Kang-Tae Kim (POSTECH)

Title: 小平(코다이라) 소멸정리와 GAGA

Abstract: 포스텍의 요청에 따라 저는 “Géometrie Algebrique et Géometrie Analytique (GAGA)” 라는 이름으로 알려진 분야의 위대한 업적인 Kodaira Vanishing Theorem (1953) 에 관하여 이번 학기에 포스텍에서 대학원 강의를 하였습니다. 대수기하학과 복소해석기하학의 절묘한 조화를 보여 주는 주제입니다. 벡터 다발에 값을 가지는 미분 형식에 관한 돌보(Dolbeault) 코호몰로지 소멸 정리이므로, 필요한 개념과 정리, 정의, 증명 및 연구 방법론 등을 소개하되 가능한 1953년 전후에 개발된 小平邦彦 (코다이라 쿠니히코) 교수의 방법론에 충실하게, 구성된 이론을 현대적인 개선을 조금 가미하고 요약하여 발표합니다. 이번 학기를 통해 다듬어진 저의 강의록 “GAGA via The Kodaira Vanishing Theorem” 파일을 (한글, pdf) 학술회의에서 무료 배부하겠습니다.

    강의록 파일: click here to download 

     PPT 파일: click here to download

    Seungjae Lee (Kyungpook National University)

Title: Global extension of holomorphic jets on a complex submanifold

Abstract: I will discuss holomorphic jets and their global extension. The geometry of complex submanifolds determines the global extendibility of holomorphic jets to a holomorphic object on a complex manifold. First, I will introduce some previous works by Grauert and Griffiths, and, if time permits, I will present some interesting results. 

    Hoseob Seo (IBS-CCG)

Title:  A survey of multiplier ideal sheaves on toric varieties 

Abstract: The multiplier ideal sheaf is an important object in algebraic geometry. In both the algebraic and analytic categories, its computation requires several subtle steps and is often quite challenging. However, when the ambient space or the input has many symmetries, it can be computed using combinatorial methods. In this talk, we survey a generalization of Blickle's criterion for multiplier ideal sheaves on toric varieties due to J. An.

    Sungmin Yoo (Incheon National University)

Title: Convergence of sequences of the Bergman type volume forms 

Abstract:  Following the Yau-Tian-Donaldson conjecture, the construction of sequences of Bergman-type metrics on a polarized manifold has been studied by many mathematicians including Tian, Donaldson, Tsuji, Berman, Berndtsson, and others. In this talk, I will survey the relevant results on this topic, including my recent findings on the uniform convergence of Tsuji's Bergman sequence on a uniformly squeezing domain. I will also introduce several remaining problems in this area. 

    Jihun Yum (Gyeongsang National University)

Title: Schwarz lemma for the Bergman metric 

Abstract:   The Schwarz(-Pick) lemma in one complex variable states that, for a holomorphic function f : D → D between two unit discs, f ∗ g_D ≤ g_D, where g_D denotes the Bergman metric of D. This is generalized by Shing-Tung Yau in the following sense. 

• (M, g_M): a complete Kahler manifold with K1 ≤ Ric(g_M) 

• (N, g_N ): a Hermitian manifold with holomorphic bisectional curvature(g_N ) ≤ K2 < 0 

• f : M → N holomorphic 

Then f ∗ g_N ≤ (K1/K2) g_M. In this presentation, we discuss Schwarz(-type) lemma for the Bergman metric.