix multiplication tells us that we only need 7 multiplications (not 8) when we multiply two 2 by 2 matrices, in other words, the 2 by 2 matrix multiplication tensor has rank 7. Usually, the study of rank complexities of a tensor is achieved throughout a flattening method that derives a certain matrix from the given tensor. The Koszul flattening method, introduced by Landsberg and Ottaviani, is a simple and powerful method that works for a tensor of order 3 using the exterior product. It has several applications in the study of lower bounds of tensor ranks and Waring ranks for various tensors (of order 3) appearing in algebra and geometry, including the matrix multiplication tensor and the determinant/permanent polynomial for 3 by 3 matrices. 

We first discuss a successive usage of Koszul flattening for tensors of higher orders introduced by Hauenstein-Oeding-Ottaviani-Sommese. As applications, I will discuss some observations on the lower bounds on tensor ranks of the determinant and permanent as tensors of order n. These results greatly improve lower bounds on the border ranks of those tensors for n at least 4. This is a joint work with Jong In Han and Jeong-Hoon Ju.

• Kang-Hyurk Lee (Gyeongsang National University) 

Title: The potential rescaling method in the uniformization of negatively curved Kähler manifolds  

Abstract: In the study of complex model domains, the affine rescaling method has been a fundamental approach to classify them.  A typical application of this method is to show the existence of 1-parameter family of automorphisms on a domain with noncompact automorphism group; for instance, the Bedford-Pinchuk theorem on finite type domains and Frankel's theorem on convex domains. Especially, Frankel’s theorem implies that a bounded convex domain in the complex Euclidean space admitting a compact quotient should be a bounded symmetric domain. 

Recently, we introduced the potential rescaling method to replace the affine rescaling method for the generalization of Frankel’s theorem, so for the uniformization of negatively curved compact Kähler manifolds. This method is to rescale a potential of Käher-Einstein metric by automorphisms and construct a specific potential which generates a complete holomorphic vector field. This gives an intrinsic generalization of the Wong-Rosay theorem which says that a smoothly bounded domain with a compact quotient is biholomorphic to the unit ball. In this talk, I will introduce the potential rescaling method and recent problems for the uniformization. 

• Hoseob Seo (IBS-CCG) 

Title: Analytic multiplier ideal sheaves and adjoint ideal sheaves

Abstract: Various kinds of multiplier ideals played important roles in complex geometry and algebraic geometry. In this talk, we recall the notions of multiplier ideal sheaves and adjoint ideal sheaves in both algebraic and analytic categories and their basic properties. One of applications of these multiplier ideal sheaves is to characterize singularities of log pairs of varieties. In this point of view, we introduce the notion of the Ohsawa measure and present relations between singularities of pairs and integrability of the Ohsawa measure. This talk is based on joint work with Dano Kim.