Course on variation of Hodge structures, genus 0 mirror symmetry, and beyond

Practical infos

• Meetings: Monday 10:10am-12:00pm (weekly), Wednesday 1:00pm-2:50pm (bi-weekly, odd-numbered weeks), 三教501

• Evaluation: Final take-home exam

• Course progress.

• Course materials will be posted on https://course.pku.edu.cn

Syllabus

This course begins with an introduction to Variation of Hodge Structures (VHS) in complex algebraic geometry, then explores enumerative applications via mirror symmetry. Later we will move to Cohomological Field Theories (CohFTs) and Givental-Teleman’s quantization.

Part 1: Variation of Hodge Structures

This part establishes the geometric foundation, emphasizing the deformation of complex structures, period integrals, and the Gauss-Manin connection from a complex analytic and algebraic geometry perspective.

• Topics: Hodge decomposition, primitive cohomology, period domains and period maps, Griffiths transversality, the Gauss-Manin connection, local systems, and an introduction to degenerations (mixed Hodge structures).
• Primary References:
  • C. Voisin, Hodge Theory and Complex Algebraic Geometry I & II.
  • J. Carlson, S. Müller-Stach, C. Peters, Period Mappings and Period Domains.
  • P. Griffiths, Topics in Transcendental Algebraic Geometry.

Part 2: Genus 0 Mirror Symmetry for the Quintic

Applying the tools from Part 1 to Calabi-Yau threefolds, specifically focusing on the quintic threefold to demonstrate the predictive power of mirror symmetry.

• Topics: Calabi-Yau manifolds, the A-model and B-model for the quintic threefold, a quick review of Gromov-Witten invariants (genus 0), the mirror map, quantum cohomology, and the calculation of rational curves on the quintic.
• Primary References:
  • D. Cox, S. Katz, Mirror Symmetry and Algebraic Geometry.

Part 3: Cohomological Field Theories and Givental’s Quantization

A modern generalization of the structures seen in mirror symmetry to a broader algebraic framework.

• Topics: Definition and examples of Cohomological Field Theories (CohFTs), topological field theories (2D TQFTs), Frobenius manifolds, the Witten-Kontsevich theorem (briefly), and Givental’s quantization formalism (action of the symplectic loop group on CohFTs).
• Primary References:
  • R. Pandharipande, Cohomological field theory calculations (ICM 2018).
  • Y. P. Lee, Notes on Axiomatic Gromov-Witten Theory and Formalism.
  • A. Givental, Semisimple Frobenius structures at higher genus, IMRN (2001).
  • C. Teleman, The structure of 2D semi-simple field theories, Inventiones Mathematicae (2012).