Joint workshop on algebraic and complex geometry I
(2024 Summer)

Organizers

• 김영락 (부산대)

• 김인균 (KIAS HCMC)

• 김현규 (KIAS)

• 박경동 (경상국립대)

• 이강혁 (경상국립대)

Focus Lecturers

• 김인균 (KIAS HCMC)

• 서호섭 (IBS-CCG)

• 최영준 (부산대)

Contributed Lecturers

• 김재현 (이화여대)

• 조예원 (경상국립대)

• 염지훈 (경상국립대)

Participants

• 권민성 (KAIST)

• 김동현 (연세대)

• 김신영 (연세대)

• 김영락 (부산대)

• 김유정 (부산대)

• 김인균 (KIAS HCMC)

• 김재현 (이화여대)

• 김정섭 (KIAS)

• 박경동 (경상국립대)

• 박수현 (부산대)

• 변지수 (경남대)

• 서호섭 (IBS-CCG)

• 엄기윤 (KAIST)

• 염지훈 (경상국립대)

• 원준영 (이화여대)

• 이강혁 (경상국립대)

• 이대원 (이화여대)

• 장성욱 (연세대)

• 전호연 (서울대)

• 정지원 (부산대)

• 조민서 (부산대)

• 조예원 (경상국립대)

• 조용화 (경상국립대)

• 최영준 (부산대)

• 최준호 (KIAS)

• 최현일 (부산대)

• 홍규식 (전주대)

• 황택규 (경상국립대)

Title & Abstract

• 김인균 (KIAS HCMC)
  Title :  Singularities of Pairs
  
  Abstract : The class of all pairs (X,D), where X is a variety and D is a Q-linear combination of divisors, is usually referred to as the log category. The terminology seems to derive from the observation that differential forms on a variety X with logarithmic poles along a divisor D should be thought of as analogs of holomorphic differential forms. In this talk, we will study the basic theory of discrepancies. We will discuss the important issues of how discrepancies can be computed from a log resolution. Next, we will prove the very useful inversion of adjunction theorem, which illuminates how discrepancies behave under restriction to hypersurfaces. Finally, we will study the formula for the log canonical threshold of a curve on a surface.
  
  
  

• 서호섭 (IBS-CCG)
  Title : Plurisubharmonic functions and analytic multiplier ideal sheaves
  
  Abstract : Plurisubharmonic functions are fundamental objects in complex geometry and several complex variables. In this series of lectures, we will review the notion of plurisubharmonic functions and their important applications, including pseudoconvexity and Hörmander's d-bar estimates. Given their strong connection with algebraic geometry, we will explore analytic multiplier ideal sheaves and their applications to vanishing theorems and surjectivity theorems from both analytic and algebraic view points. If time permits, we will see that analytic multiplier ideal sheaves may behave more peculiarly than algebraic ones in terms of jumping numbers.

References:

1. J.-P. Demailly, Complex Analytic and Differential Geometry
2. J.-P. Demailly, Transcendental Methods in Algebraic Geometry
3. R. Lazarsfeld, Positivity in Algebraic Geometry
4. L. Hörmander, An Introduction to Complex Analysis
5. S. G. Krantz, Function Theory of Several Complex Variables

• 최영준 (부산대)
  Title : Hodge theory and its application in Complex Geometry
  
  Abstract : Hodge theory is an important tool in complex geometry, providing a method for studying cohomology groups using partial differential equations. In this lecture series, we will introduce the basics of Hodge theory, starting with an overview of Hermitian differential geometry. We will explore applications such as the Hodge decomposition theorem and the Serre duality theorem. Finally, we will discuss the celebrated Kodaira vanishing theorem.

References: 

1. Griffiths and Harris, Principles of Algebraic Geometry
2. J.-P. Demailly, Complex Analytic and Differential Geometry
3. R. O. Wells, Differential Analysis on Complex Manifolds 

• 김재현 (이화여대)
  Title : Some linear systems on del Pezzo surfaces
  
  Abstract : For an arbitrary divisor on a nonsingular projective variety, we consider a vector space consisting of certain rational functions defined on the variety. There exists a well-established correspondence between effective divisors linearly equivalent to given divisor and elements in the vector space up to constant factor. The projectivizations of subspaces in such vector spaces are referred to as linear systems. Then it is well-known that linear systems provide various information for the geometry of the variety. In this talk, we will briefly introduce some linear systems of rational curves on del Pezzo surfaces and present recent works related to them. 

• 조예원 (경상국립대)
  Title : Introduction to singular Kähler-Einstein metrics
  
  Abstract : In 2009, Eyssidieux-Guedj-Zeriahi constructed singular Kähler-Einstein metrics on a compact klt pair with negative or trivial first Chern class, generalizing the works of Aubin and Yau. Their work was largely motivated by the Minimal Model Program developed by Birkar-Cascini-Hacon-Mckernan, and it led to the existence of the SKE metric of negative scalar curvature on the canonical model of a smooth projective variety of general type. Later, the notion of the SKE metric also found some applications in the uniformization theory of complex varieties. In this talk, I shall summarize the construction of SKE metrics and my recent work with Y.-J. Choi on the regularity of SKE potentials.

• 염지훈 (경상국립대)
  Title : Introduction to the Bergman geometry
  
  Abstract : The Poincaré metric on the unit disc D, which is invariant under all biholomorphisms (or conformal maps) of D, is the most fundamental Riemannian metric in differential geometry. We first introduce the Bergman metric on a bounded domain in $\mathbb{C}^n$, which can be regarded as a generalization of the Poincaré metric. Then we introduce some fundamental theorems that may show how to use the curvatures of the Bergman metric to characterize bounded domains in $\mathbb{C}^n$ (or complex manifolds). At last, my recent work related to them will be presented.