We determine contact homology algebra of a subcritical Stein-fillable contact manifold whose first Chern class vanishes. We also compute the genus-0 one point correlators and gravitational descendants of compactly supported closed forms of their subcritical Stein fillings. This is a step towards determining the full potential function of the filling as defined in [EGH00]. These invariants also give a canonical presentation of the cylindrical contact homology. With respect to this presentation, we determine the degree-2 differential in the Bourgeois-Oancea exact sequence of [BO09]. As a further application, we proved that if a Kähler manifold M 2n admits a subcritical polarization and c 1 vanishes in the subcritical complement, then M is uniruled.Theorem 1.1 ([Yau04]). Let (M 2n , ∂M), n ≥ 2, be a subcritical Stein domain of finite type, and ξ the maximal complex subbundle on ∂W . If c 1 (ξ) = 0, thenGiven a graded vector space V , let Λ(V ) denote the tensor algebra generated by V modulo the graded commuting relations, and V [n] the graded vectors space consisting of elements of V with a grading shift of positive n.Date: June, 2012.1 4 JIAN HE Reeb dynamics are described in more detail, essentially summarizing the previous work of Yau, and we prove Theorems 1.2 and 1.3 modulo the technical Proposition 3.15; in section 4 we apply Theorem 1.3 to subcritical polarizations; in section 5 we prove Proposition 3.15; and in the last section we discuss connections with the Bourgeois-Oancea exact sequence and prove Theorem 1.5. Symplectic Field Theory and DescendantsIn this section we will give an extremely brief overview of aspects of symplectic field theory and define the invariants we wish to compute. See [EGH00] for a more complete discussion. Throughout this section we assume the polyfold theory of Hofer, Zehnder and Wysocki, [HWZ06], [HWZ07], [HWZ08], which forms the analytical foundation of SFT. In other words, there exists a abstract perturbation scheme under which all moduli spaces are branched manifolds with boundaries and corners of the expected dimension.Let (V 2n−1 , ξ) be a contact manifold with a contact 1-form α, i.e., (dα) n−1 ∧ α is a volume form and ξ = Ker(α). The Reeb vector field is the unique vector field R such thatThe flow of the Reeb vector field preserves the contact structure ξ. A (possibly multiply covered) Reeb orbit γ is non-degenerate if the linearized Poincaré return map of the Reeb flow has no eigenvalue equal to 1. For a generic choice of α, there are countably many closed Reeb orbits, all of which are non-degenerate. Let κ γ denote the multiplicity of the orbit γ.Definition 2.1. A Reeb orbit is good if it is not an even multiple of another orbit γ such that the linearized Poincaré return map along γ has an odd total number of eigenvalues (counted with multiplicity) in the interval (−1, 0).Remark 2.2. All orbits appearing in this paper are easily seen to be good.If c 1 (ξ) = 0, then the Reeb orbits admit a consistent Z-grading. Define the index of a Reeb orbit γ to be µ(γ) = µ(γ) + (n − 3...

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In this paper we prove a recursion relation between the the one-point genus-0 gravitational descendants of a Stein domain (M, ∂M ). This relation is best described by the degree −2 map D in the linearized contact homology of ∂M , arising from the Bourgeois-Oancea exact sequence between symplectic homology of M and linearized contact homology of ∂M . All one-point genus-0 gravitational descendants can be reduce to the one-point Gromov-Witten invariants via iterates of D.

Given a surface with boundary and some points on the boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on the surface. If only bigons are allowed, then one obtains the notion of arc diagrams, whose enumeration is known to have a rich structure. We show that the count of polygon diagrams on surfaces with any genus and number of boundary components exhibits similar structure. In particular it is almost polynomial in the number of points on the boundary components, and the leading coefficients of those polynomials are intersection numbers on compactified moduli spaces of curves.