What are some known techniques and examples of exotic symplectic structures on a fixed smooth closed 4-manifolds [by exotic I mean two symplectic structures that are not symplectomorphic]. This sounds like a very vague question, but let me elaborate my concern here.
Chern-class corresponding to the symplectic structure is an important tool to distingusih different symplectic structures on a fixed smooth 4-manifold. And for that reason I want to understand one example of smooth closed 4-manifold $X$ with two different symplectic structures $\omega_1$ and $\omega_2$ whose corresponding Chern-classes are identical.
On the other hand if we remove the closedness condition, then one can construct exotic symplectic structures on $\mathbb R^4$ by starting with $\mathbb R^3$ with two contact structure $\xi_{std}$ and $\xi_1$ and consider their symplectizations. Check this nice work of Casal's for details https://arxiv.org/pdf/1402.7099v1.pdf .
It will be great if someone can explain me some known techniques and ideas along this line.
