Today I gave the first of five lectures at the SFT VI workshop in Munich. Thanks to the excellent minicourses over the weekend, I could assume some background about holomorphic curves and Seiberg-Witten theory (and I’ll be able to assume some background about contact geometry when we come to that later). I reviewed the definition of Taubes’s Gromov invariant of symplectic four-manifolds, and computed this for the example of cross a mapping torus (although I rushed through the latter a bit due to lack of time). More details about this are in installments 2 and 3 of the lecture notes previously posted here.

Tim Perutz asked an interesting question: What can we say about a contact three manifold if all of its Reeb orbits (are nondegenerate and) have odd Conley-Zehnder index (i.e. are elliptic or negative hyperbolic)? Taubes and I showed that if all Reeb orbits are (nondegenerate and) elliptic then the three-manifold is a lens space. I suspect that similar arguments might prove the same conclusion when there can also be negative hyperbolic orbits. However this requires checking because there are a number of subtle points in that argument.

**Intersection theory**

Chris Wendl talked about (his interpretation of) Richard Siefring’s intersection theory for punctured holomorphic curves in four-dimensional symplectizations (or more generally symplectic cobordisms). Since this is closely related to the intersection theory in the foundations of ECH, but has different purposes, I would like to try to clarify how these two approaches compare.

First let’s review the intersection theory for ECH. Suppose is somewhere injective (see the lecture notes for this notation). Recall from installment 6 of the lecture notes that there is a relative adjunction formula

where is a count of the singularities of with positive integer weights. Furthermore, the ECH index inequality (see installment 7 of the lecture notes) uses the writhe bound

Equality holds in this bound only if the multiplicities of the ends of (i.e. the covering multiplicities of the Reeb orbits associated to each end) satisfy the partition conditions described in installment 9 (in which case it is known that equality holds “generically”, e.g. for curves of index in a symplectization if is generic; this follows from part II of my gluing paper with Taubes).

Now here is my understanding of what Chris Wendl and Richard Siefring are doing. Note that if you combine the relative adjunction formula and writhe bound above, you get an upper bound for . This bound is usually not sharp, because my writhe bound is only sharp if the multiplicities of the ends of satisfy the partition conditions. On the other hand one can study the writhe more carefully to get a sharp upper bound of the form

where is the maximum conceivable value of , given the Euler characteristic of , the relative homology class of , and the multiplicities of the ends of (and the structure of the Reeb orbits). In particular, while can vary as one deforms , the number is a topological invariant depending only on , the relative homology class of , and the end multiplicities. When equality does not hold, one would interpret as the count of actual singularities , plus “virtual” singularities “hidden at infinity”. (Chris didn’t say actually this, but I am guessing that he would have said this if he had more time, since he said something analogous for the intersection of two curves.) And one would expect that equality holds generically. (Perhaps I am the only one who expects this, but I would like this to be true for something else I have been thinking about, involving ruling out certain kinds of degenerations of holomorphic curves; more about this later.)

Summary: Siefring/Wendl get a sharp writhe bound, while the writhe bound that enters into the ECH index inequality is only sharp for certain end multiplicities.

Also: all writhe bounds depend on difficult asymptotic analysis of holomorphic curves proved by Siefring. (I originally proved the ECH writhe bound only under an analytical simplifying assumption.)

I hope that was a fair comparison of the two approaches; if not, please feel free to complain here.

What you conjecture that I would have said is indeed what I would have said if I’d had more time.

About genericity: I would call it a general folk theorem (well known to the very small number of people who’ve deeply immersed themselves in Richard’s intersection theory) that the count of “intersections hidden at infinity” is generically zero. A more precise version of this statement is that for generic J, in any given moduli space of somewhere injective J-holomorphic curves, the subset consisting of curves with a positive count of singularities hidden at infinity is a countable union of submanifolds of positive codimension. A similar statement holds for the count of intersections between a pair of curves. The reason is essentially that the curves with nontrivial hidden intersections satisfy a Fredholm problem with extra exponential weights describing their relative asymptotic behavior, and the weights cause this problem to have lower Fredholm index than the usual one.

The details of this genericity statement haven’t been written down in the literature, but several years ago I worked out the relevant linear Fredholm theory far enough to convince myself that it’s correct. I never fully thought through the question of how to set up the nonlinear problem the right way so that this linear Fredholm theory gives the result we need… this strikes me as something of an analytical headache, but not one that would be deeply problematic.

OK, good, when I get a chance I’ll show you what I want this genericity for, and then maybe we can prove something.