HSE, 2020 (Fall): Complex algebraic geometry

Complex algebraic geometry and Hodge theory

Announcement and literature

Slides:

Lecture 1: Complex manifolds (September 23, 2020).
Lecture 2: Cauchy formula (September 26, 2020).
Lecture 3: Frobenius theorem (September 30, 2020).
Lecture 4: Another proof of Frobenius theorem; real structures (October 3, 2020).
Lecture 5: Real analytic manifolds (October 7, 2020).
Lecture 6: Proof of Newlander-Nirenberg theorem for real analytic manifolds (October 10, 2020).
Lecture 7: Homogeneous spaces and Kähler metrics (October 14, 2020).
Lecture 8: Levi-Civita connection (October 17, 2020).
Lecture 9: Symplectic connections (October 21, 2020).
Lecture 10: Bismut connections (October 24, 2020).
Lecture 11: Hodge $*$ operator (October 27, 2020).
Lecture 12: Supersymmetry on Kähler manifolds. (October 31, 2020).
Lecture 13: Cohomology and a circle action. (November 4, 2020).
Lecture 14: Dolbeault cohomology of a torus. (November 7, 2020).
Lecture 15: Poincaré-Dolbeault-Grothendieck lemma. (November 11, 2020).
Lecture 16: Riemann-Hilbert correspondence (November 14, 2020).

Handouts

Assignment 1: Complex structure operators (September 26, 2020)
Assignment 2: Almost complex structures (October 3, 2020)
Assignment 3: Real analytic manifold and integrability (October 10, 2020)
Assignment 4: homogeneous spaces (October 17, 2020)
Assignment 5: connection and curvature (October 24, 2020)
Assignment 6: Lie superalgebras (October 31, 2020)
Assignment 7: Stone-Weierstrass approximation theorem (November 7, 2020)
Assignment 8: Plurisubharmonic functions (November 14, 2020)

Miscellanea: test problems, exam, etc.