Taeyong Ahn (Inha University)
Title: Intersection of positive closed currents
Abstract: In this lecture, we will discuss the wedge product of positive closed currents. Positive closed currents are measure-theoretic methods to study geometric objects. For technical reasons, the theory is well developed for positive closed (1, 1)-currents. First, we will review some basic materials concerning positive closed currents and the wedge product of positive closed (1, 1)-currents. Next, we study the theory of superpotentials on the projective spaces for the higher bidegree. Also, why it is not easy to extend superpotentials to all positive closed currents on a general compact Kaehler manifold will be treated. If time allows, we will talk about the theory of density introduced by Dinh-Sibony.
References: 1) general positive closed currents: Demailly's book. 2) superpotentials on the projective space: Dinh, Tien-Cuong; Sibony, Nessim, Super-potentials of positive closed currents, intersection theory and dynamics. Acta Math. 203 (2009), no. 1, 1–82. 3) superpotentials on the compact Kaehler manifold: Dinh, Tien-Cuong; Sibony, Nessim, Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms. J. Algebraic Geom. 19 (2010), no. 3, 473–529. 4) regularization of positive closed currents: Dinh, Tien-Cuong; Sibony, Nessim, Regularization of currents and entropy. Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 6, 959–971. 5) theory of density: Dinh, Tien-Cuong; Sibony, Nessim, Density of positive closed currents, a theory of non-generic intersections. J. Algebraic Geom. 27 (2018), no. 3, 497–551. Vu, Duc-Viet, Densities of currents on non-Kähler manifolds. Int. Math. Res. Not. IMRN 2021, no. 17, 13282–13304.
Ye-Won Luke Cho (Pusan National University)
Title: Pluripotential theory and degenerate complex Monge-Ampère equations
Abstract: In recent years, the pluripotential theory in complex analysis has been successfully applied to the study of degenerate complex Monge-Ampère equations. Eyssidieux-Guedj-Zeriahi also showed in 2009 that solutions of such equations lead to the notion of singular Kähler-Einstein metrics on `mildly singular' compact algebraic varieties, generalizing the classical works of Aubin and Yau. In this talk, I would like to review the aforementioned pluripotential methods in Kähler geometry. I will also present my recent work with Y.-J. Choi on the continuity of solutions of degenerate complex Monge-Ampère equations on compact complex analytic spaces. This in particular implies that any singular Kähler-Einstein potential on a normal compact Kähler variety is continuous on the whole variety.
Kang-Tae Kim (POSTECH)
Title: Potential Function Lemma for $C'$ closed differential forms by $G^3 K$
Abstract: Poincaré lemma is usually known as "Any closed $C^\infty$ differential form is locally exact." History tells us that it is actually a theorem of Vito Volterra. In any case, I do not wish to discuss to whom the due credit has to be tributed. So we just call it "The Potential Function Lemma." On the other hand, this lemma works for closed $C^1$ differential form. But, it was not known whether the lemma holds for $C'$ forms. I will present how this works investigating how much Green's theorem tells us. This work is a collaboration with Robert E. Greene of U. C. L. A. We developed this because it is significant for the "correct" setup of complex function theory. In the presentation, I will also mention what the correct setup of Complex Analysis has to be.
Aeryeong Seo (Kyungpook National University)
Title: On holomorphic fiber bundle over compact Kahler manifold with bounded symmetric domain fibers
Abstract: In this talk, I will present the existence of plurisubharmonic exhaustion functions on the holomorphic fiber bundles with irreducible bounded symmetric domain fibers. I will talk about basic notions and ideas of proofs.
Jihun Yum (IBS-CCG)
Title: Statistical Bergman Geometry
Abstract: Let Ω be a bounded domain in $C^n$ and P(Ω) be the space of all (real) probability distributions on Ω. In general, P(Ω) is an infinite-dimensional smooth manifold and there exists a natural Riemannian pseudo-metric in P(Ω), which is called the Fisher information metric. In this talk, we first introduce a map Φ: Ω → P(Ω), which in fact becomes an isometry with respect to the Bergman metric on Ω and the Fisher information metric on P(Ω). The map Φ connects the Bergman geometry and the information geometry and makes us interpret the Bergman geometry in a statistical way. As a result, we found a new formula for the holomorphic sectional curvature of the Bergman metric, which automatically implies that the holomorphic sectional curvature is always less than or equal to 2. At last, we will prove a central limit theorem on Ω. This is joint work with Gunhee Cho at U. C. Santa Barbara.