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**Continuity of extremal transitions and flops for Calabi-Yau manifolds.**
*(English)*
Zbl 1264.32021

A Calabi-Yau manifold \(M\) is a simply connected projective manifold with trivial canonical bundle \( \mathcal{K}_{M}\cong \mathcal{O}_{M}\). By Yau theorem (Calabi conjecture) any Kähler class \(\alpha\in H^{1,1}(M, \mathbb{R})\), contains a unique Ricci-flat Kähler metric with Kähler form in \(\alpha\). The goal of the paper is to study the behavior of the Ricci-flat metrics under two types of algebro-geometric surgeries, whose definition we now recall.

Let \(M_{0}\) be a normal projective variety with singular set \(S\). A resolution of \(M_0\) is a couple \((\bar{M},\bar{\pi})\), where \(\bar{M}\) is a projective manifold and \(\bar{\pi} :\bar{M} \rightarrow M_0\) is a morphism such that \(\bar{\pi}: \bar{M}\backslash \bar{\pi}^{-1}(S)\rightarrow M_{0} \backslash S\) is biholomorphic. On the other hand, a smoothing of \(M_0\) is a couple \((\mathcal{M}, \pi)\), where \(\mathcal{M}\) is an \((n+1)\)-dimensional variety, \(\pi\) is a proper flat morphism of \(\mathcal{M}\) onto the unit disc \(\Delta \subset \mathbb{C}\), \(M_{0}=\pi^{-1}(0)\), and \(M_{t}=\pi^{-1}(t)\) is a smooth projective \(n\)-dimensional manifold for any \(t\in \Delta\backslash \{0\}\).

If \(M_{0}\) admits both a resolution \( (\bar{M}, \bar{\pi})\) and a smoothing \((\mathcal{M}, \pi)\), the process of going from \(\bar{M}\) to \(M_{t}\), \(t\neq 0\), is called an extremal transition, denoted by \(\bar{M}\rightarrow M_{0} \rightsquigarrow M_{t}\).

If \(M_{0}\) admits two different resolutions \( (\bar{M}_{1}, \bar{\pi}_{1})\) and \((\bar{M}_{2}, \bar{\pi}_{2})\) with both exceptional subvarieties of codimension at least 2, the process of going from \(\bar{M}_{1}\) to \(\bar{M}_{2}\) is called a flop, denoted by \(\bar{M}_{1} \rightarrow M_{0} \dashrightarrow \bar{M}_{2}\).

Extremal transitions and flops are the two kinds of algebro-geometric surgeries studied in the paper. They are of course very interesting in both mathematics and physics. For example, M. Reid [Math. Ann. 278, 329–334 (1987; Zbl 0649.14021)] conjectured that any two Calabi-Yau threefolds are connected by an extremal transition. See [M. Rossi, J. Geom. Phys. 56, No. 9, 1940–1983 (2006; Zbl 1106.32019)] for a survey of extremal transitions in topology, algebraic geometry and physics.

P. Candelas and X. C. de la Ossa [“Comments on conifolds”, Nuclear Phys. B 342, No. 1, 246–268 (1990)] conjectured that extremal transitions and flops should be “continuous in the space of Ricci-flat Kähler metrics”, even though they involve topologically distinct Calabi-Yau manifolds.

This conjecture can be stated very efficiently in terms of Gromov-Hausdorff as follows.

(i) If \(\bar{M}\rightarrow M_{0} \rightsquigarrow M_{t}\) is an extremal transition among Calabi-Yau manifolds, then there exists a family of Ricci-flat Kähler metrics \(\bar{g}_{s}\), \(s\in (0,1)\), on \(\bar{M}\), and a family of Ricci-flat Kähler metrics \(\tilde{g}_{t}\) on \(M_{t}\) such that \(\{( \bar{M}, \bar{g}_{s})\}\) and \(\{(M_{t}, \tilde{g}_{t})\}\) converge to a single compact metric space \((X, d_{X})\) in the Gromov-Hausdorff topology.

