1
$\begingroup$

It is well-known that the Gromov-Witten invariants and their Floer-theoretic counterpart of symplectic manifolds have rich algebraic structures. However, sometimes it's quite useful even by considering the moduli spaces of solutions of $J$-holomorphic or Floer equations. For example:

  1. Gromov's original characterization of homotopy types of the symplectomorphism groups of $\mathbb{CP}^2$ and $\mathbb{P}^1 \times \mathbb{P}^1$, by considering foliations by $J$-holomorphic curves;

  2. McDuff' construction of cohomologous but non-symplectomorphic symplectic structures, by looking at bordism classes of certain moduli spaces;

  3. Hofer's proof of the degenerate Arnol'd conjecture for symplectically aspherical manifolds;

  4. McDuff's proof of the uniqueness of the symplectic filling of standard contact $S^{2n-1}$ up to diffeomorphism, and in the similar spirit, Wendl's result on fillings of planar open book decompositions of $3$-manifolds;

  5. Abouzaid's construction of the bounding parallelizable manifold of exact Lagrangian spheres in $T^* S^{4k+1}$.

Question: modulo direct generalizations and improvements of the above results, and Floer-homotopical considerations, what are some other geometric applications of the geometry of the moduli spaces of $J$-holomorphic curves (or solutions to Floer equations)?

$\endgroup$
5
  • 1
    $\begingroup$ The result of Koll’ar and Ruan that uniruledness of compact Kaehler manifolds is invariant under symplectic deformations. $\endgroup$ Feb 23, 2022 at 23:54
  • 1
    $\begingroup$ Also you misspelled the name of “Wendl”. $\endgroup$ Feb 23, 2022 at 23:55
  • 1
    $\begingroup$ In 2, should "deformation equivalent" be "cohomologous"? $\endgroup$
    – Will Sawin
    Feb 24, 2022 at 0:16
  • $\begingroup$ To Jason and Will: thanks, I've corrected the typos. $\endgroup$ Feb 24, 2022 at 0:21
  • 2
    $\begingroup$ Gromov's original paper contains many more such results, all of which have the flavour you describe because Floer homology and GW invariants hadn't been invented! For example, the absence of exact Lagrangians in C^n. Sure you can reprove them now using Floer homology language, but the original arguments are worth reading. Also you could consider Richard Hind's classification of Lagrangian spheres in S^2 x S^2 as another item on your list arxiv.org/abs/math/0311092 $\endgroup$ Feb 24, 2022 at 1:28

1 Answer 1

2
$\begingroup$

I proved that smooth projective planes, in the generalized sense of the axiomatic theory of projective planes, which have dimension 4 are diffeomorphic to the complex projective plane, using a generalization of the notion of pseudoholomorphic curve, and symplectic structures on moduli spaces: McKay, Benjamin, Smooth projective planes. Geom. Dedicata 116 (2005), 157–202. Subsequently the classification of diffeomorphism types of smooth projective planes was completed by Kramer and Stolz.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.