Calabi-Yau Manifolds with Small Hodge Numbers
Student No.:50
Time:Mon/Wed 13:00-14:50, 2015-9-2~2015-9-28
Instructor:Philip Candelas  [Mathematical Institute,Oxford]
Place:Conference Room 4, Floor 2, Jin Chun Yuan West Building
Starting Date:2015-9-2
Ending Date:2015-9-28





CY manifolds, that is compact Kähler manifolds with vanishing first Chern class, are of interest to mathematicians and also to physicists owing to their role in string theory.

While there is no proof that the Hodge numbers of CY manifolds are bounded, All known examples satisfy |h^{1,1} +h^{2,1}| <= 502. This bound gives some idea of the topological complexity of the "generic” CY manifold. This course will review the construction of CY manifolds with small Hodge numbers. In some sense, these are among the simplest CY manifolds. I will also indicate the application of some of these manifolds in string theory.


1)     Review of CY manifolds.

2)     Simple examples of CY manifolds, CICY’s.

3)     Elements of toric geometry, reflexive polytopes and the Kreuzer-Skarke list.

4)     Quotients of CICY’s and simple quotients from the KS list.

Manifolds that lead to interesting string theory models.





Some knowledge of complex geometry. A knowledge of algebraic geometry and toric geometry, while an advantage, is not essential.





For complex differential geometry: any of the standard texts. 

For CY manifolds: “Calabi-Yau Manifolds and Related Geometries” by Gross, Huybrechts and Joyce, Springer 2003.

For applications to string theory: “Supersting Theory” vol I, by Green, Schwarz and Witten, Cambridge 1987.