(ii) If \(\bar{M}_{1} \rightarrow M_{0} \dashrightarrow \bar{M}_{2}\) is a flop between Calabi-Yau manifolds, then there are Ricci-flat Kähler metrics \(\bar{g}_{i,s}\), \(s\in (0,1)\) on \(\bar{M}_{i}\) (\(i=1,2\)) and a compact metric space \((X, d_{X})\), such that \[ ( \bar{M}_{1}, \bar{g}_{1,s}) \longrightarrow (X, d_{X}) \longleftarrow ( \bar{M}_{2}, \bar{g}_{2,s}), \quad s\rightarrow 0. \] (The convergence is in the sense of Gromov-Hausdorff.) The paper is devoted to the proof of these versions of the conjecture. The following are two main results. (Recall that a Calabi-Yau variety is a simply connected projective normal variety \(M_{0}\) with trivial canonical sheaf and only canonical singularities.)

Theorem 1.1. Let \(M_{0}\) be a Calabi-Yau \(n\)-variety with singular set \(S\). Assume that \(M_{0}\) admits a smoothing \(\pi: \mathcal{M}\rightarrow \Delta\), such that the relative canonical bundle \(\mathcal{K}_{\mathcal{M}/\Delta}\) is trivial and \( \mathcal{M}\) admits an ample line bundle \(\mathcal{L}\). For any \(t\in \Delta\backslash \{0\}\), let \(\tilde{g}_{t}\) be the unique Ricci-flat Kähler metric on \(M_{t}=\pi^{-1}(t) \) with Kähler form \(\tilde{\omega}_{t}\in c_{1}(\mathcal{L})|_{M_{t}} \). Assume that \(M_{0}\) admits also a crepant resolution \((\bar{M}, \bar{\pi})\). Let \(\{\bar{g}_{s}\}_{s\in (0, 1]}\) be a family of Ricci-flat Kähler metrics with Kähler forms \(\bar{\omega}_{s}\), such that \(\lim\limits_{s\rightarrow 0}[\bar{\omega}_{s}]=\bar{\pi}^{*} c_{1}(\mathcal{L})|_{M_{0}} \) in \(H^{1,1}(\bar{M}, \mathbb{R} )\). Then there exists a compact length metric space \((X, d_{X})\) such that \[ \lim_{t\rightarrow 0}d_{GH}((M_{t}, \tilde{g}_{t}),(X, d_{X}))= \lim_{s\rightarrow 0}d_{GH}((\bar{M}, \bar{g}_{s}),(X, d_{X}))=0. \] Furthermore, \((X, d_{X})\) is isometric to the metric completion \(\overline{(M_{0}\backslash S,d_{g})} \) where \(g\) is a Ricci-flat Kähler metric on \(M_{0}\backslash S\), and \(d_{g}\) is Riemannian distance function of \(g\).

Theorem 1.2. Let \(M_{0}\) be an \(n\)-dimensional Calabi-Yau variety with singular set \(S\), and let \(\mathcal{L}\) be an ample line bundle on \(M_0\). Assume that \(M_{0}\) admits two crepant resolutions \((\bar{M}_{1}, \bar{\pi}_{1})\) and \((\bar{M}_{2}, \bar{\pi}_{2})\). Let \(\{\bar{g}_{\alpha,s}\}_{s\in (0,1]}\) be a family of Ricci-flat Kähler metrics on \(\bar{M}_{\alpha} \) with Kähler classes \(\lim\limits_{s\rightarrow 0}[\bar{\omega}_{\alpha,s}]=\bar{\pi}_{\alpha}^{*} c_{1}(\mathcal{L}) \), \(\alpha=1,2\). Then there exists a compact length metric space \((X, d_{X})\) such that \[ \lim_{s\rightarrow 0}d_{GH}((\bar{M}_{1}, \bar{g}_{1,s}),(X, d_{X}))= \lim_{s\rightarrow 0}d_{GH}((\bar{M}_{2}, \bar{g}_{2,s}),(X, d_{X}))=0. \] Furthermore, \((X, d_{X})\) is isometric to the metric completion \(\overline{(M_{0}\backslash S,d_{g})} \) where \(g\) is a Ricci-flat Kähler metric on \(M_{0}\backslash S\), and \(d_{g}\) is the Riemannian distance function of \(g\).

