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Riemann Surfaces Riemann Surfaces Joachim Wehler Ludwig-Maximilians-Universitat¨ Munchen¨ Department of Mathematics Winter Term 2019/20 - Lecture Notes DRAFT, Release 2.38 2 c Joachim Wehler, 2019, 2020, 2021 I prepared these notes for the students of my lecture and the subsequent seminar on Riemann surfaces. The two courses took place during the winter term 2019/20 and the summer term 2020 at the mathematical department of LMU (Ludwig-Maximilians-Universitat)¨ at Munich. The lecture in class presented the content of these notes up to and including Section 10.1. The following chapters served as the basis for the seminar. Compared to the oral lecture in class and to the seminar talks these written notes contain some additional material. These lecture notes are permanently continued. Some undefined references in the actual release possibly refer to a forthcoming chapter of a later release. These lecture notes are based on the book Lectures on Riemann Surfaces by O. Forster [8]. It serves as textbook for both courses. These notes make no claim of any originality in the presentation of the material. Instead they are just a commentary of selected chapters from [8] - and as mostly the case, the commentary is longer than the original! In addition, Chapter 12 relies on some notes of an unpublished lecture by O. Forster on Kahler¨ manifolds. I thank all participants who pointed out errors during the lecture in class and typos in the notes. Please report any further errors or typos to [email protected] Release notes: • Release 2.38: Chapter 8. Proof of Proposition 8.7 corrrected. • Release 2.37: Chapter 11, Proposition 11.6, proof corrected. Minor revisions. • Release 2.36: Chapter 11, Remark 11.3, transition function corrected. • Release 2.35: Chapter 10, Remark 10.21 added, renumbering. • Release 2.34: Minor revisions. • Release 2.33: References updated, minor revisions. • Release 2.32: Chapter 15, Remark 15.17 and Remark 15.19 added, minor revisions, renumbering. • Release 2.31: Chapter 15, Section 15.2 Corollary 15.14 expanded, Remark 15.16 removed, renumbering. • Release 2.30: Chapter 15, Section 15.2 Proof of Proposition 15.12 corrected, Corollary 15.15 added, renumbering. • Release 2.29: Chapter 15, Section 15.2 Remark 15.16 added, renumbering. • Release 2.28: Chapter 15, Section 15.1 update. • Release 2.27: Chapter 15, Section 15.1 removed for update. • Release 2.26: Minor revision, correction of some typos. • Release 2.25: Chapter 9, Proposition 9.7 added, renumbering. Chapter 11, Remark 11.21 expanded. Chapter 15, Remark 15.23 expanded. Index updated. • Release 2.24: Chapter 10, Remark 10.25 added, renumbering. Chapter 11, Remark 11.21 expanded. Chapter 12, Remark 12.30 updated. Chapter 13 added. Chapter 15 added. 3 • Release 2.23: Chapter 12, Section 12.3. Remark 12.30 updated, proof of Theorem 12.32 simplified. • Release 2.22: Chapter 14 added. • Release 2.21: Chapter 12 completed, Section 12.3 corrected. • Release 2.20: Chapter 12, Section 12.3 removed. • Release 2.19: Chapter 10, Remark 10.14 added and subsequent renumbering. Chapter 12 completed. • Release 2.18: Chapter 12, Section 12.1 and 12.2 added. • Release 2.17: Chapter 11, Proof of Theorem 11.8 changed. • Release 2.16: Chapter 10, Proposition 10.13. Minor revision of the proof. • Release 2.15: Chapter 10, Section 10.2. Minor revisions and renumbering. • Release 2.14: Minor revisions and typos corrected. • Release 2.13: Chapter 11, Section 11.3. Minor revisions and typos corrected. • Release 2.12: Chapter 11 added. • Release 2.11: Chapter 10, Lemma 10.12 added. Minor revisions. • Release 2.10: Chapter 10 completed. • Release 2.9: Typos corrected. In Definition 6.10 indices changed. • Release 2.8: Chapter 4, Lemma 4.1 modified. Typos corrected. • Release 2.7: Chapter 4, Lemma 4.1 and Corollary 4.2 minor modifications. Index added. Typos corrected. • Release 2.6: Chapter 10, Section 10.1 added. • Release 2.5: List of results completed. • Release 2.4: Chapter 9 completed. • Release 2.3: Chapter 9, Section 9.1 added. • Release 2.2: Chapter 8 added. • Release 2.1: Chapter 7 completed. • Release 2.0: Chapter 7, Section 7.1, Proof of Corollary 7.6 expanded, Section 7.2 added. List of results continued. • Release 1.9: Chapter 1, Definition 1.2 expanded. Chapter 7, Proof of Lemma 7.4 expanded. • Release 1.8: Chapter 7, Section 7.1 added. List of results continued. • Release 1.7: Chapter 6 completed. Chapter 5, Remark 5.7 added. • Release 1.6: Chapter 6, Section 6.2 added. • Release 1.5: Chapter 6, Section 6.1 added. List of results created • Release 1.4: Chapter 4, Definition 4.18 corrected with respect to shrinking. • Release 1.3: Chapter 5 completed. • Release 1.2: Chapter 4 completed. • Release 1.1: Chapter 4, Section 4.2 added. • Release 1.0: Chapter 4, Section 4.1 created. • Release 0.9: Chapter 3, Remark 3.31 added. • Release 0.8: Chapter 3 renumbered and completed. • Release 0.7: Chapter 3, Section 3.2 added. • Release 0.6: Chapter 3, Section 3.1 created. • Release 0.5: Chapter 2 completed. • Release 0.4: Chapter 2, Section 2.2 added. 