Instructor Info

Name: Maria Angelica Cueto
Email: cueto.5@osu.edu
Office: Math Tower (MW) 636
Office Phone: 688 5773

Office Hours

W 2:00pm-3:00pm
in Math Tower (MW) 636

or by appointment

Time and Location

Lecture: M-W 10:20am-11:40am
in University Hall (UH) 43.

Riemann surfaces are intimately tied to ``multivalued functions'' and were originally conceived in 1850's by Karl Weierstrass as visualizing aids for such functions. Since then, these geometric objects have been extensively studied and have acquired fundamental importance in several branches of mathematics and physics, including complex potential theorem, geometry (conformal, differential and algebraic), and combinatorial/algebraic topology, to name a few.

This one semester graduate topics course will serve as an introduction to the subject, focusing on compact Riemann surfaces and maps between them. We will loosely follow the textbooks listed in the References. The compact case is especially interesting since it leads to concrete formulas for counting maps (e.g., Riemann-Hurwitz and Riemann-Roch Theorems). Moreover, this case is known to be essentially algebraic (a consequence of Riemann-Roch): compact Riemann surfaces are smooth projective complex algebraic curves.

By participating in this course, students will learn about this beautiful subject and will gain some insight into deep yet technical results in algebraic geometry by means of concrete hands-on examples.

Prerequisites: Some experience with basic Algebraic Topology and Complex Analysis will be helpful (we will review the relevant results). No knowledge of Algebraic Geometry will be assumed.

[References]       [Homework and Final presentations]      [Topics]      [Lecture Notes]

References

 The literature on Riemann surfaces is vast. We will be using complementary references to cover the material discussed in class. Links to electronic copies available through the OSU library are provided whenever possible (access requires connection via an OSU proxy, e.g., by being on campus)

Main textbook: Lectures on Riemann surfaces, by Otto Forster (Springer Graduate Texts in Mathematics Series, GTM 81). Available online through OSU library.

Other textbooks and resources:
  • Algebraic Curves and Riemann Surfaces by Rick Miranda, AMS Graduate Studies in Mathematics, Volume 5).
  • Introduction to Algebraic Curves by Phillip A. Griffiths (AMS, Translations of mathematics Monographs, Volume 76).
  • Riemann surfaces and algebraic curves: a first course in Hurwitz theory by Renzo Cavalieri and Eric Miles (Cambridge University Press, 2016). Available online through OSU library.
  • Riemann surfaces by Simon Donaldson (Oxford Graduate Texts in Mathematics, 22).
  • Compact Riemann surfaces by Raghavan Narasimhan (Lectures in Mathematics, ETH Zürich.) Available online through OSU library.

Back to Top

Homework and Final presentations

Students will be encourage to work on exercises mentioned in class and posted regularly on Carmen, as the semester progresses. Participants are encourage to work in teams, but individual solutions must be submitted for grading and credit.

Each student is expected to work on and submit at least some of the problems (chosen by the students). Only one solution per problem will be uploaded to Carmen for grading. Solutions should be uploaded as a pdf file (preferrably produced in LaTeX).

As an optional complementary assignment, students are welcome to give a 30 minute presentation in class on a topic complementing the material discussed in class. Topics will be selected in agreement with the instructor.

Back to Top

Tentative topics

The following is a tentative list of topics that will be covered during the semester. For a list of topics cover each class, see the corresponding handwritten notes in the section entitled Lectures.

