$\DeclareMathOperator\SL{SL}$Let $M_g$ be the moduli space of curves, $\Omega M_g$ the total space of the bundle of holomorphic 1-forms and $\pi: \Omega M_g\to M_g$ the natural projection. On $\Omega M_g$ there's an action of $\SL_2(\mathbb{R})$. When the image $\pi(\SL_2(\mathbb{R})\cdot x)\subset M_g$ of an orbit is a closed algebraic curve of $M_g$ it's called a Teichmüller curve. This is the most common definition of Teichmüller curve. On the other hand sometimes they are also referred to as affine invariant submanifolds (i.e. complex submanifolds of a stratum of $\Omega M_g$ which are $\SL_2(\mathbb{R})$-invariant and linear in the period coordinates). My confusion stems from the following fact: if $(X,\omega)$ generates a Teichmüller curve then its stabiliser under the action of $\SL_2(\mathbb{R})$ is a lattice and the orbit would have real dimension equal to $\dim \SL_2(\mathbb{R})=3$, so I don't see how to interpret a Teichmüller curve as an affine invariant submanifold.
2 Answers
The point here is that you have not only an $SL(2,\mathbb R)$ action, but a $GL(2,\mathbb R)$ action as well. So the orbit is locally complex affine subspace of complex dimension $2$.
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$\begingroup$ Thank you, this is helpful. So in fact when people speak about "Teichmuller curves inside a stratum" they refer to an object of complex dimension 2? $\endgroup$– AngyFeb 23, 2018 at 11:33
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$\begingroup$ I am not sure how to interpret this phrase, is it written somewhere? On the other hand $\Omega M_g$ indeed has various strata preserved by the action. Also, you can embedd your Teichmuller curve in $\Omega M_g/\mathbb C^*$ (as a complex $1$-dim submanifold of course, the complex two-dimensional surface will then by a $\mathbb C^*$-bundle over it ) $\endgroup$ Feb 23, 2018 at 11:47
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$\begingroup$ I agree. Sometimes Teichmuller curves are cited as examples of affine invariant submanifolds and I wasn't sure about the details of this phrase. $\endgroup$– AngyFeb 23, 2018 at 11:56
Veech coined the term "Teichmueller curve" and McMullen popularized it. A Teichmueller curve is a curve in the classical Riemann moduli space that is totally geodesic with respect to the Teichmueller metric. It is known that the restriction of the metric to the curve has constant negative curvature, and this leads one to $SL_2(R)$. (See McMullen - Billiards and Teichmüller curves.) But the cotangent bundle of this curve naturally immerses into the cotangent bundle of the Riemann moduli space which, by Serre duality, is the moduli space of quadratic differentials. In many (all?) interesting cases the latter surface is actually contained in a locus that consists of squares of abelian differentials. The usual stratification of these moduli spaces is by the divisor data of the quadratic/abelian differential.
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$\begingroup$ Does it have something to do with spectral curves in Hitchin's integrable systems ? $\endgroup$ Apr 9 at 16:52
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$\begingroup$ @ Alexander Chernov. Hmmm....I don't know. The Teichmueller curve lies in the classical Riemann moduli space, and Hitchen considers the moduli space of stable $G$-bundles. Do the latter moduli spaces have a Kobayashi metric? If so, then a `Teichmueller curve' would be a curve that is totally geodesic w.r.t. this metric. $\endgroup$ Apr 9 at 19:53
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$\begingroup$ Okay, I see, seems different things - in Hitchin's setup - spectral curve(S) (not one , but family) can be mapped to cotangent bundle of moduli space , probably non of them is contrained onto zero section = moduli space itself. $\endgroup$ Apr 9 at 20:03