Exact orbifold fillings of contact manifolds

Abstract

We study exact orbifold fillings of contact manifolds using Floer theories. Motivated by Chen-Ruan's orbifold Gromov-Witten invariants, we define symplectic cohomology of an exact orbifold filling as a group using classical techniques, i.e. choosing generic almost complex structures. By studying moduli spaces of pseudo-holomorphic curves in orbifolds, we obtain various non-existence, restrictions and uniqueness results for orbifold singularities of exact orbifold fillings of many contact manifolds. For example, we show that exact orbifold fillings of (RP 2n−1 , ξ std ) always have exactly one singularity modeled on C n /(Z/2Z) if n = 2 k . Lastly, we show that in dimension at least 3 there are pairs of contact manifolds without exact cobordisms in either direction, and that the same holds for exact orbifold cobordisms in dimension at least 5.

On the intersection form of fillings

“…As a consequence, the claim follows. That W being a manifold follows from the same argument as in [11,Theorem C].…”

“…Then after choosing a trivialization of det C ⊕ N ξ for some N ∈ N + , we can assign a rational Conley-Zehnder index to each non-degenerate Reeb orbit [11,15]. For those orbits with torsion homology classes, the Conley-Zehnder index is independent of N and the trivialization.…”

On fillings of contact links of quotient singularities

“…Hence, to prove Theorem B, we need to enhance both the pseudoholomorphic curve argument and the topological argument. On the pseudo-holomorphic curve side, we use symplectic cohomology for exact orbifolds developed by Gironella and the author [38] and study the secondary coproduct on positive symplectic cohomology, which was proposed by Seidel [63], defined by Ekholm and Oancea [27], and recently studied extensively by Cieliebak, Hingston and Oancea [20,21,23]. Then, by a result of Eliashberg, Ganatra and Lazarev [31], we can get information on the intersection form of the filling.…”

Asymptotically holomorphic theory for symplectic orbifolds

“…This idea was further explored in [PS21, CGHM + 21] which brought breakthroughs on dynamics on surfaces and C 0 symplectic geometry. Symplectic orbifolds were also used in [GZ21] to study the symplectic cobordism category of contact manifolds. From yet another point of view, they also appear in constructions of symplectic manifolds via the desingularisation process.…”

“…It is hence a natural direction of research that of extending well known and powerful techniques in smooth symplectic geometry to the orbifold setting. The first set of tools are pseudoholomorphic and Floer theories, which have been the object of study for instance in [CR02,CP14,GZ21]. This paper is devoted to extend the techniques in [Don96] to the orbifold setting.…”