line bundles on smooth affine variety - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T18:00:59Z http://mathoverflow.net/feeds/question/21259 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/21259/line-bundles-on-smooth-affine-variety line bundles on smooth affine variety Vladimir 2010-04-13T21:05:29Z 2010-04-17T16:41:28Z <p>Let L be a line bundle on a smooth affine variety X (say, over complex numbers). Is it true that L always admits a FLAT algebraic connection?</p> http://mathoverflow.net/questions/21259/line-bundles-on-smooth-affine-variety/21262#21262 Answer by Bugs Bunny for line bundles on smooth affine variety Bugs Bunny 2010-04-13T21:27:50Z 2010-04-13T21:27:50Z <p>No, there is no reason for an an $O$-module, even locally free rank 1, to be a $D$-module.</p> http://mathoverflow.net/questions/21259/line-bundles-on-smooth-affine-variety/21264#21264 Answer by Torsten Ekedahl for line bundles on smooth affine variety Torsten Ekedahl 2010-04-13T21:39:58Z 2010-04-13T21:39:58Z <p>No, any line bundle with a flat connection has a trivial rational Chern class. Now, take any smooth connected projective variety $X$ for which the Chern classes of line bundles form a group of rank $r$ larger than $1$. Removing an irreducible ample divisor $D$ from $X$ gives a smooth affine variety for which the Chern classes form a group of rank $r-1$. A specific example is $\mathbb P^1\times\mathbb P^1$ but there are lots of others of any dimension $>1$.</p> http://mathoverflow.net/questions/21259/line-bundles-on-smooth-affine-variety/21675#21675 Answer by Vladimir for line bundles on smooth affine variety Vladimir 2010-04-17T16:41:28Z 2010-04-17T16:41:28Z <p>I knew the argument with the truncated De Rham complex, but couldn't cook up an example - thank you, Torsten. </p>