Lecture Description
The theme of these lectures will be /Period Integral Calculus". Period integrals are
analytic objects that one can use to study deformations of algebraic varieties. The goal
is to nd simple algebraic/combinatorial ways to characterize them, and then use their
characterizations to answer questions about algebraic varieties. These questions include
computing deformation invariants (like Gromov-Witten invariants), local monodromy of
singularities, period mappings, and Abel-Jacobi maps. Here is a tentative outline of topics
to be covered:
(1) Motivations: examples from mirror symmetry, Hodge theory, D-modules, special
functions
(2) Calabi-Yau bundles, classication, and their connection to Poincare residues
(3) Period integrals, period sheaves, and their dierential systems
(4) The theory of tautological systems; old and new examples
(5) Brief overview of representations of complex reductive groups, Borel-Weil theory
(6) Homogeneous spaces, and descriptions of their tautological systems
(7) If time permits: Holonomic rank, explicit solutions to tautological systems
References for covered material
(1) Period Integrals and Tautological Systems, by B.Lian, R.Song & S.T. Yau, arXiv
1105.2984, to appear in Journ. EMS.
(2) Period Integrals of CY and General Type Complete Intersections, by B.Lian, &
S.T. Yau, arXiv 1105.4872, to appear in Invent. Math.
(3) Picard-Fuchs Equations for Relative Periods and Abel-Jacobi Map for Calabi-Yau
Hypersurfaces, by B. Lian, S. Li & S.T. Yau, arXiv 0910.4215, to appear in Am.
Journ. Math.
(4) Additional references will be provided, and lecture notes will be available.
References for background material:
(1) Principle of algebraic geometry, by P. Griths and J. Harris. Knowledge of de-
nitions and theorems in Chapters 0 and 1 will be assumed. Knowledge of Chapter 2
would help, but not essential. Or
(2) Hodge theory & complex algebraic geometry Volume I, by C. Voisin. Knowledge