  • Basics on Riemann surfaces, covering spaces and fundamental groups
  • Examples: Riemann surfaces from multivalued functions; complex algebraic curves.
  • Maps between Riemann surfaces; local behavior
  • Analytic continuation of germs.
  • Riemann surfaces associated to holomorphic differential 1-forms
  • Algebraic Riemann surfaces associated to finite field extensions of C(z).
  • Universal covers of Riemann surfaces, quotients and Galois correspondence
  • Integration of differential forms, Stokes theorem, periods
  • Sheaves on Riemann surfaces and Čech (sheaf) cohomology. Examples. Leray's theorem.
  • Finiteness theorem of cohomology for compact Riemann surfaces. Corollaries. Definition of genus via cohomology.
  • Divisors on compact Riemann surfaces. Riemann-Roch Theorem, Riemann Hurwitz, Serre duality. Existence of non-constant meromorphic functions on compact Riemann surfaces.
  • Abel-Jacobi Theory. Algebraic vs. Topological genus. The Jacobian of a compact Riemann surfaces and the Abel-Jacobi map.
  • Elliptic curves
Depending on the interest of the participants, possible extra topics could include:
  • Applications of Riemann Roch (canonical embeddings, hyperelliptic curves, inflection and Weiestrass points).
  • Hurwitz Theory (counting covers between compact Riemann surfaces with prescribed ramification profiles).
  • a bird's-eye view of non-compact Riemann surfaces and the main techniques involved in their study (e.g., Riemann's Uniformization Theorem, Fuchsian groups, etc.)

