I am interested in stacks of vector bundles on varieties and how deformations of the variety (including non-commutative ones) reflect themselves in deformations of the stack of vector bundles.

I will denote by $X$ the space I am interested in. Lets take it to be either $Spec(\mathbb{C}[x,y]/m^n)$ where $m$ is the ideal $(x,y)$ or the formal completion of $Spec(\mathbb{C}[x,y])$ at the origin. Lets consider the stack of line bundles on $X$. Up to isomorphism, all line bundles are trivial so this stack is just $[pt/G]$ where $G=R^{\times}$ and $R= \mathbb{C}[[x,y]]$ or $R= \mathbb{C}[x,y]/m^n$.

Here, we think of $R^{\times}$ as an algebraic group. As far as I understand, the families of stacks over $Spec(\mathbb{C}[\epsilon]/\epsilon^{2})$ which restrict to this stack $[pt/G]$ on the central fiber (mod equivalence) are given by $H^{2}(G,\mathfrak{g})$, where $\mathfrak{g}$ is the Lie Algebra of $G$, and the action is adjoint which here is trivial.

The space $H^{2}(G,\mathfrak{g})$ seems to be unpleasantly large. If I let $K$ be the kernel of the projection map of $G$ to $\mathbb{C}^{\times}$ and $\mathfrak{k}$ be the lie algebra of $K$ then I get

$H^{2}(G,\mathfrak{g}) \cong Alt_{\mathbb{C}}^{2}(\mathfrak{k}, \mathfrak{g})= \wedge^{2}\mathfrak{k}^{\vee} \otimes_{\mathbb{C}} \mathfrak{g}$

One sees this using short exact sequence of groups $0\to \mathfrak{k} \stackrel{exp}\to G \to \mathbb{C}^{\times} \to 1. $

The deformations of $G$ as a group seem to be badly behaved, in particular this space is way larger than the deformations of $R$ as an algebra. So it seems like most of the deformations of the stack are not modular (do not come from deformations of $X$). In the case that $R= \mathbb{C}[[x,y]]=\mathfrak{g}$, this space $H^{2}(G,\mathfrak{g})$ is not even a finitely generated $R$ module.

Should this be remedied in some way by changing the set up? If instead of groupoids of vector bundles, I could keep the same objects but allow more morphisms. In other words, I can consider stacks in abelian categories of coherent sheaves. This would seem to improve the situation.

  • $\begingroup$ I see now that the group $K$ that comes from projection of $G$ onto the constants is also a ring called the Witt Ring of $R$, this could be useful somehow. $\endgroup$ – Oren Ben-Bassat Mar 29 '11 at 14:10

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