CplxGeom [PhD Math UniFi 2017]

Part 1. (march-may)

Lecture 1. (Friday March 03, 2017, 09:00-11:00, Aula Tricerri.) Analytic functions of several complex variables. [Voi1, Chapter 1]

Lecture 2. (Thursday March 09, 2017, 17:00-19:00, Aula 7.) Categories; Abelian categories. [Voi1, Chapter 4]

Lecture 3. (Thursday March 16, 2017, 17:00-19:00, Aula 7) Derived functors; presheaves and sheaves. [Voi1, Chapter 4]

Lecture 4. (Thursday March 23, 2017, 17:00-19:00, Aula 7.) Cohomology of sheaves. [Voi1, Chapter 4]

Lecture 5. (Monday April 03, 2017, 08:30-10:30, Aula Tricerri.) G-structures, almost-complex structures, differential geometry of complex manifolds. [Voi1, Chapter 2], [BHPV, Chapter VI]

Lecture 6. (Tuesday April 11, 2017, 08:30-10:30, Aula CDM-214.) Examples of complex manifolds; Fr\"olicher spectral sequence. [Voi1, Chapter 8]

Lecture 7. (Thursday April 20, 2017, 09:00-11:00, Aula CDM-227.) Leray-Serre spectral sequence; Riemannian and Hermitian manifolds. [Voi1, Chapter 5]

Lecture 8. (Thursday May 11, 2017, 09:00-11:00, Aula Tricerri.) Elliptic Hodge theory for Riemannian and Hermitian manifolds. [Voi1, Chapter 5], [Ara, Chapter 8], [BDIP, Chapter I.3]

Lecture 9. (Friday May 19, 2017, 09:00-12:00, Aula 1.) Hodge theory for compact Kaehler manifolds. [Voi1, Chapters 3-6]

Lecture 10. (Thursday May 25, 2017, 09:00-12:00, Aula 1.) Hodge conjecture. [Voi1, Chapter 7, Chapter 11], [Lew, Chapters 5-6-7]

Part 2. (october-november)

Lecture 11. Yau's estimates for non-homogeneous Monge-Ampère equations, I.

Lecture 12. Yau's estimates for non-homogeneous Monge-Ampère equations, II.

Lecture 13. Yau's estimates for non-homogeneous Monge-Ampère equations, III.

Lecture 14. Yau's estimates for non-homogeneous Monge-Ampère equations, IIII.

Seminars.

Francesco Pediconi, "Deformazioni di strutture complesse". (Tuesday September 12, 2017, xx:00-xx:00, Aula X).

Pietro Gheri, "TBA". (XX XXXX XX, 2017, xx:00-xx:00, Aula X).

Luca Sodomaco, "TBA". (XX XXXX XX, 2017, xx:00-xx:00, Aula X).

Possible seminars.

Hodge theory of compact complex surfaces and Lamari criterion. [BHPV, Chapter IV]

Elliptic Hodge theory. [BDIP, Chapter I.3], [Voi1, Chapter 5]

Relative Hodge theory, and deformations of complex structures. [Voi1, Chapters 9-10]

Vuletescu and Voisin counterexamples to Hodge conjecture for non-projective Kaehler manifolds. [Lew]

Serre's Algebraic Geometry and Analytic Geometry. [Ara, Chapters 15-16]

Kodaira embedding and balanced metric. [Voi, Chapter 7]

...

References.

  • notes
  • [Voi1] Voisin, "Hodge Theory and Complex Algebraic Geometry 1", Cambridge, 2002.
  • [BHPV] Barth, Hulek, Peters, Van de Ven, "Compact Complex Surfaces", Second Enlarged Edition, Springer, 2004.
  • [BDIP] Bertin, Demailly, Illusie, Peters, "Introduction to Hodge theory", AMS-SMF, 2002.
  • [CEGL] Cattani, El Zein, Griffiths, Le (eds.), "Hodge Theory", Princeton, 2014.
  • [Lew] Lewis, "A Survey of the Hodge Conjecture", Second Edition, AMS, 1997.
  • [Wel] Wells, "Differential Analysis on Complex Manifolds", Springer, 2008.
  • [Ara] Arapura, "Algebraic Geometry over the Complex Numbers", 2012.
  • [Hor] Hormander, "An Introduction to Complex Analysis in Several Variables", North-Holland,1990.
  • [GR] Gunning, Rossi, "Analytic Functions of Several Complex Variables", Prentice-Hall, 1965.