Abstract
We construct a multiplicative spectral sequence converging to the symplectic cohomology ring of any affine variety X, with first page built out of topological invariants associated to strata of any fixed normal crossings compactification \((M,{\mathbf {D}})\) of X. We exhibit a broad class of pairs \((M,{\mathbf {D}})\) (characterized by the absence of relative holomorphic spheres or vanishing of certain relative GW invariants) for which the spectral sequence degenerates, and a broad subclass of pairs (similarly characterized) for which the ring structure on symplectic cohomology can also be described topologically. Sample applications include: (a) a complete topological description of the symplectic cohomology ring of the complement, in any projective M, of the union of sufficiently many generic ample divisors whose homology classes span a rank one subspace, (b) complete additive and partial multiplicative computations of degree zero symplectic cohomology rings of many log Calabi-Yau varieties, and (c) a proof in many cases that symplectic cohomology is finitely generated as a ring. A key technical ingredient in our results is a logarithmic version of the PSS morphism, introduced in our earlier work Ganatra and Pomerleano, arXiv:1611.06849.
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Notes
When \({\mathbf {D}}= D\) is a smooth divisor, this is standard and in the literature, compare [Sei08a, eq. (3.2)].
We remind the reader that for any (convergent) spectral sequence associated to a filtered dg or \(A_\infty \) algebra \(F^{p}C^{\bullet }\), the algebra structure on the final \(E_\infty \) page, the associated graded algebra \(gr_F(H^*(C^{\bullet }))\), need not be isomorphic as rings to \(H^*(C^{\bullet })\) even if it is additively isomorphic: in general \(H^*(C^{\bullet })\) could be a non-trivial deformation of \(gr_F(H^*(C^{\bullet }))\).
With respect to the (homological shadow of) action filtration inducing the spectral sequence (1.1).
The terminology “topological pair” was introduced in our earlier work [GP16], where its usage is slightly less general than what is used here.
An alternative naming convention would be to say a pair \((M,{\mathbf {D}})\) is r-topological if (for some \(J_0\)) it contains no m-pointed relative \(J_0\)-holomorhic spheres for all \(m \le r\). We expect such generalized notions to be useful in the study of higher-arity operations on symplectic cohomology, such as \(A_\infty \) or \(L_{\infty }\) structures.
Recall \({\text {min}}_{{\mathbf {D}}} R^{\vec {\epsilon }} > 1\), because \(R^{\vec {\epsilon }} =1\) along \(\partial \bar{X}_{\vec {\epsilon }}\).
Once more, this requires establishing that relevant Floer trajectories are a priori bounded away from \({\mathbf {D}}\), which is a consequence of the maximum principle mentioned earlier.
Let \(\underline{S}:= u_2 \setminus \bar{S}\). By Stokes, \(E_{top}(\underline{S})=A_a(x_2)-(\int _{\partial \bar{S}}u^*\theta - H_{s,t}dt)\). As the topological energy of \(u_1 \cup u_2\) is \(A_{a}(x_2)-A_{a}(x_1)\), the equation holds.
We do this so that our conventions for cohomological spectral sequences match those found in standard textbooks such as [McC01].
We thank Alessio Corti and Nicolo Sibilla for suggesting this point of view.
In [GP16], we do not make any assumptions on the Nijenhuis tensor of our almost complex structure, however the arguments explained there apply without modification.
We do this to rely on standard gluing results in the literature on (local) Floer cohomology. It is likely not necessary.
It is still true that \(\gamma \) and \(\gamma '\) must have the same Hamiltonian action.
Both the ball \(B_r(s_0)\) and definition of area are with respect to the the metric induced by \(\omega \) and J.
By restricting to the Floer solutions in the complement of \(z=0\), one can also place these maps into the more general framework of Morse-Bott Fredholm theory as explained in §6.1 of [DL18].
Recall that these orbits y may be degenerate; compare the discussion just below Lemma 4.6.
In particular, no sphere bubbling can occur in the limiting process.
Or \({\mathbb {Z}}/2k{\mathbb {Z}}\)-graded, or fractionally-graded when \(c_1(X)\) is torsion, etc. though we leave these details to the reader.
The argument given in [FT17, Lemma 4.9] begins by noting that it suffices to verify (5.1) for the orders relative each smooth component \(D_i\) of \({\mathbf {D}}\) individually, immediately reducing to the case of a single smooth divisor. Once in the smooth case, the result appears in various places; see e.g., [IP03, Proposition 7.3] for a separate treatment.
The only difference is that, given that the points \(z_1\) and \(z_2\) collide, any possible stable sphere bubble will have at most 3 (rather than 2) external marked points. The same analysis thus requires the extra “multiplicatively” topological hypothesis to rule such bubbles out.
This notion of log Calabi-Yau pair is stricter than the usual usage in birational geometry [GHK15a].
This follows because each curve \(u \in \mathcal {M}_{0,2}(A,\mathbf {v}_I)^{o}\) is necessarily somewhere injective.
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Acknowledgements
Our use of normal crossings-adapted symplectic and Liouville structures and subsequent Morse-Bott analysis borrows extensively from work of Mark McLean [McL16a, McL12a]. We would like to thank him for explanations of his work as well as several other conversations related to this paper. We also benefited greatly from discussions with Strom Borman, Luis Diogo, Yakov Eliashberg, Eleny Ionel, Samuel Lisi, and Nick Sheridan at an early stage of this project; we thank all of them. Finally, we would like to thank an anonymous referee for suggestions that improved the exposition of this article.
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S. G. was partially supported by the National Science Foundation through a postdoctoral fellowship—Grant Number DMS-1204393—and Agreement Number DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
D. P. was supported by EPSRC, Imperial College, University of Cambridge, and UMass Boston.
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Ganatra, S., Pomerleano, D. Symplectic cohomology rings of affine varieties in the topological limit. Geom. Funct. Anal. 30, 334–456 (2020). https://doi.org/10.1007/s00039-020-00529-1
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DOI: https://doi.org/10.1007/s00039-020-00529-1