2.1. Generalized Gibbons-Hawking ansatz [05CA]

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2.1. Generalized Gibbons-Hawking ansatz

Suppose $T^{n}$ , a $n$ -dimensional (real) torus, acts freely on an $N$ -(complex) dimensional Kähler manifold $M$ by Hamiltonian holomorphic isometries. Then $M$ can be considered as a principal $T^{n}$ -bundle over a real manifold of dimension $2N-n$ . The generalized Gibbons-Hawking ansatz expresses the Kähler and Ricci-flat conditions as differential equations in the $n$ moment map coordinates and $N-n$ holomorphic coordinates on the Kähler quotient.

Let $\mathfrak{t}$ denote the Lie algebra of the torus Lie group $T^{n}$ , and let $\mathfrak{t}_{\mathbb{Z}}$ be the natural integral lattice in $\mathfrak{t}$ . We will fix a basis in $\mathfrak{t}_{\mathbb{Z}}$ . This defines affine coordinates $u_{i}$ on the dual space $\mathfrak{t}^{*}\cong\mathbb{R}^{n}$ . Let $Y$ be either $\mathbb{C}^{N-n}$ or $(\mathbb{C}^{*})^{N-n}\cong\mathbb{R}^{N-n}\times(S^{1})^{N-n}$ , with the affine complex coordinates $\eta_{p}=x_{p}+iy_{p}$ , where $y_{p}$ are the phase coordinates on the torus $(S^{1})^{N-n}$ in the latter case.

Consider a principal $T^{n}$ -bundle $\pi:M\to B^{\circ}$ over an open set $B^{\circ}$ in $\mathfrak{t}^{*}\times Y$ , with coordinates $(u_{i},\eta_{p},\bar{\eta}_{q})$ . Denote by $[\nu]$ its integral Chern class as an element in $H^{2}(B,\mathfrak{t}_{\mathbb{Z}})$ .

Theorem 2.1 (cf. [PP91]).

Let $V^{ij}$ , respectively $W^{pq}$ , be real symmetric, respectively hermitian, positive definite matrices of smooth functions on $B^{\circ}$ , locally given by some potential function $\Phi$ :

(1)

V^{ij}=\frac{\partial^{2}\Phi}{\partial u_{j}\partial u_{j}},\quad W^{pq}=-4\frac{\partial^{2}\Phi}{\partial\eta_{p}\partial\bar{\eta}_{q}},\qquad 1\leq i,j\leq n\quad n+1\leq p,q\leq N.

Then the following $\mathfrak{t}$ -valued 2-form is closed:

(2)

F_{j}=\sqrt{-1}\left(\frac{1}{2}\frac{\partial W^{pq}}{\partial u_{j}}d\eta_{p}\wedge d\bar{\eta}_{q}+\frac{\partial V^{ij}}{\partial\eta_{p}}du_{i}\wedge d\eta_{p}-\frac{\partial V^{ij}}{\partial\bar{\eta}_{q}}du_{i}\wedge d\bar{\eta}_{q}\right).

Suppose, in addition, that $\det V^{ij}=\det W^{pq}$ and $(F_{1},\dots,F_{n})$ is in the cohomology class $2\pi[\nu]$ . Then there exist a connection on the bundle $M\to B$ with associated 1-forms $A_{i}$ and the curvature $(F_{1},\dots,F_{n})$ such that $M$ is a Kähler manifold with Ricci-flat metric given by

(3)

h=(V^{-1})^{ij}dz_{i}\otimes d\bar{z}_{j}+W^{pq}d\eta_{p}\otimes d\bar{\eta}_{q},

where $dz_{j}=V^{ij}du_{i}+\sqrt{-1}\cdot A_{j}$ and $d\eta_{p}$ form a basis of holomorphic 1-forms. The holomorphic $N$ -form and the Kähler form:

(4)

\Omega=\wedge_{j=1}^{k}dz_{j}\bigwedge\wedge_{p=1}^{l}d\eta_{p},\qquad\omega=du_{j}\wedge A_{j}+\frac{\sqrt{-1}}{2}\cdot W^{pq}d\eta_{p}\wedge d\bar{\eta}_{q}

are compatible in the sense that $\Omega\wedge\bar{\Omega}=const\cdot\omega^{N}$ .

Proof.

