Higher Dimensional Geometry (2015 Spring)  

This is the first semester of a one year class on higher dimensional geometry. 

We aim to discuss various aspects of birational geometry in higher dimension with the minimal model program (MMP) in the central position. In the first semester, the materials include vanishing theorems, bend and break, singularity theory, multiplier ideals/adjoint linear system with focus on Fujita conjecture, the fundamental MMP theorems like Cone theorems and Contraction theorems, Kollár injectivity Theorem and so on.

The plan is after this semester’s survey we can finish the preparation and in the next semester we will move into more advanced topics including the proof of the finite generation of pluricanonical rings. 

Prerequisite

Being familiar with basic language on varieties, sheaves, cohomology, linear system etc. Basic surface theory is also assumed. A solid knowledge of Hartshorne's book will be enough. 

Reference

[Laz04] Lazarsfeld, Robert; Positivity in algebraic geometry I, II.

[Kol86] Kollár, János; Higher direct image I, II. 

[Kol97] Kollár, János; Singularity of Pairs. 

[KM98] Kollár, János; Mori, Shigefumi; Birational geometry of algebraic varieties.  Section 1-3. 

[Kol08] Kollár, János; Exercises in the birational geometry of algebraic varieties. arXiv:0809.2579. 

[PS07] Peters, Chris; Steenbrink, Joseph; Mixed Hodge Structures. 

Classroom/Schedule

理教 412 Tuesday 10:00-11:50 and Thursday (Odd weeks) 8:00-9:50.   

Syllabus 

Part 1 Basics on minimal model program ([KM98] Chapter 2 and 3) 

3.3: Ampleness criterion and Cones

3.5: Vanishing theorems

3.10: Types of singularities in MMP

3.17: MMP for smooth surfaces

3.19: No Class 

3.24: No Class (Tianyuan workshop in Chinese Academy).

3.31: Running MMP (1) — base point free theorem 

4.2:  Running MMP (2) — non-vanishing theorem and cone theorem 

4.7: Running MMP (3) — rationality theorem 

4.14: Running MMP (4) — MMP process

4.16: An application: KSBA moduli theory

4.21: Discussion on Exercises

4.28: Discussion on Exercises

Part 2 Semipositivity Theorems

4.30: Basic Hodge theory

5.5: Variation of Hodge structure ([PS07], Chapter 10) 

5.12: Degeneration of Hodge structure I: one parameter semistable family ([Kol86], Theorem 2.6; [PS07], Chapter 11)

5.14: Degeneration of Hodge structure II: general case ([Kol86], Theorem 2.6; [PS07], Chapter 11)

5.19: Degeneration of Hodge structure III: Mixed Hodge Structure ([PS07], Chapter 11)

5.26: Injectivity Theorem ([Kol86]).

5.28: No Class

6.2: Introduction to rationally connected varieties (given by Zhiyu Tian)

6.9 and 11: No Class (Morgan Brown’s lecture on Berkovich spaces).

6.16: Semipositivity of push-forwards of pluricanonical rings

To be continued in 2015 Fall.  

Exercise

I put all homework here.  

Final

Final is here.