4 • Release 0.3: Chapter 2, Section 2.1 created. • Release 0.2: Chapter 1 completed. • Release 0.1: Chapter 1, Section 1.1 created. Contents Introduction ....................................................... 1 Part I General Theory 1 Riemann surfaces and holomorphic maps ......................... 7 1.1 The concept of the Riemann surface . .7 1.2 Holomorphic maps . 14 2 The language of sheaves......................................... 21 2.1 Presheaf and sheaf . 21 2.2 The stalk of a presheaf . 26 2.3 General sheaf constructions . 33 3 Covering projections ........................................... 39 3.1 Branched and unbranched covering projections . 40 3.2 Proper holomorphic maps. 54 3.3 Analytic continuation . 59 4 Differential forms .............................................. 75 4.1 Cotangent space . 75 4.2 Exterior derivation . 83 4.3 Residue theorem . 89 5 Dolbeault and de Rham sequences ............................... 99 5.1 Exactness of the Dolbeault sequence . 99 5.2 Exactness of the de Rham sequence . 105 6 Cohomology ...................................................109 6.1 Cechˇ Cohomology and its inductive limit . 110 6.2 Long exact cohomology sequence . 119 6.3 Computation of cohomology groups . 123 v vi Contents Part II Compact Riemann Surfaces 7 The finiteness theorem ..........................................135 7.1 Topological vector spaces of holomorphic functions. 135 7.2 Hilbert spaces of holomorphic cochains . 141 7.3 Finiteness of dim H1(X;O) and applications . 152 8 Riemann-Roch theorem .........................................157 8.1 Divisors . 157 8.2 The Euler characteristic of the sheaves OD ..................... 164 9 Serre duality ..................................................169 9.1 Dualizing sheaf and residue map . 169 9.2 The dual pairing of the residue form . 179 9.3 Applications of Serre duality and Riemann-Roch theorem . 190 10 Vector bundles and line bundles .................................195 10.1 Vector bundles . 195 10.2 Line bundles and Chern classes . 201 10.3 The divisor of a line bundle . 223 11 Maps to projective spaces .......................................233 n 11.1 The projective space P ..................................... 233 11.2 Very ample invertible sheaves and projective embeddings . 235 11.3 Tori and elliptic curves . 248 12 Harmonic theory ...............................................257 12.1 De Rham cohomology with harmonic forms . 257 12.2 Harmonic forms on Hermitian manifolds . 270 12.3 The example of Riemann surfaces . 284 Part III Open Riemann Surfaces 13 Distributions ..................................................309 13.1 Definitions and elementary properties . 309 13.2 Weyl’s Lemma about harmonic distributions . 313 14 Runge approximation ..........................................321 14.1 Prerequisites from functional analysis . 321 14.2 Runge sets . 325 14.3 Approximation of holomorphic functions. 328 15 Stein manifolds ................................................341 15.1 Mittag-Leffler problem and Weierstrass problem . 341 15.2 Triviality of holomorphic line bundles . 350 15.3 Open Riemann surfaces and Stein manifolds . 359 Contents vii List of results ......................................................367 References .........................................................373 Index .............................................................375 Introduction A first course in complex analysis considers domains in the plane C, and studies their holomorphic and meromorphic functions. But already the introduction of the point at infinity shows the necessity for a wider scope and calls for the introduction of the complex projective space P1. The projective space is compact. Hence it does not embedd as a domain into C. Instead, the projective space is a first non-trivial example of a compact complex manifold. These lecture notes deal with Riemann surfaces, i.e. complex manifolds of com- plex dimension = 1. A manifold is a topological space which is covered by open subsets homeomorphic to an open subset of the plane or of a higher-dimensional affine space. To obtain respectively a smooth or a complex structure on the mani- fold it is required that the local homeomorphisms transform respectively in a smooth or holomorphic way. According to the definition topology covers the global structure of the manifold which can be quite different from the plane. While analysis determines the type of the manifold which is defined by the transformation type of the local home- omorphisms. These
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