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Suppose $M$ is a compact smooth manifold and $V$ is a compact complex manifold. I want to show that the spaces $C^{k,\alpha}(M,V)$ and $W^{k,p}(M,V)$ (the latter for $kp>\dim M$) are complex analytic Banach manifolds, compatibly with the tautological complex structure on the tangent spaces of these smooth Banach manifolds (for $f:M\to V$, the tangent space at $f$ is the space of sections of $f^*T_V$ of regularity $C^{k,\alpha}$ or $W^{k,p}$).

I saw this statement in a paper of Douady (available here: http://www.numdam.org/item/SB_1964-1966__9__7_0). It is stated in the beginning of section 3 without any indication of proof.

Why does the manifold of maps have a complex structure if the target is complex?

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  • $\begingroup$ Did you check the "convenient setting of global analysis" huge book by Kriegl-Michor? Typically these types of facts are found there. $\endgroup$ Sep 4 '18 at 18:17
  • $\begingroup$ I just took a look at the section on manifolds of maps, but was unable to find this fact. I think my answer below settles the question though. $\endgroup$ Sep 4 '18 at 21:52
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The usual way to give charts on $C^{k,\alpha}(M,V)$ or $W^{k,p}(M,V)$ is to fix a connection (usually coming from a metric) on the tangent bundle $T_V$, and then using exponentiation with respect to this. It is easy to see that what we need for holomorphicity of this chart is the holomorphicity of the individual exponential maps $\exp_x:T_xV\to V$ (at least near $0$). Below, I show how to construct a smooth map $e:T_V\to V$ (defined on a neighborhood of the zero section) such that each $e_x:=e|_{T_xV}$ maps $0$ to $x$, has derivative equal to identity at $0$ and is holomorphic near $0$. This will clearly suffice.

Let $x\in V$ be any point. Pick a local holomorphic coordinate system $z_1,\ldots,z_n$ near $x$ which identifies $x$ with $0$ and a neighborhood $U$ of $x$ with an open set $\Omega\subset \mathbb C^n$. Without loss of generality, let us assume that $B_2=\{z\in\mathbb C^n\;|\;\|z\|\le 2\}\subset\Omega$. Choose a cutoff function $\chi:\mathbb C^n\times\mathbb C^n\to\mathbb R$ which is $1$ in a neighborhood of $B_1\times B_1$ and zero outside $B_2\times B_2$. Then, defining $\sigma(z,w) = (z,\chi(z,w)\cdot(w-z))$ gives a function $\sigma:\mathbb C^n\times \mathbb C^n\to\mathbb C^n\times\mathbb C^n = T_{\mathbb C^n}$ which has the following properties. For each $z$, $\sigma(z,z)=(z,0)$, $w\mapsto\sigma(z,w)$ is holomorphic for $w$ close to $z$ (for $z\in B_1$) and $\frac{\partial\sigma}{\partial w}(z,z) = (\text{Id},\text{Id})$ for all $z\in B_1$.

We can view $\sigma$ (after extension by zero) as a map $\sigma:U\times V\to T_U\subset T_V$ with the following properties. For all $y\in U$ and $y'\in V$, we have $\sigma(y,y')\in T_yV$ with $\sigma(y,y)=0$, $\sigma(y,\cdot):V\to T_yV$ holomorphic near $y$ (for $y$ close to $x$) and finally, $\nabla\sigma(y,\cdot)|_y:T_yV\to T_yV$ is the identity map (for $y$ close to $x$). As a result, we conclude that given any point in $V$, we can find for it an open neighborhood $U$ and a smooth map $\sigma_U:U\times V\to T_U$ with the following properties: (1) $\sigma_U(y,y')\in T_yV$ for all $y\in U$, $y'\in X$, (2) For any $y\in U$, denoting $y'\mapsto\sigma_U(y,y')$ by $\sigma_{U,y}:V\to T_yV$, we have $\sigma_{U,y}(y) = 0$ and $\nabla\sigma_{U,y}|_y = \text{Id}:T_yV\to T_yV$, and, (3) $\sigma_{U,y}$ is holomorphic in a neighborhood of $y$.

Now, choose a finite open cover of $V$ by such open $U$'s (each equipped with a corresponding $\sigma_U$). Let $\{\rho_U\}$ be a partition of unity on $V$ subordinate to this cover. Now, define $\sigma:V\times V\to T_V$ by the equation \begin{align*} \sigma(y,y') = \sum_U\rho_U(y)\sigma_{U}(y,y') \end{align*} Note that $\sigma$ maps the diagonal of $V\times V$ identically onto the zero section of $T_V$ and that its derivative along the diagonal is the identity map. Thus, applying the inverse function theorem, we get an open neighborhood $\mathcal U\subset T_V$ of the zero section and a smooth map $\tilde\sigma:\mathcal U\to V\times V$ which is inverse to $\sigma$. Let $e:\mathcal U\to V$ be the projection of $\tilde\sigma$ onto the second coordinate of $V\times V$. This $e$ is the desired fibrewise holomorphic exponential map.

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