2017 Advanced topics in Geometry F1

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Academic unit or major
Graduate major in Mathematics
Honda Nobuhiro 
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Media-enhanced courses
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Course description and aims

We explain basic materials in the theory of complex manifolds. We introduce basic examples of complex manifolds, and then introduce the concept of harmonic differential forms. These establish basic results in geometry. For complex manifolds, the concept of Dolbeault groups naturally makes sense, and we can represent their elements by harmonic forms. If the manifolds are moreover Kaehler, harmonic forms as a real manifold and a complex manifold agree, and it induces a constraint on the topology of such manifolds.

Student learning outcomes

・to be familiar with basic examples of complex manifolds and basic concept such as first Chern class and blow-up,
・to understand cohomology and harmonic forms
・to know (Hodge) decomposition of differential forms on a closed real manifold and compact complex manifolds


complex manifold, complex projective variety,

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Standard lecture course

Course schedule/Required learning

  Course schedule Required learning
Class 1 holomorphic functions, complex manifolds, complex projective space, algebraic variety Details will be provided during each class session.
Class 2 Weierstrass preparation theorem, intersection of analytic cylcles
Class 3 harmonic forms and the Hodge decomposition of differential forms
Class 4 harmonic forms and Hodge decomposition (the case of complex manifolds)
Class 5 Dolbeault cohomology, Dolbeault's theorem, Hodge decomposition on compact Kaehler manifolds
Class 6 holomorphic line bundle, chern class, positivity, vanishing theorem
Class 7 computations of Dolbeault cohomology
Class 8 blowup and blow down, birational transformation



Reference books, course materials, etc.

P. Griffiths, J. Harris, "Principles of Algebraic Geometry", Wiley-Interscience

Assessment criteria and methods

Assignments (100%).

Related courses

  • MTH.B505 : Advanced topics in Geometry E1
  • MTH.E532 : Special lectures on advanced topics in Mathematics H
  • MTH.C301 : Complex Analysis I

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Basic knowledge on geometry and complex analysis is required. MTH.B505 is assumed to be taken.

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