Let $M$ be a smooth variety of even dimension over $\mathbb C$. I am interested in necessary or sufficient conditions such that $X$ is a deformation of a family of cotangent bundles.
In other words, when do we have (or at least locally over $\mathbb C[[t]]/t^2$):
- A smooth map $f: \mathcal{M} \to \mathbb A^1$ with $0$-fiber $f^{-1}(0) \cong M$.
- A smooth map $g: \mathcal{U} \to \mathbb A^1 -0$, such that its relative cotangent bundle $T^*\mathcal{U} \cong \mathcal{M}|_{\mathbb A^1-0}$ over $\mathbb A^1 -0$.
One may ask similar questions for (symplectic) manifolds.
I believe that this is a strong restriction on the topology of $M$, but $M$ is not necessarily a cotangent bundle. And the zero section $\mathcal{U} \to T^* \mathcal{U}$ may extend to $\mathcal{M}$. Motivations come from deformations of symplectic manifolds to cotangent bundles. Thank you for any references or nontrivial examples.
