COMPLEX GEOMETRY Course notes
COMPLEX GEOMETRY Course notes
COMPLEX GEOMETRY Course notes
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TABLE OF CONTENTS<br />
1 <strong>COMPLEX</strong> ANALYSIS 1<br />
1.1 Complex Analysis in one variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1<br />
1.2 Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4<br />
1.3 Complex Analysis in several variables . . . . . . . . . . . . . . . . . . . . . . . . . . 6<br />
2 RIEMANN SURFACES 9<br />
2.1 Complex manifolds, Lie groups and Riemann surfaces . . . . . . . . . . . . . . . . 9<br />
2.2 Holomorphic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11<br />
2.3 Meromorphic functions and differentials . . . . . . . . . . . . . . . . . . . . . . . . . 12<br />
2.4 Weierstrass P -function on C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15<br />
2.5 Dimension on Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16<br />
2.6 Covering spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18<br />
2.7 The Riemann surface of an algebraic function . . . . . . . . . . . . . . . . . . . . . 20<br />
2.8 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21<br />
2.9 Topology of Riemann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24<br />
2.10 Product structures on ⊕ i Hi dR<br />
(Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27<br />
2.11 Questions about (compact) Riemann surfaces . . . . . . . . . . . . . . . . . . . . . 31<br />
2.12 Harmonic differentials and Hodge decompositions . . . . . . . . . . . . . . . . . . . 32<br />
2.13 Analysis on the Hilberts space of differentials . . . . . . . . . . . . . . . . . . . . . 34<br />
2.14 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36<br />
2.15 Proof of Weyl’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37<br />
2.16 Riemann Extension Theorem and Dirichlet Principle . . . . . . . . . . . . . . . . 39<br />
2.17 Projective model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40<br />
2.18 Arithmetic nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41<br />
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