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 Teichmuller curve. This is the most common definition of Teichmuller 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 Teichmuller 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 Teichmuller curve as an affine invariant submanifold.
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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 $2$.
answered Feb 23 '18 at 1:58
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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? $\endgroup$ – Angy Feb 23 '18 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 ) $\endgroup$ – Dmitri Panov Feb 23 '18 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. $\endgroup$ – Angy Feb 23 '18 at 11:56
