Complex Manifolds
A.Y. 2018/2019
Learning objectives
Learn some basic tools and methods in the theory of complex manifolds.
Expected learning outcomes
Students will learn some basic tools and results in the theory of complex manifolds including vector bundles, sheaves and their cohomology. Special attention will be given to 1 dimensional tori and the adjunction formula.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
· Complex manifolds: the holomorphic tangent space and differential forms of type (p,q).
· Submanifolds and embeddings.
· Complex tori of dimension 1, the Weierstrass p-function.
· Addition on elliptic curves, moduli of elliptic curves.
· Holomorphic vector bundles, examples and the canonical bundle.
· The adjunction formula and the line bundle defined by a codimension one submanifold.
· Sheaves, stalks, homomorphisms of sheaves and the sheaf generated by a presheaf.
· Soft sheaves, cohomology of sheaves and the long exact cohomology sequence.
· The abstract de Rham theorem, de Rham cohomology, algebraic topology and sheaves.
· Submanifolds and embeddings.
· Complex tori of dimension 1, the Weierstrass p-function.
· Addition on elliptic curves, moduli of elliptic curves.
· Holomorphic vector bundles, examples and the canonical bundle.
· The adjunction formula and the line bundle defined by a codimension one submanifold.
· Sheaves, stalks, homomorphisms of sheaves and the sheaf generated by a presheaf.
· Soft sheaves, cohomology of sheaves and the long exact cohomology sequence.
· The abstract de Rham theorem, de Rham cohomology, algebraic topology and sheaves.
Professor(s)