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? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T08:10:15Z http://mathoverflow.net/feeds/question/51289 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51289/can-the-class-of-the-canonical-bundle-be-recovered-from-the-total-space-of-the-co 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? Ben Webster 2011-01-06T07:53:27Z 2011-01-07T01:00:40Z <p>This is a somewhat speculative question, so bear with that (or not, as is your preference). </p> <p>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. </p> <p>Now, $T^*X$ is homotopy equivalent to $X$, so this also a well-defined class in $H^2(T^*X;\mathbb Z)$. </p> <blockquote> <p>Is there some canonical way of getting this class which only uses the geometry of $T^*X$?</p> </blockquote> <p>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 </p> <blockquote> <p>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).</p> </blockquote> <p><strong>EDIT:</strong> 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.</p> <p>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?</p> http://mathoverflow.net/questions/51289/can-the-class-of-the-canonical-bundle-be-recovered-from-the-total-space-of-the-co/51291#51291 Answer by Torsten Ekedahl for 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? Torsten Ekedahl 2011-01-06T09:07:15Z 2011-01-06T09:07:15Z <p>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 $2\dim X$.</p>