In my last post and in the comments I misremembered and garbled various things about automatic transversality. Let me try to clean things up here.
As Chris Wendl pointed out, a correct statement is the following. Let be a (completed) four-dimensional strong symplectic cobordism between contact manifolds, and let be an almost complex structure on satisfying the usual conditions. If is a -holomorphic curve in , let denote the genus of , let denote the number of ends of at positive hyperbolic orbits, and let denote the Fredholm index of .
Theorem. Let be an immersed -holomorphic curve in with ends at nondegenerate Reeb orbits. Suppose that . Then is automatically transverse (regardless of whether or not is generic).
Chris Wendl has a paper proving more general versions of this in which does not have to be immersed and can have ends at Morse-Bott Reeb orbits. (I have used the Morse-Bott version before.) But let me just explain the proof of the above, in my notation.
Proof. To simplify notation, assume all ends of are positive; it is easy to fix the notation when this is not the case.
Since is immersed, it has a well-defined normal bundle , and we can regard the linearized Cauchy Riemann operator as a map from sections of to sections of . Suppose that is not transverse. Then the formal adjoint has a nonzero kernel. Let be a section of which is an element of . It is a standard fact that every zero of is isolated and has negative multiplicity. (This follows for example from the Carleman similarity principle.) Also the zero set of is compact (this follows from stuff about the asymptotics of which I can review later). So the algebraic count of zeros of is negative:
Now the algebraic count of zeroes of is given in terms of the relative first Chern class of by the formula
Here is a trivialization of over the ends of , and denotes the sum of the winding numbers of around the ends with respect to the trivialization . Indeed, one can regard this formula as the definition of the relative first Chern class .
We can take the trivialization of over the ends to be induced by a trivialization of the contact plane field over the ends, which we also denote by , together with the canonical trivialization of over the ends. Then
It also follows from the asymptotics of which I didn’t explain here that
Here indexes the ends of , denotes the Reeb orbit at the end of , and denotes the Conley-Zehnder index with respect to . Note that is even if and only if is positive hyperbolic. So if denotes the total number of ends of , we can rewrite the above inequality as
Next recall that the Fredholm index is given by the formula
We now put everything together as follows:
So if is not transverse. QED.
It is easy to see that automatic transversality can fail for an index cylinder with both ends at positive hyperbolic orbits (contrary to a foolish claim that I made in the comments to the previous post). Just think about the symplectic fixed point Floer homology for the indentity map on a surface, perturbed by a Morse function. (This is not quite the contact geometry setting but it is close.) It is possible to have a flow line between two index critical points (e.g. for the height function on a torus standing on its side), and this gives rise to a holomorphic cylinder of index zero whose ends are at positive hyperbolic orbits.