Vector bundles on some non-projective surfaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T04:44:34Z http://mathoverflow.net/feeds/question/43700 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43700/vector-bundles-on-some-non-projective-surfaces Vector bundles on some non-projective surfaces Alexander Braverman 2010-10-26T17:27:19Z 2010-10-26T17:27:19Z <p>Let \$X\$ be a smooth projective curve over a field \$k\$ and let \$L\$ be a line bundle on \$X\$. I will denote by \$S\$ the total space of \$L\$ -- this is a smooth surface over \$k\$ containing \$X\$ (as the zero section). Let \$S^0\$ be the complement to the zero section. Let also \$G\$ be some reductive group (you can assume that \$G\$ is \$GL(n)\$ or \$SL(n)\$ if you wish). I need some finiteness results about moduli spaces of \$G\$-bundles on \$S\$ and \$S^0\$. Specifically, I would like to prove (or disprove) the following statements:</p> <p>1) Assume that \$deg(L)&lt;0\$. Consider all \$G\$-bundles \$F\$ on \$S\$ whose restriction to \$X\$ is fixed (i.e. it is isomorphic to a fixed \$G\$-bundle on \$X\$). Is it true that when \$k\$ is finite, the number of isomorphism classes of such bundles is finite? For general \$k\$, is it true that this (non-algebraic) stack can be covered by a scheme of finite type over \$k\$? I think that it is easy to prove this when we replace \$S\$ by the formal neighbourhood of \$X\$ in \$S\$ (by deformation theory), but is it true for \$S\$ itself?</p> <p>2) Consider now \$G\$-bundles on \$S^0\$ and assume that \$deg(L)\neq 0\$ (note that \$S^0\$ doesn't change when we replace \$L\$ by \$L^{-1}\$, so you may as well assume that \$deg(L)&lt;0\$). In the case when \$k\$ is finite I would like to prove the number of isomorphism classes of such bundles is finite. This is easy if \$G=GL(1)\$ (i.e. when we talk about line bundles); also it is not difficult when \$X={\mathbb P}^1\$ (in this case the number of isomorphism classes is finite for any \$k\$ -- not necessarily finite). For general \$k\$ there should probably be a statement that the (non-algebraic) stack \$Bun_G(S^0)\$ can be covered by a scheme of finite type over \$k\$ (but again, I just need the statement for the finite field).</p>