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Let us assume first of all that we are in the affine case (we can worry about globalization later) and that we have $X$ affine over $S$, where $S$ is some unspecified scheme (but in practice probably the spectrum of a field), with $X=\mathrm{Spec}(A)$ (thus $A$ is an $\mathcal{O}_S$-algebra). We are emphatically not assuming $X$ to be smooth over $S$.

Assume that we are given a map $\mathfrak{g}\to\mathrm{Der}_S(A)$ of Lie algebras and that we are viewing $\mathfrak{g}$ as a Lie-sub-algebra of $\mathrm{Der}_S(A)$.

In analogy with the differential-geometric case we can interpret this as a distribution on $X$ and so we can ask: what are the integral subschemes of this distribution? Specifically, is there, through every point, a unique integral subscheme? And even more importantly, what can go wrong in the singular points and can we "integrate" this action to an analogue of a Lie groupoid?

I'm seriously betting the answer to most of the above is a resounding "NO!" but I'm curious to know what can go wrong and what is known to go wrong? In short: what is known concerning this? Can one form something like "$X/\mathfrak{g}$"?

Finally, let me iterate that I'm not assuming $X$ to be $S$-smooth.

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1 Answer 1

I don't think that there is any hope for "integral subschemes" in the usual sense even in the nonsingular case, because it is exceptionally rare for a polynomial vector field to have algebraic trajectories. For example, consider an "irrational line on a torus" given by the vector field

$$z\frac{\partial}{\partial z}-\lambda w\frac{\partial}{\partial w}$$

with irrational $\lambda.$ Its integral curves have local equations $z^\lambda w=C,$ hence these integral curves are not algebraic.

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Sure. Or, perhaps even more simply, in the x y plane look at $\frac{\partial}{\partial x}-y\frac{\partial}{\partial y}$. –  Tom Goodwillie Sep 4 '10 at 14:09
    
This is an interesting and delicate question, especially in an arithmetic context. One version of it is Grothendieck's $p$-curvature conjecture. See J-B Bost, Algebraic leaves of algebraic foliations over number fields, Publ. Math. IHES a few years ago, where he looks at the integrability of vector fields on abelian varieties over number fields. –  inkspot Sep 4 '10 at 15:57

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