7.2. Tian-Yau metrics [055U]

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7.2. Tian-Yau metrics

In this subsection we briefly review the complete Ricci-flat Kähler metrics, constructed in [TY90] on the complement of a smooth anti-canonical divisor in a Fano manifold. We will state without proof some facts on the asymptotics of these metrics. Interested readers are referred to [HSVZ18], Section 3 for details.

Let $Y$ be an $n$ dimensional Fano manifold, $D$ a smooth anti-canonical divisor in $Y$ , and denote $Z=Y\setminus D$ . By adjunction formula $D$ itself is Calabi-Yau, and we can find a Ricci-flat Kähler metric $\omega_{D}\in 2\pi c_{1}(L_{D})$ , where $L_{D}$ is the restriction of $K_{Y}^{-1}$ to $D$ . Fixing a defining section $S$ of $D$ , we can view $S^{-1}$ as a holomorphic $n$ -form $\Omega_{Z}$ on $Z$ with a simple pole along $D$ . Rescaling suitably we may assume the Poincaré residue of $\Omega_{Z}$ gives a holomorphic volume form $\Omega_{D}$ on $D$ satisfying the normalization condition (4.1).

As before we can fix the hermitian metric on $L_{D}$ whose curvature form is $-\sqrt{-1}\omega_{D}$ and we also fix a smooth extension to $Y$ with strictly positive curvature. Then

(7.44)

\omega_{Z}\equiv\frac{n}{n+1}\sqrt{-1}\partial\bar{\partial}(-{\log|S|^{2}})^{\frac{n+1}{n}}

defines a Kähler form on a neighborhood of infinity in $Z$ . The Tian-Yau metric $\omega_{TY}$ on $Z$ is then obtained by solving a Monge-Ampère equation with reference metric $\omega_{Z}$ . Let $\mathcal{C}$ be the Calabi model space constructed using $(D,L_{D},\omega_{D})$ , as in Section 2.2.

Proposition 7.4 ([TY90], see also [HSVZ18]).

There is a smooth function $\phi$ on $Z$ such that $\omega_{TY}\equiv\omega_{Z}+\sqrt{-1}\partial\bar{\partial}\phi$ is a complete Ricci-flat Kähler metric on $Z$ solving the Monge-Ampère equation

(7.45)

\omega_{TY}^{n}=\frac{1}{n\cdot 2^{n-1}}(\sqrt{-1})^{n^{2}}\Omega_{Z}\wedge\overline{\Omega}_{Z}.

Moreover, there is a diffeomorphism $\Phi:\mathcal{C}\setminus K^{\prime}\rightarrow Y\setminus K$ , where $K\subset Z$ is compact and $K^{\prime}=\{|\xi|\geq\frac{1}{2}\}$ and constant $\delta_{Z}>0$ , such that the following asymptotics hold uniformly for all $z$ large

(1)

(7.46) $|\nabla_{g_{Z}}^{k}\phi|_{g_{Z}}=O(e^{-\delta_{Z}(-\log|S|^{2})^{1/2}})\ \text{for all}\ k\geq 0.$

(2)

(7.47)

|\nabla_{g_{\mathcal{C}}}^{k}(\Phi^{*}J_{Z}-J_{\mathcal{C}})|_{g_{\mathcal{C}}}=O(e^{-(\frac{1}{2}-\epsilon)z^{n}})\ \text{for all}\ k\geq 0,\epsilon>0.

(3)

(7.48)

|\nabla_{g_{\mathcal{C}}}^{k}(\Phi^{*}\Omega_{Z}-\Omega_{\mathcal{C}})|_{g_{\mathcal{C}}}=O(e^{-(\frac{1}{2}-\epsilon)z^{n}})\ \text{for all}\ k\geq 0,\epsilon>0.

(4)

(7.49) $|\nabla_{g_{\mathcal{C}}}^{k}(\Phi^{*}\omega_{TY}-\omega_{\mathcal{C}})|_{g_{\mathcal{C}}}=O(e^{-{\delta_{Z}}z^{n/2}}).$
(5)

There is a constant $C>0$ such that

(7.50) $C^{-1}z\leq\Phi^{*}((-\log|S|^{2})^{\frac{1}{n}})\leq Cz.$

In particular, the space $(Z,\omega_{TY})$ is $\delta_{Z}$ -asymptotically Calabi in the sense of Definition 5.1. For later purposes we also need a simple observation regarding the asymptotics of $\omega_{TY}$ . Fix a local holomorphic chart $\{U,w_{1},\cdots,w_{n}\}$ centered at a point $p\in D$ , i.e. $w_{i}(p)=0$ for all $i$ , and such that $S$ is locally defined by $w_{1}=0$ . Define a cylindrical type Kähler metric as follows

(7.51)

\omega_{cyl}\equiv\sum_{j\geq 2}\sqrt{-1}dw_{j}\wedge d\bar{w}_{j}+\sqrt{-1}|w_{1}|^{-2}dw_{1}\wedge d\bar{w}_{1}.

By a straightforward computation we get

Lemma 7.5.

On $U\setminus D$ , there is a constant $C>0$ such that

(7.52)

C^{-1}(-\log|S|^{2})^{\frac{1}{n}-1}\omega_{cyl}\leq\omega_{TY}\leq C(-\log|S|^{2})^{\frac{1}{n}}\omega_{cyl},

and for all $k\geq 1$ , there are constants $C_{k},m_{k}>0$ such that

(7.53)

|\nabla^{k}_{\omega_{cyl}}\omega_{TY}|_{\omega_{cyl}}\leq C_{k}(-\log|S|^{2})^{m_{k}}.

Using this Lemma, later when we do estimates for quantities using the Tian-Yau metric, we can do computations using the cylindrical metric which becomes much simpler, and in the end we only get an error which is of polynomial order in $-\log|S|^{2}$ .