Misha Verbitsky

Instituto Nacional de Matemática Pura e Aplicada

Hyperkähler manifolds

13:30-15:00, Mondays, 17:00-18:30 Fridays, Sala 236, January-February 2023

Slides:

• Lecture 1: Holonomy (March 13).
• Lecture 2: Levi-Civita connection (March 17).
• Lecture 3: Frobenius theorem (March 20).
• Lecture 4: Riemann-Hilbert correspondence (March 24).
• Lecture 5: de Rham theorem about decomposition of holonomy (March 27).
• Lecture 6: Berger list of irreducible holonomy groups (March 31).
• Lecture 7: Hyperkahler manifolds (April 3).
• Lecture 8: Bogomolov decomposition theorem (April 7).
• Lecture 9: Clifford algebras (April 10).
• Lecture 10: Spin structures (April 14).
• Lecture 11: Dirac operator (April 17).
• Lecture 12: Bochner vanishing (April 21).
• Lecture 13: Symplectic reduction (April 24, April 27).
• Lecture 14: Geometric invariant theory (May 1)
• Lecture 15: Hyperkahler reduction (May 5)
• Lecture 16: Complex subvarieties and the proof of Newlander-Nirenberg (May 8)
• Lecture 17: Trianalytic subvarieties (May 8)
• Lecture 18: Twistor spaces and equivariant bundles (May 12, May 15)
• Lecture 19: Kobayashi-Hitchin correspondence (May 18)
• Lecture 20: Yang-Mills connections (May 22)
• Lecture 21: Hyperholomorphic bundles (May 26)
• Lecture 22: Hodge theory and supersymmetry on hyperkahler manifolds (May 29)
• Lecture 23: The period map and the BBF form (June 23)
• Lecture 24: The Fujiki formula and its applications (June 26)
• Lecture 25: Lie algebra acting on cohomology (June 30)

Assignments

• Assignment 1: Kähler structures (March 13).
• Assignment 2: Holonomy and torsion (March 20).
• Assignment 3: Riemannian holonomy (March 27).
• Assignment 4: Quaternionic Hermitian structures (April 3).
• Assignment 5: Calabi-Yau manifolds and invariants theory (April 10).
• Assignment 6: Clifford algebras (April 17).
• Assignment 7: Symplectic reduction (April 24).
• Assignment 8: Geometric invariant theory (May 1).
• Assignment 9: Moore and Dieudonne determinants (May 8).
• Assignment 10: Calibrations and Wirtenger's inequality (May 15).
• Assignment 11: Holomorphic vector bundles (May 22).
• Assignment 12: Hodge theory and supersymmetry (May 29).
• Assignment 13: Verma modules (June 5).
• Assignment 14: Hodge structures (June 12).

Miscellanea

• Class assignments results (updated)
• Exam problems
• Personal problem sets, obtained randomly with this program.
• Final results (pending)