G E O M E T R Y    I I ,    S P R I N G   2 0 2 0

Course Information

• Ceiba
• Lectures: Wednesday, 10:20 ~ 11:30, and Friday 10:20 ~ 11:30; at Astro-Math 101
• Office hours: 14:00 ~ 15:00 every Tuesday; at Astro-Math 458
• Grading scheme:
  • Homework 40%
         You have two jokers: the lowest two grades will be discarded.
  • Midterm 30%
  • Final 30%
• Graders: 李宸寬 and 郭茗崴
• Course prerequisite:
  • (undergraduate) analysis and algebra
  • geometry of surfaces in the 3-space
  • manifold, differential form
• Main Topics:
  • minimal surface
  • vector bundle: connection, curvature and characteristic class
  • basic theory for complex manifold
• Textbooks:
  • [L] H. Blaine, Jr. Lawson, Lectures on minimal submanifolds. Vol. I. mathscinet
  • [S] Richard M. Schoen, Topic class on minimal surfaces. (notes taken by Xin Zhou)
  • [M] Shigeyuki Morita, Geometry of differential forms. mathscinet
  • [K] James Morrow and Kunihiko Kodaira, Complex manifolds. mathscinet

Lecture summaries and references

• (Week 1)  Review on the geometry of surfaces. First variational formula.  note
• (Week 2)  Weierstrass representation. Simons identity.  note
• (Week 3)  monotonicity formula, volume minimizing property of entire graph, second variational formula, stability.  note
• (Week 4)  isoperimetric inequality.  note
• (Week 5)  some phenomena in higher codimensions.  note
• (Week 6)  differential forms, Stokes theorem, de Rham cohomology.  note
• (Week 7)  vector bundles.  note
• (Week 8)  connection, curvature.  note
• (Week 9)  Pontryagin class and Chern class.  note
• (Week 10)  Euler class.  note
• (Week 11)  introduction to complex manifold.  note
• (Week 12)  some examples of complex manifolds.  note
• (Week 13)  Hermitian metric, Chern connection, Kähler metric.  note
• (Week 14)  curvature of Kähler metric, Ricci curvature and canonical bundle.  note
• (Week 15)  Hodge theory for Kähler manifolds.  note
                    Kodaira Bochner formula, Kodaira vanishing theorem.  note

Homework

• Homework 01: due March 13.
• Homework 02: due March 20.
• Homework 03: due March 27.
• Homework 04: due April 10.
• Homework 05: due April 17.
• Homework 06: due April 24.
• Take-home midterm: due May 1.
• Homework 07: due May 8.
• Homework 08: due May 15.
• Homework 09: due May 22.
• Homework 10: due May 29.
• Homework 11: due June 5.
• Homework 12: due June 12.
• Take-home final: due June 26.

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