Adiabatic limits of Ricci-flat Kähler metrics.

*(English)*Zbl 1208.32024If \(X\) is a compact Kähler manifold with \(c_1(X)_{\mathbb{R} } =0\) (the author calls this a Calabi-Yau manifold for short), by Yau’s solution of the Calabi-Yau conjecture, there exists a unique Ricci-flat Kähler metric in each Kähler class and this metric depends smoothly on the Kähler class. The author wishes to understand what happens to the Ricci-flat metrics as the class goes to the boundary of the Kähler cone. In previous work [J. Eur. Math. Soc. (JEMS) 11, No. 4, 755–776 (2009; Zbl 1177.32015)], he studied the case when the limit class has positive volume. In this paper, he considers the case when the limit volume is zero, restricting to the particular but very interesting case of a Calabi-Yau manifold that admits a holomorphic fibration to a lower dimensional space such that the limit class is the pullback of a Kähler class from the base.

More precisely, let \(X\) be an \(n\)-dimensional Calabi-Yau manifold and let \(f : X \rightarrow Z\) be a holomorphic map to a compact Kähler manifold \(Z\). Set \(Y:=f(X)\) and assume that \(Y\) be normal and of dimension \(m\) with \(0< m <n\) and that \(f: X\rightarrow Y\) have connected fibres. Let \(\omega_X\) and \( \omega_Z\) be Kähler metrics on \(X\) and \(Z\), respectively. Then \(\omega_0=f^* \omega_Z\) is a smooth nonnegative (1,1)-form and \([\omega_0]\) lies on the boundary of the Kähler cone of \(X\), while for any \(t\in (0,1]\) the class \( [\omega_0] + t [\omega_X]\) lies inside the cone. Denote by \(\tilde{\omega}_t\) the unique Ricci-flat Kähler metric belonging to the class \( [\omega_0] + t [\omega_X]\). One can find an analytic subvariety \(S\subset X\) such that \(Y_{\text{sing}} \subset f(S)\) and \(f: X-S \rightarrow Y-f(S)\) is a submersion. For any \(y\in Y-f(S)\), the fibre \(X_y : = f^{-1}(y)\) is a smooth Calabi-Yau manifold of dimension \(n-m\). So, \(f: X-S \rightarrow Y-f(S)\) is a family of Calabi-Yau manifolds, and, on the base \(Y-f(S)\), a Weil-Petersson metric denoted by \(\omega_{WP}\) is defined [see, e.g., A. Fujiki and G. Schumacher, Publ. Res. Inst. Math. Sci. 26, No. 1, 101–183 (1990; Zbl 0714.32007)].

Theorem 1.2. There is a smooth Kähler metric \(\omega\) on \(Y\backslash f(S)\) such that, for \(t \to 0\), the Ricci-flat metrics \(\tilde{\omega}_t\) converge to \(f^*\omega\) both as currents and in the \(C^{1,\beta}_{loc}(X-S)\) topology of potentials for any \(\beta \in (0,1)\). The metric \(\omega\) satisfies \( \mathrm{Ric}(\omega)=\omega_{WP}\) on \(Y- f(S)\). Moreover, for any \(y\in Y- f(S)\), the metrics \(\tilde{\omega}_t|_{X_y}\) converge to zero in the \(C^{1}\) topology of metrics, uniformly as \(y\) varies in a compact set of \(Y- f(S)\).

The case when \(f:X\to Y=\mathbb{P}^1\) is an elliptically fibered \(K3\) surface with \(24\) singular fibers of type \(I_1\) goes back to M. Gross and P. M. H. Wilson [J. Differ. Geom. 55, No. 3, 475–546 (2000; Zbl 1027.32021)]. The equation \(\mathrm{Ric}(\omega)=\omega_{WP}\) has first been considered explicitly by J. Song and G. Tian [Invent. Math. 170, No. 3, 609–653 (2007; Zbl 1134.53040)]. Other related papers include J. Fine [J. Differ. Geom. 68, No. 3, 397–432 (2004; Zbl 1085.53064)] and J. Stoppa [J. Differ. Geom. 83, No. 3, 663–691 (2009; Zbl 1203.32006)].

More precisely, let \(X\) be an \(n\)-dimensional Calabi-Yau manifold and let \(f : X \rightarrow Z\) be a holomorphic map to a compact Kähler manifold \(Z\). Set \(Y:=f(X)\) and assume that \(Y\) be normal and of dimension \(m\) with \(0< m <n\) and that \(f: X\rightarrow Y\) have connected fibres. Let \(\omega_X\) and \( \omega_Z\) be Kähler metrics on \(X\) and \(Z\), respectively. Then \(\omega_0=f^* \omega_Z\) is a smooth nonnegative (1,1)-form and \([\omega_0]\) lies on the boundary of the Kähler cone of \(X\), while for any \(t\in (0,1]\) the class \( [\omega_0] + t [\omega_X]\) lies inside the cone. Denote by \(\tilde{\omega}_t\) the unique Ricci-flat Kähler metric belonging to the class \( [\omega_0] + t [\omega_X]\). One can find an analytic subvariety \(S\subset X\) such that \(Y_{\text{sing}} \subset f(S)\) and \(f: X-S \rightarrow Y-f(S)\) is a submersion. For any \(y\in Y-f(S)\), the fibre \(X_y : = f^{-1}(y)\) is a smooth Calabi-Yau manifold of dimension \(n-m\). So, \(f: X-S \rightarrow Y-f(S)\) is a family of Calabi-Yau manifolds, and, on the base \(Y-f(S)\), a Weil-Petersson metric denoted by \(\omega_{WP}\) is defined [see, e.g., A. Fujiki and G. Schumacher, Publ. Res. Inst. Math. Sci. 26, No. 1, 101–183 (1990; Zbl 0714.32007)].

Theorem 1.2. There is a smooth Kähler metric \(\omega\) on \(Y\backslash f(S)\) such that, for \(t \to 0\), the Ricci-flat metrics \(\tilde{\omega}_t\) converge to \(f^*\omega\) both as currents and in the \(C^{1,\beta}_{loc}(X-S)\) topology of potentials for any \(\beta \in (0,1)\). The metric \(\omega\) satisfies \( \mathrm{Ric}(\omega)=\omega_{WP}\) on \(Y- f(S)\). Moreover, for any \(y\in Y- f(S)\), the metrics \(\tilde{\omega}_t|_{X_y}\) converge to zero in the \(C^{1}\) topology of metrics, uniformly as \(y\) varies in a compact set of \(Y- f(S)\).

The case when \(f:X\to Y=\mathbb{P}^1\) is an elliptically fibered \(K3\) surface with \(24\) singular fibers of type \(I_1\) goes back to M. Gross and P. M. H. Wilson [J. Differ. Geom. 55, No. 3, 475–546 (2000; Zbl 1027.32021)]. The equation \(\mathrm{Ric}(\omega)=\omega_{WP}\) has first been considered explicitly by J. Song and G. Tian [Invent. Math. 170, No. 3, 609–653 (2007; Zbl 1134.53040)]. Other related papers include J. Fine [J. Differ. Geom. 68, No. 3, 397–432 (2004; Zbl 1085.53064)] and J. Stoppa [J. Differ. Geom. 83, No. 3, 663–691 (2009; Zbl 1203.32006)].

Reviewer: Alessandro Ghigi (Milano)