Example 1.7 . [03YY]

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Example 1.7.

( $\mathbb{C}^{N}$ Harvey-Lawson example) The affine space $\mathbb{C}^{N}$ with the standard Euclidean metric $\omega=\frac{\sqrt{-1}}{2}\sum_{i=0}^{N-1}dz_{i}\wedge d\bar{z}_{i}$ and holomorphic volume form $\Omega=\sqrt{-1}^{N-1}dz_{0}\wedge\ldots dz_{N-1}$ admits a diagonal $T^{N-1}$ -action, where the $k$ -th circle factor acts by

e^{i\theta_{k}}\cdot(z_{0},z_{1},\ldots,z_{N-1})=(e^{-i\theta_{k}}z_{0},z_{1},\ldots,e^{i\theta_{k}}z_{k},z_{k+1},\ldots,z_{N-1}).

The corresponding moment coordinates are

\mu_{i}=\frac{1}{2}(|z_{i}|^{2}-|z_{0}|^{2}),\quad i=1,2,\ldots N-1,\text{ and }\eta=z_{0}z_{1}\ldots z_{N-1}.

This defines a $T^{N-1}$ -bundle away from the singular locus $\bigcup\{z_{i}=z_{j}=0\}$ . Special cases include (1.1)(1.3). The inverse matrices are

(V^{-1})^{ij}=|z_{0}|^{2}+\delta_{ij}|z_{i}|^{2},\quad W^{-1}=|z_{0}z_{1}\ldots z_{N-1}|^{2}\left(\frac{1}{|z_{0}|^{2}}+\ldots+\frac{1}{|z_{N-1}|^{2}}\right),

viewed as functions of $\mu_{i}$ and $\eta$ . The special Lagrangian fibration described by Remark 1.6 is the well known Harvey-Lawson example.

We notice in particular when $N=3$ that the discriminant locus of the singular $T^{2}$ -bundle is given by $\mathfrak{D}\subset\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\{0\}\subset\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{C}_{\eta}$ as in (1.2). This is not an accidental feature of the Euclidean metric:

Lemma 1.6.

Let $\mathbb{C}^{3}$ be equipped with the holomorphic volume form $\Omega$ above, and let $\omega$ be any $T^{2}$ -invariant Kähler form with infinite volume on the singular loci $\{z_{i}=z_{j}=0\}$ for any $i,j$ . Then the discriminant locus of the singular $T^{2}$ -bundle is $\mathfrak{D}\subset\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{C}_{\eta}$ in moment coordinates $\mu_{1},\mu_{2}$ and $\eta$ .

Proof.

The discriminant locus is the image of the singular locus $\mathcal{C}_{ij}=\{z_{i}=z_{j}=0\}$ under the moment map. We shall focus on $\mathcal{C}_{01}$ . The holomorphic moment coordinate $\eta$ depends only on $\Omega$ and the $T^{2}$ action, so $\eta=z_{0}z_{1}z_{2}$ as before and vanishes on $\mathcal{C}_{01}$ . The symplectic moment coordinates are defined by $d\mu_{i}=-\iota_{\frac{\partial}{\partial\theta_{i}}}\omega$ , and are normalised to be zero at $(z_{1},z_{2},z_{3})=0$ . In particular since the Hamiltonian vector field $\frac{\partial}{\partial\theta_{1}}$ vanishes on $\mathcal{C}_{01}$ , the moment $\mu_{1}$ must be the constant zero on $\mathcal{C}_{01}$ . Furthermore $\mu_{2}>0$ on $\mathcal{C}_{01}$ by considering the weight of the remaining $S^{1}$ action at the fixed point, so the image of $\mathcal{C}_{01}$ is contained in $\mathfrak{D}_{1}\cup\{0\}$ . The infinite volume condition and the formula

0\leq\int_{\mathcal{C}_{01}\cap\{\mu_{2}<m\}}\omega=-2\pi\int_{\mu_{2}=m}^{0}d\mu_{2}=2\pi m,\quad\forall m\geq 0,

ensure that $\mu_{2}$ stretches to infinity, so $\mathfrak{D}_{1}\cup\{0\}$ is the image of $\mathcal{C}_{01}$ . Likewise the image of $\mathcal{C}_{02}$ is $\mathfrak{D}_{2}\cup\{0\}$ and the image of $\mathcal{C}_{12}$ is $\mathfrak{D}_{3}\cup\{0\}$ . ∎

Remark 1.9.

The same method shows the complex geometry of the positive vertex in Section 1.1.6 is compatible with the discriminant locus described in Section 1.1.3.