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We use Floer theory to describe invariants of symplectic $\mathbb{C}^*$-manifolds admitting several commuting $\mathbb{C}^*$-actions. The $\mathbb{C}^*$-actions induce filtrations by ideals on quantum cohomology, as well as filtrations on Hamiltonian Floer cohomologies, and we prove relationships between these filtrations. We also carry this out in the equivariant setting, in particular $\mathbb{C}^*$-actions then give rise to Hilbert-Poincar\'{e} polynomials on ordinary cohomology that depend on Floer theory. For semiprojective toric manifolds, we obtain an explicit presentation for quantum and symplectic cohomology in the Fano and CY setting, both in the equivariant and non-equivariant setting., Comment: 38 pages, 1 image

We prove that Floer theory induces a filtration by ideals on equivariant quantum cohomology of symplectic manifolds equipped with a $\mathbb{C}^*$-action. In particular, this gives rise to Hilbert-Poincar\'e polynomials on ordinary cohomology that depend on Floer theory. En route, the paper develops structural properties of filtrations on three versions of equivariant Floer cohomology. We obtain an explicit presentation for equivariant symplectic cohomology in the Calabi-Yau and Fano settings., Comment: 72 pages, 12 figures, fixed Theorem 1.10 and 14.3, new Remark 1.12 about the persistence module, very minor other changes

Using Morse-Bott-Floer spectral sequences, we describe a filtration by ideals on quantum cohomology for symplectic manifolds with a Hamiltonian $S^1$-action that extends to a pseudoholomorphic $\mathbb{C}^*$-action. These spaces include all Conical Symplectic Resolutions, in particular all Quiver Varieties. Our Morse-Bott-Floer spectral sequences give explicit descriptions of birth-death phenomena of the barcode of the persistence module associated to the $\mathbb{C}^*$-action, defined in our earlier paper. This paper contains the foundational work to rigorously construct a filtration on Floer complexes, announced in that earlier paper. We also include a substantial appendix on Morse-Bott-Floer theory, where a large part of the technical difficulties of the paper are dealt with. We compute a plethora of explicit examples, each highlighting various features, for Springer resolutions, ADE resolutions, and several Slodowy varieties of type A. We also consider certain Higgs moduli spaces, for which we compare our filtration with the well-known P=W filtration., Comment: 90 pages, 28 figures, minor corrections, added details in the proof of Prop B.6

We construct a filtration by ideals on quantum cohomology for symplectic manifolds with a Hamiltonian $S^1$-action that extends to a pseudoholomorphic $\mathbb{C}^*$-action. These spaces include all Conical Symplectic Resolutions, in particular all Quiver Varieties. In particular, we obtain a family of filtrations on singular cohomology for any Conical Symplectic Resolution, that is sensitive to the choice of $\mathbb{C}^*$-action. The symplectic form is rarely exact at infinity for these spaces, so substantial foundational work is carried out to rigorously define Floer theory, in particular symplectic cohomology. Using Floer theory, we construct a periodic persistence module, giving rise to a graded periodic barcode associated to the $\mathbb{C}^*$-action. This encodes birth-death phenomena of Floer invariants. Our filtrations can be viewed as a Floer-theoretic analogue of Atiyah-Bott filtrations, arising from stratifying a manifold by gradient flowlines of a Morse-Bott function, but they are distinct from those and they can detect non-topological properties of the quantum product., Comment: 72 pages, 4 figures. Minor changes. Added comments about persistence modules and barcodes. The toric discussion is summarized in the Introduction, but the proofs appearing in an earlier arxiv version will appear in a forthcoming separate paper

We prove that symplectic cohomology for open convex symplectic manifolds is invariant when the symplectic form undergoes deformations which may be non-exact and non-compactly supported, provided one uses the correct local system of coefficients in Floer theory. As a sample application beyond the Liouville setup, we describe in detail the symplectic cohomology for disc bundles in the twisted cotangent bundle of surfaces, and we deduce existence results for periodic magnetic geodesics on surfaces. In particular, we show the existence of geometrically distinct orbits by exploiting properties of the BV-operator on symplectic cohomology., Comment: 48 pages, 1 figure. Results are unchanged. Additional details were added in the Introduction and in the proof of Prop.3.9. To appear in: International Journal of Mathematics

We prove the generalised McKay correspondence for isolated singularities using Floer theory. Given an isolated singularity \C^n/G for a finite subgroup G in SL(n,\C) and any crepant resolution Y, we prove that the rank of positive symplectic cohomology SH_+(Y) is the number of conjugacy classes of G, and that twice the age grading on conjugacy classes is the \Z-grading on SH_+(Y) by the Conley-Zehnder index. The generalised McKay correspondence follows as SH_+(Y) is naturally isomorphic to ordinary cohomology H(Y), due to a vanishing result for full symplectic cohomology. In the Appendix we construct a novel filtration on the symplectic chain complex for any non-exact convex symplectic manifold, which yields both a Morse-Bott spectral sequence and a construction of positive symplectic cohomology., Comment: 39 pages, 2 figures; Version 3: Sec.3 and Sec.4.7 more detailed proofs, Sec.6.1 new Figure, Lemma 7.1 sign correction. To appear in: Journal of Differential Geometry

