Vorlesung: Gromov-Witten Theory, Wintersemester 2022-23

Lecture summaries / reading suggestions / exercises (updated every week) Announcements The lecture and Übung on 25.01.2023 (and also my office hour) are cancelled due to illness. To make up for the two weeks that were lost to Covid, there will be occasional extra lectures on Wednesdays 15:15-16:45 in 1.114, starting in the week of 06.12.2022. See the announcements on the moodle for details. 01.11.2022 The time and place of the Wednesday lectures has now been changed permamently to 13:15-14:45 in room 1.114. 24.10.2022 The lecture on Wednesday 26.10.2022 will take place at an exceptional time and place: 13:15-14:45 in Room 1.114. 18.10.2022 The conclusion of this morning's discussion was that there should be no scheduling changes for this course. 17.10.2022: The course will be taught in hybrid format, with in-person lectures and problem sessions broadcast live in a Zoom meeting and recorded. This should make it unnecessary to attend the class in person if you have any symptoms that could indicate a Covid infection, and I ask that you not do so (even if you have a negative self-test). The links for the Zoom meeting and video repository can be found on the moodle (see link and enrolment key below). 29.09.2022: The problem class (Übung, newly scheduled for Tuesdays at 9:15) will not meet in the first week of the semester, and it might also be scheduled for a new time starting from the second week. This will be discussed in the first lecture. General information Instructor: Prof. Chris Wendl (for contact information and office hours see my homepage) Moodle: https://moodle.hu-berlin.de/course/view.php?id=114268 The enrolment key is: enumerate Important: You must join the moodle for the course in order to receive occasional time-sensitive announcements, e.g. if a lecture has been cancelled or rescheduled. HU students can access moodle using their HU username and password. Non-HU users can access it by following the above link and then setting up a HU Moodle Account with their external e-mail address as a username. You will need to enter the enrolment key printed above. Time and place: Lectures on Tuesdays 11:15-12:45 in room 1.012 and Wednesdays 13:15-14:45 in room 1.114 (RUD25) Problem Class (Übung) Wednesdays 15:15-16:45 in room 1.114 (RUD25) (new time and place as of January!) Language: The course will be taught in English.

Prerequisites:
I will assume that all students are comfortable with the essentials of differential geometry (smooth manifolds, vector fields and Lie bracket, differential forms and Stokes' theorem, de Rham cohomology, connections on vector bundles), as well as some algebraic topology (fundamental group, singular homology and cohomology) and functional analysis (continuous linear operators on Banach spaces, the standard L^p-spaces). Some previous knowledge of additional topics from topology (homological intersection theory, the first Chern class) and functional analysis or PDE theory (Fourier transforms, distributions, Sobolev spaces) will sometimes be helpful, though the relevant results can be taken as black boxes when necessary. For students who have not seen any symplectic geometry before, I will give a concise overview of the subject in the first one or two problem classes.

Contents:
Gromov-Witten theory lies in the intersection of three subbranches of mathematics: symplectic geometry, algebraic geometry, and mathematical physics. This course will focus mainly on the symplectic perspective, but it may also be of interest to students and researchers from the other two subjects.

Symplectic manifolds were invented around the turn of the 20th century as the natural geometric setting in which to study Hamilton's equations of motion from classical mechanics. The subject of symplectic geometry has developed considerably since then, and it retains a close connection with theoretical physics despite being technically a branch of “pure” mathematics. In particular, the subfield known as symplectic topology, which deals with “global” rather than “local” properties of symplectic manifolds, has witnessed an explosion of activity since the introduction of techniques from elliptic PDE theory in the 1980s. The most spectacular advances came from Gromov's theory of pseudoholomorphic curves, which has led to a wide assortment of algebraic invariants of symplectic manifolds, some of them related to structures that physicists study in quantum field theory or string theory. One example of this is the Gromov-Witten invariants, which are interpreted as counts of holomorphic curves satisfying specified constraints in a symplectic manifold. Since many interesting examples of symplectic manifolds are also algebraic varieties, the Gromov-Witten invariants are also heavily studied by algebraic geometers and can be viewed as a modern approach to enumerative problems (i.e. generalizations of the question “how many lines are there through two points?”) that have been studied in algebraic geometry since the 19th century.

The first goal of this course will be to establish the basic analytical underpinnings of the Gromov-Witten invariants: we will study the local and global structure of moduli spaces of Riemann surfaces and holomorphic curves, elliptic regularity theory for the nonlinear Cauchy-Riemann equation, Fredholm theory, the Riemann-Roch formula, transversality results via the Sard-Smale theorem, and Gromov's compactness theorem for pseudoholomorphic curves. These ingredients are sufficient to give a mathematically rigorous definition of the Gromov-Witten invariants for symplectic manifolds that satisfy a technical condition known as “semi-positivity”, which is always satisfied for manifolds of dimension at most six. Once this is established, there are various additional topics we might discuss, depending on the time available and interests of the class:

• The Kontsevich-Manin axioms for GW-invariants, and Kontsevich's recursion formula, which gives a beautiful answer to a fundamental question about the enumeration of rational curves in complex projective space.
• Quantum cohomology, a “deformation” of the usual cohomology ring of a manifold, in which the product depends on counts of holomorphic curves.
• Symplectic Calabi-Yau 6-manifolds, a setting in which each GW-invariant is determined by obstruction bundles over the moduli spaces of branched covers of finitely-many embedded curves.
• Advanced transversality methods: what can be done to define the GW-invariants for symplectic manifolds that are not semi-positive? (There are various approaches to this - I am likely to limit myself to the ones that I actually understand.)

Literature:
I will not be writing detailed lecture notes for this course, but will write up a brief summary of what was covered at the end of each week, including reading suggestions and exercises. A considerable amount of the material we'll cover is contained in notes that I have written for other courses in the past, notably:

• Lectures on Holomorphic Curves in Symplectic and Contact Geometry
• Lectures on Symplectic Field Theory
• Holomorphic Curves in Low Dimensions (Springer LNM, 2018)

• McDuff and Salamon: J-holomorphic Curves and Symplectic Topology (2nd edition, AMS, 2012)

• Ana Cannas da Silva: Lectures on Symplectic Geometry (Springer LNM, 2008)

• McDuff and Salamon: Introduction to Symplectic Topology (3rd edition, Oxford University Press, 2017)

Homework:
I will assign exercises sometimes. Sometimes I will discuss them in the problem class. They will not be graded.

Grades:
Since this is an advanced course, I have a fairly relaxed attitude about grades. If you come to the course with adequate prerequisites and stay with it for the whole semester, you can come to my office at the end for a conversation (let's pretend that's the English translation of “mündliche Prüfung”). The format is as follows: you pick one particular coherent topic from the course to focus on, typically the contents of four to six lectures, and we will talk about that. If you demonstrate that you learned something interesting from the course, you'll get a good grade.