## ECH lecture notes posted

The ECH lecture notes are now posted at arXiv:1303.5789. Here is a brief guide to what in these lecture notes is new, i.e. has not appeared in previous papers. (Some of this material was in blog postings last July.)

Section 1 is an introduction to the basic structure of ECH and its application to symplectic embedding problems. There is nothing really new here; however I’m hoping it will be useful for people who haven’t seen this material before.

Section 2, after reviewing some basics about holomorphic curves, computes Taubes’s Gromov invariant in the $S^1$-invariant case and uses the result to motivate the choice of generators of the ECH chain complex. This is new (in the sense that it hasn’t appeared in print; of course this was known from the beginning of the development of ECH), and I hope it will demystify the definition of ECH a bit.

Section 3 gives more details about the definition of ECH and computes the ECH of an ellipsoid. There is nothing new here, although again I hope it will be useful for readers who haven’t seen it before and don’t want to dig through the original papers.

Section 4 computes the U map on the ECH of a (nearly round) ellipsoid, outlines the computation of the ECH of T^3, and then explains the computation of the ECH capacities of toric domains. The U map computation hasn’t appeared before. Of course the computation of the ECH of $T^3$ was discussed extensively in my paper with Michael Sullivan. Actually there is a connection with tropical geometry which is discussed in general in Brett Parker’s thesis and which he gave a talk about long ago, and a similar approach appears in Taubes’s work on holomorphic curves in ${\mathbb R}\times S^1\times S^2$, but I think there should be another expository article explaining this beautiful picture for the $T^3$ case. The computation of ECH capacities of toric domains appeared before in my paper on Quantitative ECH, but the explanation here is maybe a little simpler.

Section 5 gives an introduction to the foundations of ECH. Since the foundations are quite complicated, hopefully this introduction will make it not so hard to get some idea of what is involved. Also, section 5.2 discusses the “topological complexity” (roughly speaking the genus) of the holomorphic curves counted by ECH; while this isn’t new, I have hopefully spelled it out more simply than before.

Section 6 compares ECH with SFT, responding to various frequently asked questions.

Finally there are various exercises and some of them have answers in the appendix.

Also, there are a few new things about the terminology. For example, I now use the term “holomorphic currents” to describe unions of somewhere injective holomorphic curves with positive integer multiplicities. This is appropriate for ECH because when multiple covers arise we do not care about a covering map; we only care about the degree of the cover. It would also make sense to refer to the generators of the ECH chain complex as “Reeb currents”, but I didn’t do that.