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．Seminars

	An Extension Theorem for LC Pairs

October 14, 2016
R202, Astronomy-Mathematics Building, NTU

Invited Speakers:
Tsz On Mario Chan (National Taiwan University)
Shin-Yao Jow (National Tsing Hua University)
Zheng-yu Hu (Zhejiang University)
Wan Keng Cheong (National Cheng Kung University)

Organizers:
Jungkai Chen (National Taiwan University& NCTS)
Ching-Jui Lai (National Cheng Kung University)

Program (10:00~17:00)

	Time
Speaker: Shin-Yao Jow (NTHU) Title: Is the Fujita-Zariski Eecomposition the Same as the Nakayama-Zariski Decomposition? Abstract: Let $X$ be a smooth projective variety. If $D$ is a big $\mathbb{R}$ -divisor on $X$ , there are two ways to characterize the positive part $P(D)$ of the Zariski decomposition of $D$ : item $P(D)$ is the largest nef $\mathbb{R}$ -divisor $\le D$ .item $P(D)$ is the nef $\mathbb{R}$-divisor $\le D$ such that $H^0(X,\lfloor mP(D) \rfloor)=H^0(X,\lfloor mD\rfloor)$ for all positive integer $m$ . end{enumerate} If $D$ is big, the positive part defined by either of these conditions agrees. If $D$ is merely pseudoeffective, these two conditions can be suitably generalized, but it is no longer clear that they define the same decomposition of $D$ . We will give an introduction and share some premature thought about this problem.	10:30-11:20
Speaker: Tsz On Mario Chan (NCTS) Title: An Extension Theorem for LC Pairs Abstract: As shown in the work of Demailly--Hacon--Pu aun, the conjectural nonvanishing theorem and extension theorem for semi-log-canonical (or even semi-dlt) pairs imply the existence of good minimal models for klt pairs. The extension theorem for plt pairs is also shown to hold true in the same work. The extension theorem for the dlt case with a stronger assumption is then proved by Gongyo and Matsumura. In this talk, a proof of the extension theorem for (semi-)log-canonical pairs is presented. The proof follows the lines in the proof of Demailly--Hacon--Pu aun and makes use of the techniques in the latest $L^2$ -extension theorem of Demailly.This is a joint work with Young-Jun Choi in Greonoble.	11:30-12:20
Speaker: Zhengyu Hu (NCTS) Title: Log Canonical Pairs over Varieties with Maximal Albanese Dimension Abstract: Let $(X,B)$ be a log canonical pair over a normal variety $Z$ with maximal Albanese dimension. If $K_X+B$ is relatively abundant over $Z$ , for example, $K_X+B$ is relatively big over $Z$ , then we prove that $K_X+B$ is abundant. In particular, the subadditvity of Kodaira dimensions $\kappa(K_X+B) \geq \kappa(K_F+B_F)+ \kappa(Z)$ holds, where $F$ is a general fiber, $K_F+B_F= (K_X+B)\|_F$ , and $\kappa(Z)$ means the Kodaira dimension of a smooth model of $Z$ . I will also discuss the log Iitaka conjecture for log canonical pairs.]	14:00-14:50
Speaker: Wan Keng Cheong (NCKU) Title: On SYM-HILB Correspondence Abstract: The crepant resolution conjecture predicts a SYM-HILB correspondence between the orbifold quantum cohomology of the symmetric product of $S$ and the quantum cohomology of the Hilbert scheme of points on $S$ , where $S$ is any smooth surface. I would like to present my current project on the correspondence for a certain class of surfaces.	15:00-15:50