Theorem 4.4 [0321]

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Theorem 4.4

Let $f:X\rightarrow{\bf P}^{1}$ be an elliptically fibred K3 surface with a holomorphic section and 24 singular fibres over $\Delta=\{p_{1},\ldots,p_{24}\}$ as above. Then there exists open sets $U_{1}^{i}\subseteq U_{2}^{i}\subseteq{\bf P}^{1}$ , $i=1,\ldots,24$ , each diffeomorphic to a disc, $U^{i}_{j}\cap\Delta=\{p_{i}\}$ , positive constants $D_{1},\ldots,D_{6}$ and $\epsilon_{0}$ such that, for all $\epsilon<\epsilon_{0}$ , there exists a Kähler metric $\omega_{\epsilon}$ on $X$ with the following properties:

(1)

\int_{X}\omega_{\epsilon}^{2}=\int_{X}(\mathop{\rm Re}\Omega)^{2}=\int_{X}(\mathop{\rm Im}\Omega)^{2}

(2)

\int_{X_{b}}\omega_{\epsilon}=\epsilon

(3)

\omega_{\epsilon}|_{f^{-1}({\bf P}^{1}\setminus\bigcup_{i}U_{2}^{i})}=\omega_{SF}

(4) $\omega_{\epsilon}|_{f^{-1}(U_{1}^{i})}=T_{\sigma_{i}}^{*}\omega_{OV},$ where $\omega_{OV}$ is an Ooguri–Vafa metric and $T_{\sigma_{i}}$ denotes translation by some holomorphic section $\sigma_{i}$ .

(5) If $F_{\epsilon}=\log\left({\Omega\wedge\bar{\Omega}/2\over\omega_{\epsilon}^{2}}\right)$ , then

\|F_{\epsilon}\|_{C^{0}}\leq D_{1}e^{-D_{2}/\epsilon}

and

\|\Delta F_{\epsilon}\|_{C^{0}}\leq D_{1}e^{-D_{2}/\epsilon}

where $\Delta$ denotes the Laplacian with respect to the metric $\omega_{\epsilon}$ .

(6)

inf_{v}\{Ric(v,v)\,|\ |v|_{\omega_{\epsilon}}=1\}\geq-D_{3}e^{-D_{4}/\epsilon}.

(7) With the Riemannian metric induced by $\omega_{\epsilon}$ , $Diam(X)\leq D_{5}\epsilon^{-1/2}$ .

(8) If $R$ denotes the Riemann curvature tensor, then

\|R\|_{C^{0}}\leq D_{6}\epsilon^{-1}\log\epsilon^{-1},

\|R\|_{C^{0}}\rightarrow\infty

\epsilon\rightarrow 0

and on any non-singular fibre, there exists a constant $C$ depending on the fibre such that

\|R\|\leq C\epsilon.