First we note that the local potential description of $V$ and $W$ by (1) insures that

(5)

dF_{j}=\frac{\sqrt{-1}}{2}\cdot d\left(\frac{\partial W^{pq}}{\partial u_{j}}\right)\wedge d\eta_{p}\wedge d\bar{\eta}_{q}\\ +\sqrt{-1}\cdot d\left(\frac{\partial V^{ij}}{\partial\eta_{p}}\right)\wedge du_{i}\wedge d\eta_{p}-\sqrt{-1}\cdot d\left(\frac{\partial V^{ij}}{\partial\bar{\eta}_{q}}\right)\wedge du_{i}\wedge d\bar{\eta}_{q}\\ =\frac{\sqrt{-1}}{2}\left(\frac{\partial^{2}W^{pq}}{\partial u_{i}\partial u_{j}}+4\frac{\partial^{2}V^{ij}}{\partial\eta_{p}\partial\bar{\eta}_{q}}\right)du_{i}\wedge d\eta_{p}\wedge d\bar{\eta}_{q}=0.

Moreover, if $\theta_{j}$ denote the coordinates on the torus fiber such that $\partial/\partial\theta_{j}$ are the Hamiltonian vector fields, then the connection 1-forms can be written up to exact forms on $B^{\circ}$ in terms of the local potential $\Phi$ :

(6)

A_{j}=d\theta_{j}+\sqrt{-1}\left(\frac{\partial^{2}\Phi}{\partial u_{j}\partial\eta_{p}}d\eta_{p}-\frac{\partial^{2}\Phi}{\partial u_{j}\partial\bar{\eta}_{q}}d\bar{\eta}_{q}\right).

And one can see explicitly that $F_{j}=dA_{j}$ .

The integrability of the complex structure follows from the fact that the differential ideal generated by $(1,0)$ -forms is closed:

(7)

d(dz_{j})=dV^{ij}\wedge du_{i}+\sqrt{-1}\cdot dA_{j}\\ =-\frac{\partial V^{ij}}{\partial u_{k}}du_{i}\wedge du_{k}-\frac{\partial V^{ij}}{\partial\eta_{p}}du_{i}\wedge d\eta_{p}-\frac{\partial V^{ij}}{\partial\bar{\eta}_{q}}du_{i}\wedge d\bar{\eta}_{q}\\ -\left(\frac{1}{2}\frac{\partial W^{pq}}{\partial u_{j}}d\eta_{p}\wedge d\bar{\eta}_{q}+\frac{\partial V^{ij}}{\partial\eta_{p}}du_{i}\wedge d\eta_{p}-\frac{\partial V^{ij}}{\partial\bar{\eta}_{q}}du_{i}\wedge d\bar{\eta}_{q}\right)\\ =\left(\frac{1}{2}\frac{\partial W^{pq}}{\partial u_{j}}d\bar{\eta}_{q}-2\frac{\partial V^{ij}}{\partial\eta_{p}}du_{i}\right)\wedge d\eta_{p},

where we have only used $\frac{\partial V^{ij}}{\partial u_{k}}=\frac{\partial V^{kj}}{\partial u_{i}}$ .

It is equally easy to verify the Kähler condition:

(8)

d\omega=-du_{j}\wedge dA_{j}+\frac{\sqrt{-1}}{2}\cdot dW^{pq}\wedge d\eta_{p}\wedge d\bar{\eta}_{q}\\ =-\sqrt{-1}\left(\frac{1}{2}\frac{\partial W^{pq}}{\partial u_{j}}du_{j}\wedge d\eta_{p}\wedge d\bar{\eta}_{q}\right)+\frac{\sqrt{-1}}{2}\frac{\partial W^{pq}}{\partial u_{j}}du_{j}\wedge d\eta_{p}\wedge d\bar{\eta}_{q}=0.

Finally, the Ricci-flatness is manifest since $\det(h)=\det V^{-1}\cdot\det W=1$ in the complex coordinates $dz_{j},d\eta_{p}$ . ∎

We are interested in applying the Gibbons-Hawking ansatz to description of the metrics on the toric Calabi-Yau hypersurfaces. Given a torus fibration of such hypersurface near the large complex structure point one may approximate the true Calabi-Yau metric by non-compact solutions of GH equations over different regions of the base. Away from the discriminant locus a semi-flat metric gives a good approximation. We try to argue that as the tori shrink to zero size the metric behavior near singular fibers can also be approximated by certain $y$ -periodic solutions of GH ansatz. This local description is the main subject of the paper.