Geometry arising from hyperplane arrangements
This course explores how projective hyperplane arrangements provide a bridge among moduli theory, birational geometry, and complex differential geometry. It is organized into two parts:
1) The first part introduces GIT, Baily–Borel, and Looijenga compactifications of moduli spaces which admits a complex ball quotient description. Then I will compare the singularities arising in the GIT and Baily–Borel compactifications from the perspective of the minimal model program.
2) The second part develops Dunkl systems on certain hyperplane arrangement complements and introduces Thurston’s notion of an (X,G)-cone-manifold. Then I will show that the metric completions of the relevant arrangement complements carry natural cone-manifold structures.
1) The first part introduces GIT, Baily–Borel, and Looijenga compactifications of moduli spaces which admits a complex ball quotient description. Then I will compare the singularities arising in the GIT and Baily–Borel compactifications from the perspective of the minimal model program.
2) The second part develops Dunkl systems on certain hyperplane arrangement complements and introduces Thurston’s notion of an (X,G)-cone-manifold. Then I will show that the metric completions of the relevant arrangement complements carry natural cone-manifold structures.
讲师
日期
2026年09月15日 至 2027年01月12日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周二 | 13:30 - 16:05 | Shuimo-LG28 | - | - | - |
修课要求
Basic knowledge of algebraic and complex geometry.
课程大纲
(1) Hyperplane arrangements: two guiding problems
(2) GIT, Baily–Borel, and Looijenga compactifications of ball quotient moduli spaces
(3) Log pairs and MMP singularities: from logarithmic exponents to discrepancies
(4) Affine and projective structures, Dunkl systems, and Thurston’s (X,G)-cone-manifolds
(5) Metric completions in the elliptic, parabolic, and hyperbolic cases, with the Deligne–Mostow–Thurston example
(2) GIT, Baily–Borel, and Looijenga compactifications of ball quotient moduli spaces
(3) Log pairs and MMP singularities: from logarithmic exponents to discrepancies
(4) Affine and projective structures, Dunkl systems, and Thurston’s (X,G)-cone-manifolds
(5) Metric completions in the elliptic, parabolic, and hyperbolic cases, with the Deligne–Mostow–Thurston example
参考资料
[1] Dali Shen, The MMP singularities of GIT versus Baily-Borel compactifications for the ball quotient case.
[2] Dali Shen, Kahler cone-manifolds arising from a projective arrangement.
[3] E. Looijenga, Compactifications defined by arrangements. I. The ball quotient case.
[4] W. Couwenberg, G. Heckman, and E. Looijenga, Geometric structures on the complement of a projective arrangement.
[5] W. Thurston, Shapes of polyhedra and triangulations of the sphere.
[6] J. Kollár and S. Mori, Birational geometry of algebraic varieties.
[2] Dali Shen, Kahler cone-manifolds arising from a projective arrangement.
[3] E. Looijenga, Compactifications defined by arrangements. I. The ball quotient case.
[4] W. Couwenberg, G. Heckman, and E. Looijenga, Geometric structures on the complement of a projective arrangement.
[5] W. Thurston, Shapes of polyhedra and triangulations of the sphere.
[6] J. Kollár and S. Mori, Birational geometry of algebraic varieties.
视频公开
不公开
笔记公开
不公开
语言
中文
, 英文
讲师介绍
Dali Shen is an assistant professor at BIMSA currently. His research is focused on algebraic geometry and complex geometry. He obtained his PhD from Utrecht University. Before joining BIMSA, he held postdoc positions at IMPA and TIFR.