We define a class of non-compact Fano toric manifolds, called admissible toric manifolds, for which Floer theory and quantum cohomology are defined. The class includes Fano toric negative line bundles, and it allows blow-ups along fixed point sets. We prove closed-string mirror symmetry for this class of manifolds: the Jacobian ring of the superpotential is the symplectic cohomology (not the quantum cohomology). Moreover, SH(M) is obtained from QH(M) by localizing at the toric divisors. We give explicit presentations of SH(M) and QH(M), using ideas of Batyrev, McDuff and Tolman. Assuming that the superpotential is Morse (or a milder semisimplicity assumption), we prove that the wrapped Fukaya category for this class of manifolds satisfies the toric generation criterion, i.e. is split-generated by the natural Lagrangian torus fibres of the moment map with suitable holonomies. In particular, the wrapped category is compactly generated and cohomologically finite. The proof uses a deformation argument, via a generic generation theorem and an argument about continuity of eigenspaces. We also prove that for any closed Fano toric manifold, if the superpotential is Morse (or a milder semisimplicity assumption) then the Fukaya category satisfies the toric generation criterion. The key ingredients are non-vanishing results for the open-closed string map, using tools from the paper by Ritter-Smith (we also prove a conjecture from that paper that any monotone toric negative line bundle contains a non-displaceable monotone Lagrangian torus). We also need to extend the class of Hamiltonians for which the maximum principle holds for symplectic manifolds conical at infinity, thus extending the class of Hamiltonian circle actions for which invertible elements can be constructed in SH(M)., Comment: 71 pages (51 pages + appendices). Version 3 (updated compared to the published version): fixed a typo in Theorem 1.5(2), made notation consistent in (1.6) and (1.7), fixed SH computation in Cor 4.16, added two footnotes: one for Lemma 1.6, and one in Section 3.2

We build the wrapped Fukaya category W(E) for any monotone symplectic manifold, convex at infinity. We define the open-closed and closed open-string maps. We study their algebraic properties and prove that the string maps are compatible with the eigenvalue splitting of W(E). We extend Abouzaid's generation criterion from the exact to the monotone setting. We construct an acceleration functor from the compact Fukaya category which on Hochschild (co)homology commutes with the string maps and the canonical map from quantum cohomology QH(E) to symplectic cohomology SH(E). We define the QH(E)- and SH(E)-module structure on the Hochschild (co)homology of W(E) which is compatible with the string maps. The module and unital algebra structures, and the generation criterion, also hold for the compact Fukaya category F(E), and also hold for closed monotone symplectic manifolds. As an application, we show that the wrapped category of any monotone negative line bundle over any projective space is proper (cohomologically finite). For any monotone negative line bundle E over a toric Fano variety, we show that SH(E) is non-trivial and that W(E) contains an essential non-displaceable monotone Lagrangian torus., Comment: 78 pages, 31 figures. Version 7: Corrected the claim in 12.3. Final publication in Selecta Mathematica available at Springer via http://dx.doi.org/10.1007/s00029-016-0255-9

Let M be the total space of a negative line bundle over a closed symplectic manifold. We prove that the quotient of quantum cohomology by the kernel of a power of quantum cup product by the first Chern class of the line bundle is isomorphic to symplectic cohomology. We also prove this for negative vector bundles and the top Chern class. We explicitly calculate the symplectic and quantum cohomologies of O(-n) over P^m. For n=1, M is the blow-up of C^{m+1} at the origin and symplectic cohomology has rank m. The symplectic cohomology vanishes if and only if the first Chern class of the line bundle is nilpotent in quantum cohomology. We prove a Kodaira vanishing theorem and a Serre vanishing theorem for symplectic cohomology. In general, we construct a representation of \pi_1(Ham(X,\omega)) on the symplectic cohomology of symplectic manifolds X conical at infinity., Comment: 53 pages; version 3: improved discussion of maximum principle for negative vector bundles. The final version is published in Advances in Mathematics

We construct the TQFT on symplectic cohomology and wrapped Floer cohomology, possibly twisted by a local system of coefficients, and prove that the TQFT respects Viterbo restriction maps and the canonical maps from ordinary cohomology. We also construct the module structure of wrapped Floer cohomology over symplectic cohomology. These constructions yield new applications in symplectic topology relating to the Arnol'd chord conjecture and to exact contact embeddings. We prove that if a Liouville domain M admits an exact embedding into an exact convex symplectic manifold X, and the boundary of M is displaceable in X, then the symplectic cohomology of M vanishes and the chord conjecture holds for any Lagrangianly fillable Legendrian lying in the boundary. The TQFT respects the isomorphism between the symplectic cohomology of a cotangent bundle and the homology of the free loop space, so it recovers the TQFT of string topology. Finally, we use the TQFT to prove that symplectic cohomology vanishes iff Rabinowitz Floer cohomology vanishes., Comment: 86 pages (56 + appendix), 19 figures. The final version is to appear in Journal of Topology. Version 5 is the last arxiv version: exposition has been improved and more details are now provided in many proofs. The maximum principles and no escape Lemmas are now in the final section for ease of reference

We prove that the only exact Lagrangian submanifolds in an ALE space are spheres. ALE spaces are the simply connected hyperkahler manifolds which at infinity look like C^2/G for any finite subgroup G of SL(2,C). They can be realized as the plumbing of copies of the cotangent bundle of a 2-sphere according to ADE Dynkin diagrams. The proof relies on symplectic cohomology., Comment: 35 pages, 3 figures, minor changes and corrected typos

Let L be an exact Lagrangian submanifold inside the cotangent bundle of a closed manifold N. We prove that if N satisfies a mild homotopy assumption then the image of \pi_2(L) in \pi_2(N) has finite index. We make no assumption on the Maslov class of L, and we make no orientability assumptions. The homotopy assumption is either that N is simply connected, or more generally that \pi_m(N) is finitely generated for each m \geq 2. The result is proved by constructing the Novikov homology theory for symplectic cohomology and generalizing Viterbo's construction of a transfer map between the homologies of the free loopspaces of N and L., Comment: 27 pages; added two new sections (non-simply connected cotangent bundles, unorientable setup). The final version is published in Geometry & Topology 13, 2009