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!


1 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!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.