Back to Top

Lectures
  • Lecture 1 (Overview, basic definitions and examples of Riemann surfaces), January 9, 2023.
  • Lecture 2 (Review of basics on complex holomorphic and meromorphic functions; definitions and main properties of holomorphic and meromorphic functions on Riemann surfaces; holomorphic functions betwen Riemann surfaces and their local behavior, definition of branching numbers), January 11, 2023.
  • Lecture 3 (Curves, Loops; Homotopies between curves, free homotopies; Fundamental groups; Connected, pathwise connected, locally pathwise connected topological spaces; Simply connected spaces). Reading assignment.
  • Lecture 4 (Branch points of non-constant holomorphic functions; Topology interlude: lifting (or factoring) continuous functions relative to local homomorphisms with prescribed initial conditions, local liftings, uniqueness under connectedness conditions; lifting homotopies between curves relative to local homomorphism), January 18, 2023.
  • Lecture 5 (Curve lifting property, covering maps, proper maps, degree of proper non-constant holomorphic functions between Riemann surfaces), January 23, 2023.
  • Lecture 6 (Universal coverings, sufficient conditions for being a universal cover of a connected manifold, construction of universal coverings for connected manifolds), January 25, 2023.
  • Lecture 7 (Deck transformations, Galois coverings and Galois correspondence of coverings: identifying intermediate covers of the universal cover as subgroups of Deck transformations), January 30, 2023.
  • Lecture 8 (Classification of coverings of the punctured disc; basics on sheaves; topology on the set associated to a sheaf; terminology for "sections" of a presheaf), February 1, 2023.
  • Lecture 9 (Analytic continuation along paths, Weierstrass construction of analytic continuation; Riemann surfaces associated to analytic continuations; maximal analytic continuation), February 6, 2023.
  • Lecture 10 (Examples of anallytic continuation; Elementary symmetric functions of meromorphic functions with respect to unbranched covers; extensions to proper holomorphic branched covers tested on elementary symmetric functions), February 8, 2023.
  • Lecture 11 (Algebraic Functions: Field extensions arising from proper degree n holomorphic maps between Riemann surfaces; proof of existence of algebraic functions), February 13, 2023.
  • Lecture 12 (Algebraic Functions II: proof of uniqueness of algebraic functions; Galois correspondence between algebraic functions and Galois extension of function fields), February 15, 2023.
  • Lecture 13 (Examples of algebraic functions of degree 2, Puiseux expansions), February 20, 2023.
  • Lecture 14 (Basic definitions of cotangent space to Riemann surfaces, germs of smooth and differential 1- and 2-forms, complex of differential forms, pullback of differential forms under holomorphic maps), February 27, 2023.
  • Lecture 15 (Integration of 1-forms; closed and exact closed forms; construction of Riemann surfaces through the sheaf of primitives to closed 1-forms; periods of closed 1-forms; characterization of exactness of closed 1-forms via vanishing of all periods), March 1, 2023.
  • Lecture 16 (Čech cohomology I: Čech cohomology of a covering with values on a sheaf; behavior under refinement of coverings; Čech cohomology of a topological space by inductive limit; injectivity for first cohomology. Examples: Čech cohomology of a Riemann surface with values on differentiable k-forms for k = 0, 1, (1,0), (0,1) and 2), March 6, 2023.
  • Lecture 17 (Čech cohomology II: computation for locally constant sheaves C and Z for simply-connected Riemann surfaces; Leray coverings and Leray's theorem. Computation of the first cohomology group of X with values on the sheaf of holomorphic functions for X = DR (open disc), C and P1), March 8, 2023.
  • Lecture 18 (Integration of differentiable 2-forms, Stokes' Theorem in the plane for discs and annuli, Dolbeault's Theorem). Reading assignment.
  • Lecture 19 (Finiteness Theorem for comohology of compact Riemann surfaces; corollaries involving meromorphic functions with prescribed values or poles at given finite sets of points, vanishin of certain restrictions), March 20, 2023.
  • Lecture 20 (Divisors on Riemann surfaces; basic definitions, principal and canonical divisors, linear equivalence; examples; degree of divisors on compact Riemann surfaces; the sheaf OD associated to a divisor D), March 22, 2023.
  • Lecture 21 (The Riemann-Roch Theorem I: Leray covers for sheaves OD; statement of Riemann Roch Theorem and proof outline; Applications to find lower bounds on the dimension of global sections of OD; Skyscraper sheaves and their 0th and 1st Čech cohomology groups; Morphisms of sheaves, exactness, mono and epimorphisms; a key s.e.s. for Riemann-Roch), March 27, 2023.
  • Lecture 22 (The Riemann-Roch Theorem II: Proof of the theorem; Long exact sequences on Čech cohomology from s.e.s. on sheaves, including explicit construction of connecting homomorphisms), March 29, 2023.
  • Lecture 23 (Serre Duality I: Definition of sheaves ΩD associated to divisors on Riemann surfaces, definition of the bilinear map H0(X,Ω-D)× H1(X,OD)→ H1(X,Ω) induced by multiplication map on sheaves ΩD'× OD → OD+D', for any pair of divisors D, D' on a Riemann surface; statement of Serre Duality for compact Riemann surfaces and some immediate corollaries, including the degree formula deg(K) = 2g-2 for any canonical divisor K on a genus g compact Riemann surface, and dimCH1(X,Ω)=1), April 3, 2023.
  • Lecture 24 (Serre Duality II: More corollaries: H1(X,OD)=0 if deg(D)>2g(X)-2, vanishing of first cohomology group of the sheaf of meromorphic functions on a compact Riemann Surface); explicit computation of Serre pairing for P1; definition of the residue map on 2-forms and 1-forms; Residue Theorem for meromorphic 1-forms on compact Riemann surfaces), April 5, 2023.
  • Lecture 25 (Riemann-Hurwitz formula, topological classification of compact Riemann surfaces via polygons with identified edges; Euler characteristic computation via triangulations), April 10, 2023.
  • Lecture 26 (Hyperelliptic covers for algebraic functions associated to a bivariate polynomial T2-h(z); degree-genus formula for smooth projective plane curves; linear systems: definitions and exampls, bounds on dimensions), April 12, 2023.
  • Lecture 27 (Linear systems, maps to projective spaces and embeddings; Key bounding lemma: dim OD-[p]X ≥ dim OD -1 for all p in X and D in Div(X); example of maps to projective spaces: rational normal curves; globally generated sheaves OD and a degree criterion (deg D ≥ 2 g); Degree criterion for an embedding via the complete linear system |D| (deg F ≥ 2g + 1)), April 17, 2023.
  • Lecture 28 (Maps to projective spaces, base point free linear systems, canonical maps), April 20, 2023.
  • Lecture 29 (Canonical maps for non-hyperelliptic curves of genus at least 2 are embeddings; genus 2 compact Riemann surfaces are hyperelliptics; characterization of canonical maps for hyperelliptic curves; Abel-Jacobi theory: definition of the period lattice, and theorem of Abel-Jacobi; proof Abel-Jacobi Theorem for elliptic curves), April 24, 2023.

Back to Top