Jan 27-Jul 20, 2019

Several complex variables provides robust tools for complex/algebraic geometry, and complex/algebraic geometry provide rich problems for several complex variables. “Workshop on several complex variables and complex geometry” will focus on the fundamental material and research topics in both fields. Selected topics below are chosen for the fundamental material part; and reports on the research papers will be given in the research topic part. The depth of the fundamental material part should be suitable for master students in the relevant areas.

**Fundamental material**:

1. Positive currents and Lelong numbers.

2. L^2 estimates, Ohsawa–Takegoshi Theorem, Skoda’s Theorem.

3. Optimal L^2 extension.

4. Strong openness.

Reference：

Bo Berndtsson: 1. L2-methods for the ∂ ̄-equation (Chapters 1, 2)

2. An Introduction to things ∂ ̄ (Chapters 4,5, 6)

**Research topics**:

1. J.-P. Demailly, Th. Peternell, and M. Schneider, "Pseudo-effective line bundles on compact Kähler manifolds. " Internat. J. Math. 12 (2001), no. 6, 689–741.

2. Păun, Mihai, and Shigeharu Takayama. "Positivity of twisted relative pluricanonical bundles and their direct images." Journal of Algebraic Geometry 27.2 (2018): 211-272.

3. Deng, Fusheng, et al. "New characterizations of plurisubharmonic functions and positivity of direct image sheaves." arXiv:1809.10371 (2018).

4. Guan, Qi'an, and Xiangyu Zhou. "A proof of Demailly's strong openness conjecture." Annals of Mathematics (2015): 605-616.

5. Guan, Qi'an, and Xiangyu Zhou. "A solution of an L 2 extension problem with an optimal estimate and applications." Annals of Mathematics (2015): 1139-1208.

6. Cao, Junyan, and Mihai Păun. "Kodaira dimension of algebraic fiber spaces over abelian varieties." Inventiones mathematicae 207.1 (2017): 345-387.

**Schedule:**

June
20^{th} -July 20^{th}, 2019,

Monday – Friday, Morning: 9-10 am, 10:30-11:30 am; Afternoon: 2-3 pm, 3:30-4:30 pm