Are schematic fixed points of a torus action on an affinized twistor deformation flat? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T12:07:40Z http://mathoverflow.net/feeds/question/49686 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49686/are-schematic-fixed-points-of-a-torus-action-on-an-affinized-twistor-deformation Are schematic fixed points of a torus action on an affinized twistor deformation flat? Ben Webster 2010-12-16T22:19:28Z 2011-01-30T00:05:32Z <p>This is a follow-up to <a href="http://mathoverflow.net/questions/43255/" rel="nofollow">some</a> <a href="http://mathoverflow.net/questions/43438/under-what-hypotheses-are-schematic-fixed-points-of-a-flat-deformation-themselves" rel="nofollow">earlier</a> <a href="http://mathoverflow.net/questions/43581/" rel="nofollow">questions</a> about flatness of schematic fixed points of certain deformations. Since I could never come up with good enough hypotheses in those examples, let me try a <strong>much</strong> more restrictive set of examples.</p> <p>Assume</p> <ul> <li>$X$ is a quasi-projective symplectic variety which is projective over its affinization (algebraically convex) with $H^i(X;\mathcal{O}_X)=0$ for $i>0$ over a field $k$ of characteristic 0,</li> <li>$L$ is a line bundle on $X$,</li> <li>there is a Hamiltonian action of a torus $T$ on $X$, and a commuting action of $\mathbb{G}_m$ which is conformal of weight $n$ (the symplectic form is a weight vector of weight $n$ for it) and has positive weight on all global functions.</li> </ul> <p>A <a href="http://arxiv.org/pdf/math/0312134v2" rel="nofollow">theorem of Kaledin</a> tells us that $X$ possesses an essentially unique <strong>twistor deformation</strong>, that is a flat deformation $\mathcal X\to S$ where $S=\mathrm{Spec} ( k[[t]])$ with an extension of $L$ to a line bundle $\mathcal{L}$ such that </p> <ul> <li>$\mathcal X$ is Poisson with the coordinate $t$ being Poisson central.</li> <li>$\mathcal L$ is a Poisson module with $\{t,-\}$ inducing the identity map.</li> <li>The action of $T\times \mathbb{G}_m$ extends to an action of the Lie algebra of this group on $\mathcal X$ lifting the action on $S$ where $T$ acts trivially and $\mathbb{G}_m$ acts with weight 1 on $t$ (here you see why I had to use the Lie algebras instead of the algebraic groups). </li> <li>The global functions $A=\Gamma(\mathcal X)$ are also flat over $k[[t]]$, $t$-adically complete and normal.</li> </ul> <p>If you're having trouble imagining what this looks like, the point is to "invert Hamiltonian reduction." The total space of $\mathcal{L}$ minus its 0-section is symplectic, and its induced $\mathbb{G}_m$-action (by scalar multiplication; this has nothing to do with the one we introduced before) is Hamiltonian with moment map $t$. The variety $X$ is the Hamiltonian reduction for this action. </p> <p>We can take schematic fixed points $B$ of the Lie algebra $\mathfrak t$ acting on $A$; this is the quotient of $A$ by the ideal generated by $y\cdot a-a$ for $y\in \mathfrak t, a\in A$. This is an algebra over $k[[t]]$. </p> <blockquote> <p>Is $B$ flat over $k[[t]]$?</p> </blockquote>