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What is a simplest example of a manifold $M^{2n}$ that admits two symplectic structures with isotopic almost complex structures, and such that Gromov-Witten invariants of these symplectic structures are different? (unfortunately I don't know any example...) If we don't impose the condition that almost complex structures are isotopic, such examples exist in dimension 6.

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Refined question. Is there a manifold $M^{2n}$ with two symplectic forms $\omega_1$, $\omega_2$, such that the cohomology classes of $\omega_1$ and $\omega_2$ are the same and the corresponding almost complex structures are homotopic, but at the same time the Gromov-Witten invariants are different?

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The examples in Ruan's paper "Symplectic topology on algebraic 3-folds" (JDG 1994) seem to qualify: take any two algebraic surfaces V and W which are homeomorphic but such that V is minimal and W isn't. These are nondiffeomorphic, but VxS2 and WxS2 are diffeomorphic, and Ruan gives lots of examples (starting with V equal to the Barlow surface and W equal to the 8-point blowup of CP2) where the diffeomorphism can be arranged to intertwine the first Chern classes, whence by a theorem of Wall the almost complex structures are isotopic. However, the distinction between the GW invariants between V and W (which holds because V is minimal and W isn't) survives to VxS2 and WxS2, so VxS2 and WxS2 aren't symplectic deformation equivalent.

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  • $\begingroup$ Thanks a lot! I heard about the example, but really did not thought that you can interween Chern classes by a diffeomorphism :).But it seems to be the case, ideed... $\endgroup$ – Dmitri Panov Oct 24 '09 at 9:54
  • $\begingroup$ In fact, just to be sure that I completely understood the answer. If we denote by x1,...,x9 the second cohomology classes coming from Barlow surface, and by y the class coming from S^2, it seems that the automorphism of H^2(VxS^2) , xi --> -xi, y --> y preserves the 3-form defined on H^2(VxS^2) and indeed exchanges the two Chern classes. In order to see that this gives you a diffeo of VxS^2, you need to check that first Pontriagin class is also preserved. Is this how one should proceed? $\endgroup$ – Dmitri Panov Oct 24 '09 at 13:06
  • $\begingroup$ Yes, that's right. Expressing things very slightly differently, it's enough to find an isomorphism H^*(V)->H^*(W) of the second cohomologies of the 4-manifolds which intertwines the first Chern classes--it will then automatically preserve the Pontrjagin classes of the 4-manifolds just by Euler characteristic considerations. Then the isomorphism H^*(VxS^2)->H^*(WxS^2) constructed by trivially extending the one defined on the 4-manifold level will preserve all the relevant classes. $\endgroup$ – Mike Usher Oct 24 '09 at 14:21
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Here is an answer to the REFINED question given to me by Richard Thomas. In this refined version one looks for an example such that the cohomology classes of two symplectic forms coincide.

In a later paper 1996, Duke Vol. 83 TOPOLOGICAL SIGMA MODEL AND DONALDSON TYPE INVARIANTS IN GROMOV THEORY, Ruan proved that such refined examples exist. He says in this paper that for product examples $V\times S^2$ from the paper in JDG 1994 (cited by Mike Usher) he does not know whether the classes of constructed symplectic forms can coincide as well. In fact this does not seem very plausible.

These refined examples are two $3$-dimensional Calabi-Yau manifolds, constructed by Mark Gross. The construction is described in the paper of Mark Gross (1997): "The deformation space of Calabi-Yau $n$-folds with canonical singularities can be obstructed". One $3$-dimensional Calabi-Yau is a smooth anti-canonical section of $\mathbb CP^1\times \mathbb CP^3$ and the over is a smooth anti-canonical section of the projectivsation of the bundle $O(-1)+O+O+O(1)$ over $\mathbb CP^1$.

The construction of Gross is recalled on pages 47-48 of http://xxx.soton.ac.uk/PS_cache/math/pdf/9806/9806111v4.pdf

Using Wall's theorem Ruan proves that these Calabi-Yau manifolds are differomorphic. Then he studies the quantum cohomology rings of these Calabi-Yaus and proves that they are different.

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