Homological invariants of codimension 2 contact submanifolds

Abstract

In order to ensure that (1-3) defines a differential over QOEU , we need to ensure that y V ˇC .ˇ; V / 0 whenever #M.ˇ/ ¤ 0. If M.ˇ/ is nonempty 2 and at least one of the asymptotic orbits of ˇis disjoint from V, then this is a consequence of the familiar phenomenon of positivity of intersection. Indeed, in this case, ˇadmits a y J -holomorphic representative u which is not contained in y V. Positivity of intersection then implies that y V ˇD y V u 0.The situation is more complicated when all of the asymptotic orbits of ˇare contained in V. Indeed, in this case, the y J -holomorphic representatives of ˇmay be contained in y V and positivity of intersection fails in general. However, one can show that there is a universal lower bound on the intersection number,This explains the appearance of the correction term .ˇ; V / in (1-3).In order to construct CH .Y; ; V I r/, it is not enough to define a differential: one also needs to define continuation maps, composition homotopies, etc. These maps are defined by counting curves in more complicated setups. For example, the continuation map is obtained by counting curves in a suitably marked exact relative symplectic cobordism . y X ; y ; H /. 3 More precisely, one obtains an algebra map similar to (1-3) by counting y J -holomorphic curves in . y X ; y / weighted by their intersection number with H , for a compatible almost complex structure y J which agrees with y J ˙near the ends.Unfortunately, for an arbitrary relative symplectic cobordism, a lower bound of the type (1-7) fails to hold.A key step in constructing the invariants (1-1) is to identify a sufficiently large class of relative symplectic cobordisms for which such a lower bound does hold. This leads us to introduce notions of energy for exact symplectic cobordisms and almost complex structures on exact relative symplectic cobordisms. These energy notions are developed in Section 6 and are of central importance in this paper.We prove that a lower bound as in (1-7) holds under a certain condition which relates the behavior of ṅear V ˙to the energy of y J. We also prove analogous statements for other related setups. This allows us to prove that CH .Y; ; V I r/ is well-defined, ie it does not depend on the auxiliary contact form and almost complex structure. We also prove that an exact relative symplectic cobordism . y X ; y ; H / induces a map.Y ; ; V I r / provided that a certain inequality is satisfied, where the inequality involves r ˙and the energy of the (sub)cobordism H . y X ; y /. Côté was partially supported by a Stanford University Benchmark Graduate Fellowship and by the National Science Foundation under grant DMS-1926686. 2 Geometric preliminaries 2.1 Symplectic cobordisms Let .Y; / be a closed co-oriented contact manifold. The symplectization of .Y; / is the exact symplectic manifold .SY; Y / where S Y T Y is the total space of the bundle of positive contact forms on Y (ie a point .p; ˛/ 2 T Y is in S Y if and only if ˛W T p Y ! R vanishes on p and the induced map T p Y = p ! R