This is a somewhat speculative question, so bear with that (or not, as is your preference). 


Let $X$ be a smooth projective variety, and let $\omega\_X$ be its canonical sheaf. The Euler class of this line bundle $e(\omega\_X)\in H^2(X;\mathbb Z)$ (which for simplicity we'll assume is torsion-free) defines a distinguished cohomology class. 


Now, $T^\*X$ is homotopy equivalent to $X$, so this also a well-defined class in $H^2(T^\*X;\mathbb Z)$. 



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> Is there some canonical way of getting this class which only uses the geometry of $T^\*X$?
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Of course, "only using the geometry of $T^\*X$" is not really a well-defined notion (so I apologize if I disagree with any of the answerers about what this means). I mostly mean 



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> Is there some way of describing this class which can be applied to other symplectic varieties (possibly with an extra structure, like a dilating $\mathbb{C}^\*$-action).
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**EDIT:** An example of something I would prefer not to use is that $T^\*X$ has the homotopy type of a smooth compact manifold (as I'm interested in examples where this is not the case). Sorry, Torsten.


For an extra twist, I'm most interested in the class of $\frac 12e(\omega\_X)\in H^2(X;\mathbb Q)/H^2(X;\mathbb Z)$, which one can think of as a class in $H^2(X;\mathbb Z/2\mathbb Z)$. Could this have something to do with spin structures and characteristic classes?



Can the class of the canonical bundle be recovered from the total space of the cotangent bundle if one forgets that it is a cotangent bundle?
The reduction mod $2$ of $e(X)$ is the second Stiefel-Whitney class of $X$ which by Wu's formula can be recovered from the homotopy type of $X$ (Steenrod operations and the Poincaré duality for the mod $2$ cohomology algebra of $X$). This can be read off from the mod $2$ cohomology of $T^\ast(X)$. Note that this reconstructs the class purely formally from the Steenrod action on and the multiplication of the mod $2$ cohomology of $T^\ast(X)$ and in no way involves the manifold structure of $T^\ast(X)$. The point is rather that its cohomology behaves as the cohomology of a compact manifold of dimension $2\dim X$.


