# Integrable compatible complex structures

In reading Gauduchon's notes (cannot link them, anyway it is standard material) I ran into the following construction.

Let $(M, \omega_0)$ be a compact symplectic manifold which is fixed. An almost complex structure $J$ on $M$ is said to be compatible with $\omega_0$ if $\omega_0( \cdot, J \cdot)$ is a Riemannian metric. Let $\mathcal J$ be the space of such $J$'s and let $\mathcal J_\text{int}$ be the subspace of integrable ones.

Then a Riemannian metric and a complex structure are constructed on the infinite dimensional manifold $\mathcal J$ which turns out to be Kaehler and it is proved that $\mathcal J_\text{int}$ is a Kaehler submanifold, that is its tangent space at $J$, which is constructed by deriving the equation $N_J=0$, is invariant under the complex structure on $\mathcal J$.

My question is: can it happen that $\mathcal J_\text{int}$ is made by isolated points? I can see no way to say that its tangent space has positive dimension.

Thank you

David

• I'm pretty sure it can. Kodaira and Siu have results proving the uniqueness of the complex Kahler structure on projective space, so your $\mathcal I_{\mathrm{int}}$ should be a point for the underlying smooth $M$ with the standard Fubini-Study symplectic form $\omega_0$. For a very elementary example, take the projective line $\mathbb P^1$. – Gunnar Þór Magnússon Jan 16 '14 at 16:03
• @Gunnar: uniqueness is up to isomorphism. And in the case of $P^1$, $\mathcal{I}_int=\mathcal{I}$. In general, $\mathcal{I}_int$ is invariant by the whole symplectomorphism group of $(M,\omega)$, so it is infinite dimensional once it is nonempty (the stabilisers are finite dimensional -- riemannian isometry groups). – BS. Jan 16 '14 at 16:18
• Gauduchon also introduces another space, when you start with a Kaehler structure $(J_0, \omega_0)$. Namely he defines the space $\mathcal C$ of the $J$'s which are of the form $\phi J_0$ for some $\phi \in Diffeo(M)^0$. Do you mean that this $\mathcal C$ is a point for the Kodaira-Siu examples? – David P Jan 16 '14 at 16:49
• @David Petrecca: This $\mathcal{C}$ will be at least as big as the $Ham(M,\omega_0)$-orbit of $J_0$ (i.e. infinite-dimensional). A very nice study of $\mathcal{J}_{int}$ in some special cases (which in particular discusses your set $\mathcal{C}$) can be found in the papers: arxiv.org/abs/math/0610436 (in particular section 2) and arxiv.org/abs/math/0507382 by Abreu-Granja-Kitchloo (who prove $\mathcal{J}_{int}$ is contractible if $X$ is a rational ruled complex surface). – Jonny Evans Jan 16 '14 at 22:38

The space of integrable $J$ is acted on by the diffeomorphism group, making it infinite dimensional. Symplectomorphisms will preserve the compatibility, so the orbit of an integrable $J$ under the symplectomorphism group is still infinite dimensional. The quotient by the symplectomorphism group is not known to be even an orbifold, as far as I know.
Even simpler, for genus 1, the two-torus, we know that the set of moduli of complex structures is identified with the modular surface (upper half-plane)/$Sl(2,{\mathbb Z})$ which has complex dimension 1. To be explicit, think of the standard torus as ${\mathbb R}^2/{\mathbb Z}^2$ endowed with the symplectic structure $\omega_0 = dx \wedge dy$ where $x, y$ are standard coordinates for ${\mathbb R}^2$. Consider the usual description of the moduli space of elliptic curves, eg. from Ahlfor's text on complex analysis. The Teichmuller space is the upper half plane and a point $\tau = \tau_0 + i \tau_1$ in this plane (so $\tau_1 > 0$) gets sent to the elliptic curve ${\mathbb C}/{\mathbb Z}1 \oplus {\mathbb Z} \tau$. The complex structure on this curve is induced by projection from the standard complex structure on ${\mathbb C}$. Let $A$ be the real linear invertible shear which is the identity on the x axis and maps the unit y vector ${\partial}_x$ to $(\tau_0, \tau_1)$. Then $A$ defines a diffeo.from our standard torus onto this $\tau$-torus. Pulling back the $\tau$-complex structure using $A^{-1}$ we get a new complex structure $J_{\tau}$ on our standard torus given by $J_{\tau}(\partial_x) = e_{\tau}$ and $J_{\tau}e_{\tau} = -\partial_x$ where $e_{\tau}= (-\tau_0/\tau_1)\partial_x + (1/\tau_1) \partial_y$. The $J_{\tau}$ form a continuum of inequivalent complex structures on the standard torus. (What about the metric condition? I compute the $\omega_0 ( \cdot, J_{\tau} \cdot)$ metric to be $(1/\tau_1) dx^2 + (\tau_0/\tau_1) dy^2$, a little wierd. I would've expected a quadratic form which is positive definite iff $\tau_1 > 0$, not iff both $\tau_0 , \tau_1 > 0$... . Maybe someone else can shed light on this (?))