Fachbereich Mathematik 
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Variation of Hodge Structure and tt* Geometry, University of Hamburg,  Winter Term 2017/2018

This is an advanced graduate course for master and phd students as well as researchers who are interested in complex algebraic-geometric structures which originate from the physics of supersymmetric theories. The course will discuss the variation of Hodge structure and the associated special Kähler geometry, it will then proceed to a generalization of special Kähler geometry known as the topological anti-topological fusion geometry or tt* geometry.

A good knowledge of complex geometry and familiarity with algebraic geometry are required. A background in physics of supersymmetric theories is beneficial for the course but not necessary.

Topics of the course:

  1. (1)  Intuition for variation of Hodge structure from elliptic curves

  2. (2)  Variation of Hodge structure

  3. (3)  Special Geometry

  4. (4)  ttgeometry

  5. (5)  Advanced topics, e. g. quantum cohomology, twistorial structure of ttequations, algebraic structure of ttequations, relation to Hitchin systems.


The course takes place on:
  • Thursdays, 12:15 - 13:45 in H3 (Geomatikum)

Sessions this term:

Room, Time
I-VHS for elliptic curves, notes
H3, 12:15
I-VHS for elliptic curves, notes
H3, 12:15
I-VHS for elliptic curves, notes H2, 14:15!
Exercise session, discussion sheet 1 Please contact Martin Vogrin regarding the dates
Geo 432, 12:15
I-VHS for elliptic curves, notes (page 24 corrected) H3, 12:15
II-Variation of Hodge Structure, notes H3, 12:00
Exercise session, discussion sheet 2 Geo 432, 12:15
II-Variation of Hodge Structure, notes H3, 12:05
Will take place on another date!
other date tbd
Exercise session, discussion sheet 3, fill poll  Geo 432, 12:15
II-Variation of Hodge Structure, notes H3, 12:05
III- Special Geometry, notes H3, 12:05
III- Special Geometry, notes H3, 12:05
III- Special Geometry, by Florian Beck on Freed's paper H3, 12:05
IV-tt* Geometry, notes H3, 12:05
19.01.2018 Exercise session, discussion sheet 4
Geo 233, 12:15
IV-tt* Geometry, notes , see also section 3 of this paper.
H3, 12:05
V-Advanced topics: Semi-Infinite Hodge Structures, by Martin Vogrin based on Barannikov's paper
H3, 12:05
02.02.2018 Exercise session, discussion sheet 5
Geo 233, 12:15


A. Ceresole, R. D’Auria, S. Ferrara, W. Lerche, J. Louis, and T. Regge. Picard-Fuchs equations, special geometry and target space duality. In Mirror symmetry, II, volume 1 of AMS/IP Stud. Adv. Math., pages 281–353. Amer. Math. Soc., Providence, RI, 1997.

A. Cox and S. Katz. Mirror symmetry and algebraic geometry. 2000.

James Carlson, Stefan Müller-Stach, and Chris Peters.
Period mappings and period domains, volume 85 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2003.

Sergio Cecotti and Cumrun Vafa. Topological–anti-topological fusion. Nuclear Phys. B, 367(2):359–461, 1991.

B. Dubrovin. Geometry and integrability of topological-antitopological fusion. Comm. Math. Phys., 152(3):539–564, 1993.

Daniel S. Freed. Special Kahler manifolds. Commun. Math. Phys., 203:31–52, 1999.

Martin A. Guest. From quantum cohomology to integrable systems, volume 15 of Oxford Grad- uate Texts in Mathematics. Oxford University Press, Oxford, 2008.

Claire Voisin. Hodge theory and complex algebraic geometry. I, volume 76 of Cambridge Stud- ies in Advanced Mathematics. Cambridge University Press, Cambridge, english edition, 2007. Translated from the French by Leila Schneps.

Course description

Exercise sheets
Contact Martin Vogrin regarding the exact solution discussion times and location
Sheet 1
Sheet 2
Sheet 3
Sheet 4
Sheet 5

Research projects

  Seitenanfang  Impress 2018-01-26, Murad Alim