A technical question in Feix's construction of hyperkahler metric on cotangent bundles

Question

I am now reading Feix's paper Hyperkahler metrics on cotangent bundles and I have a technical question to ask.

In his paper, for an analytic Kähler manifold $(X,J,\omega)$, Feix considered its complexification $X^c$ which in my understanding can be thought of as a neighborhood of the diagonal in $X\times\bar{X}$, where $\bar{X}$ is the complex manifold $X$ with complex structure $-J$. One can extend $\omega$ analytically to a holomorphic symplectic form $\omega^c$ on $X^c$. This $\omega^c$ determines two natural holomorphic Lagrangian foliations $L_+$ and $L_-$. Let $z_i$ and $z'_j$ be local holomorphic coordinates of $X$ and $\bar{X}$ respectively, then the leaves of $L_+$ and $L_-$ are given by $z_i\equiv const$ and $z'_j\equiv const$.

As the diagonal intersects each leaf at exactly one point, one may identify the space of leaves of $L_+$ with $X$ and the space of leaves of $L_-$ with $\bar{X}$ respectively.

From now on let us focus on $L_+$ exclusively. By shrinking $X^c$ if necessary, one may assume that $\Lambda_x$ is simply connected for any $x\in X=$ "space of leaves of $L_+$", where $\Lambda_x$ is the leaf corresponding to $x$. As a consequence of Lagrangian foliation, each $\Lambda_x$ has a natural affine structure, so it makes sense to write $V_x$ to be the space of affine functions on $\Lambda_x$. By 1-connectedness of $\Lambda_x$, each $V_x$ is a vector space of complex dimension $n+1$, where $n$ is the complex dimension of $X$. These $V_x$ patch up to a complex vector bundle $V\to X$.

Feix claims without further explanation that this bundle is holomorphic. I really would like to know the description of the holomorphic structure here. It occurs to me that the most natural frames one can think of actually have anti-holomorphic transition functions as follows:

Let $z_i,z'_j$ be local coordinates for $X^c$, the leaves of $L_+$ are $z_i\equiv const$. Fix a leaf $\Lambda_x$, it intersects the diagonal at the point whose coordinate is $z=x,z'=\bar{x}$. One can find parallel 1-forms $\theta_i$ on $\Lambda_x$ by parallel transport with respect to the flat connection with initial value specified by $\theta_i|_{(x,\bar{x})}=\textrm{d}z'_i|_{(x,\bar{x})}$. These $\theta_i$ must be a closed form and their primitives $f_i$ along with the constant function 1 form a basis of $V_x$. However, if you work with this particular frame, then under holomorphic coordinate change of $X$, the transition matrix of these frame depends anti-holomorphically on $X$.

Surely one can switch $L_+$ and $L_-$ to solve the holomorphicity problem here. But I think a bigger trouble is then introduced since in that case the map $\phi$ defined by Feix loses its holomorphicity.

Thank you!

Answer 1

I just read your question. Funny, I was also trying to read this paper carefully not long ago, and I was annoyed by exactly that kind of problem.

I think the problem is to say that "the space of leaves of $L_+$ is $X$".

In the complex manifold $X^c$ (which we can think of as $X^c = X \times \overline{X}$ indeed), the space of leaves of $L_+$ (let's call it $B_+$ like Feix), as a complex manifold , can naturally be identified with a transverse complex submanifold, at least locally. However $X$ (embedded as the diagonal in $X^c$) is not a complex submanifold!

Instead, a natural choice of a transverse complex submanifold is a leaf of $L^-$, i.e. a horizontal slice $X \times \{x_0\} \subset X \times \bar{X}$. In this sense the space of leaves $B_+$ can be identified to $X = X \times \{x_0\}$, and the projection $L_+ \to B_+ \approx X$ is holomorphic, and so is the vector bundle $V \to B_+$. Also, I think this is the only way to make sense of what Feix writes at some point later (proof of Lemma 2):

The complexified manifold $X^c$ is foliated by Lagrangian submanifolds and choosing a suitable section of the quotient map onto $B_+ \approx X$ we can identify $X$ as a transverse Lagrangian section.

So, problem solved, right? I don't think so. As far as my understanding goes there are at least two problems (that are related) with this, if you allow me to expand.

1. As you write, maybe $X^c$ (or the part you are working with) is just a neighborhood of the diagonal in $X \times \overline{X}$. For instance, if you insist that the leaves are simply connected but $X$ is not. So one can only identify $B_+$ locally with $X$.
2. A more serious problem is the question of the choice of the (local) identification of $B_+$ with $X$. How does that affect Feix's construction?

Obviously, it is crucial in Feix's proof that the twistor space she constructs is the twistor space of a hyperkähler structure on... $T^*X$! In order to justify that, she claims (a few lines down):

We also notice that the above argument allows us to identify a neighbourhood of the zero section in $\pi^{-1}(0) \subset V^*$ with a neighbourhood of the zero section of the cotangent bundle.

If what I said earlier is correct, she is (or she should be) talking about the cotangent bundle to $B_+$, and not $X$. And what worries me is that I checked: there is no way the hyperkähler structure Feix constructs is invariant under a transformation corresponding to changing the local identification $X \approx B_+$ by choosing different leaves of $L_-$ (locally) representing $B_+$.

I wish I understood better the hyperkähler structure on the cotangent bundle of a Kähler manifold, but I understand neither the intuition behind Feix's construction (why should the complex structure $I_\lambda$ correspond to the Feix's twistor space's fiber $\pi^{-1}(\lambda)$?) nor the technical details of why this construction works, and what I pointed out here is one of my main issues. I would be happy if someone explained it to me. If you figure it out please let me know!