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CODE 109050
ACADEMIC YEAR 2024/2025
CREDITS
SCIENTIFIC DISCIPLINARY SECTOR MAT/03
LANGUAGE Italian (English on demand)
TEACHING LOCATION
  • GENOVA
SEMESTER 2° Semester

OVERVIEW

The course is an introduction to the theory of complex varieties, vector bundles on them and the Hodge theory of (compact) complex varieties. In particular, we will be discussing about complex structures and differential calculus on such varieties, the sheaf cohomology and the de Rham cohomology theory, Dolbeault's theorem and the basic theory of Kaehler varieties. The end goal is to prove the Hodge decomposition theorem for compact Kaehler varieties.

AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to present an elementary introduction to the concepts and methods of modern Complex Geometry.

AIMS AND LEARNING OUTCOMES

At the end of the course the student is expected to know the basic theory of complex varieties and vector bundles on them, will be proficient in using cohomological/differential techniques to study these complex varieties, and have a good grasp on Kaehler varieties and their Hodge theory. The course is conceived as a starting point for a research carreer in one of the deepest and oldest areas of mathematics, that has attracted an immense amount of research  in the last 170 years.

PREREQUISITES

It is recommended to have prior knowledge of a course in complex analysis in one variable and Istituzioni di Geometria Superiore course. Some basic concepts from Istituzioni di Geometria Superiore 2 will be necessary during this course.

TEACHING METHODS

Traditional.

SYLLABUS/CONTENT

  • Introduction to complex varieties and vector bundles.
  • Differential forms on complex varieties.
  • Sheaf cohomology and de Rham cohomology.
  • Complex tori.
  • Kaehler varieties and their Hodge theory.

RECOMMENDED READING/BIBLIOGRAPHY

  • P. Griffiths e J. Harris: "Principles of Algebraic Geometry" , Wiley Online Books
  • D. Huybrechts: "Complex Geometry", Universitext, Springer
  • R. Lazarsfeld: "MAT 545 --- Complex Geometry", available at: https://www.math.stonybrook.edu/Courses/MAT545/201308/MAT545F13.pdf
  • C. Schnell: "Notes on complex manifolds", available at: http://www.math.sunysb.edu/~cschnell/pdf/notes/complex-manifolds.pdf
  • C. Voisin: "Hodge Theory and Complex Algebraic Geometry I", Cambridge University Press
  • C. Voisin: "Hodge Theory and Complex Algebraic Geometry II", Cambridge University Press

TEACHERS AND EXAM BOARD

LESSONS

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

Oral.

It is recommended for the students with DSA certificate, with disabilities or other special educational needs, to contact the professor at the beginning of the course and agree on the teaching and exam methods together with the compensating tools, that are suitable for each indiviual learning methods. 

ASSESSMENT METHODS

The oral exam will cosist of presenting a recent research article connected to the material of the course. The goal is to measure the knowledge of the students on the matters of the course, the student abilities to read and understand high-level research, based on the accumulated prior knowledge, and the student capabilities to explain and teach difficult research at the level of their peers. Moreover, the students will be encouraged to explain through seminars their solutions to some of the problems/exercises that will appear during the semester, thus measuring the student's skills in solving problems related to a complex material.