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Regular version of the site
2017/2018

Complex Algebraic Geometry

Category 'Best Course for Career Development'
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type: Optional course (faculty)
When: 3, 4 module
Language: English
ECTS credits: 5
Contact hours: 76

Course Syllabus

Abstract

Complex algebraic geometry was developed by W. V. D. Hodge in 1930-es and was given its modern form by A. Weil and S.-S. Chern in 1940-ies and 1950-ies. This is a discipline allowing one to get results of classical algebraic geometry using basic observations of analysis, differential geometry and topology instead of complicated algebraic computations. As an additional bonus, the methods of Hodge theory can be used to study non-algebraic objects, such as general Kahler manifolds and more complicated geometric objects
Learning Objectives

Learning Objectives

  • The objective is working knowledge of complex algebraic geometry and the ability to solve problems related to this subject.
Expected Learning Outcomes

Expected Learning Outcomes

  • After learning the course, the student should be able to solve basic problems of complex algebraic geometry using Hodge theory and complex analysis
Course Contents

Course Contents

  • Elliptic equations on manifolds and Hodge theory.
  • Kahler manifolds and algebraic manifolds
  • Hodge theory on Riemannian and Kahler manifolds
  • Poincare-Dolbeault-Grothendieck lemma and its applications
  • Line bundles, Chern connection and its curvature, $dd^c$-lemma and its applications
  • Kodaira-Nakano vanishing theorem
  • Kodaira embedding theorem
Assessment Elements

Assessment Elements

  • non-blocking Cumulative grade
    cumulative grade is proportional to the number of solved problem sheets
  • non-blocking Final exam
Interim Assessment

Interim Assessment

  • Interim assessment (4 module)
    0.3 * Cumulative grade + 0.7 * Final exam
Bibliography

Bibliography

Recommended Core Bibliography

  • Jean-Pierre Demailly. (2007). Complex analytic and differential geometry. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.495EA558

Recommended Additional Bibliography

  • Claire Voisin. (n.d.). Hodge theory and the topology of compact Kähler and complex projective manifolds. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.36CDD0CE
  • Voisin, C., & Schneps, L. (2003). Hodge Theory and Complex Algebraic Geometry II: Volume 2. Cambridge: Cambridge University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=120395