Let \(M_{0}\) be a normal projective variety with singular set \(S\). A resolution of \(M_0\) is a couple \((\bar{M},\bar{\pi})\), where \(\bar{M}\) is a projective manifold and \(\bar{\pi} :\bar{M} \rightarrow M_0\) is a morphism such that \(\bar{\pi}: \bar{M}\backslash \bar{\pi}^{-1}(S)\rightarrow M_{0} \backslash S\) is biholomorphic. On the other hand, a smoothing of \(M_0\) is a couple \((\mathcal{M}, \pi)\), where \(\mathcal{M}\) is an \((n+1)\)-dimensional variety, \(\pi\) is a proper flat morphism of \(\mathcal{M}\) onto the unit disc \(\Delta \subset \mathbb{C}\), \(M_{0}=\pi^{-1}(0)\), and \(M_{t}=\pi^{-1}(t)\) is a smooth projective \(n\)-dimensional manifold for any \(t\in \Delta\backslash \{0\}\).

If \(M_{0}\) admits both a resolution \( (\bar{M}, \bar{\pi})\) and a smoothing \((\mathcal{M}, \pi)\), the process of going from \(\bar{M}\) to \(M_{t}\), \(t\neq 0\), is called an extremal transition, denoted by \(\bar{M}\rightarrow M_{0} \rightsquigarrow M_{t}\).

If \(M_{0}\) admits two different resolutions \( (\bar{M}_{1}, \bar{\pi}_{1})\) and \((\bar{M}_{2}, \bar{\pi}_{2})\) with both exceptional subvarieties of codimension at least 2, the process of going from \(\bar{M}_{1}\) to \(\bar{M}_{2}\) is called a flop, denoted by \(\bar{M}_{1} \rightarrow M_{0} \dashrightarrow \bar{M}_{2}\).

Extremal transitions and flops are the two kinds of algebro-geometric surgeries studied in the paper. They are of course very interesting in both mathematics and physics. For example, M. Reid [Math. Ann. 278, 329–334 (1987; Zbl 0649.14021)] conjectured that any two Calabi-Yau threefolds are connected by an extremal transition. See [M. Rossi, J. Geom. Phys. 56, No. 9, 1940–1983 (2006; Zbl 1106.32019)] for a survey of extremal transitions in topology, algebraic geometry and physics.

P. Candelas and X. C. de la Ossa [“Comments on conifolds”, Nuclear Phys. B 342, No. 1, 246–268 (1990)] conjectured that extremal transitions and flops should be “continuous in the space of Ricci-flat Kähler metrics”, even though they involve topologically distinct Calabi-Yau manifolds.

This conjecture can be stated very efficiently in terms of Gromov-Hausdorff as follows.

(i) If \(\bar{M}\rightarrow M_{0} \rightsquigarrow M_{t}\) is an extremal transition among Calabi-Yau manifolds, then there exists a family of Ricci-flat Kähler metrics \(\bar{g}_{s}\), \(s\in (0,1)\), on \(\bar{M}\), and a family of Ricci-flat Kähler metrics \(\tilde{g}_{t}\) on \(M_{t}\) such that \(\{( \bar{M}, \bar{g}_{s})\}\) and \(\{(M_{t}, \tilde{g}_{t})\}\) converge to a single compact metric space \((X, d_{X})\) in the Gromov-Hausdorff topology.

(ii) If \(\bar{M}_{1} \rightarrow M_{0} \dashrightarrow \bar{M}_{2}\) is a flop between Calabi-Yau manifolds, then there are Ricci-flat Kähler metrics \(\bar{g}_{i,s}\), \(s\in (0,1)\) on \(\bar{M}_{i}\) (\(i=1,2\)) and a compact metric space \((X, d_{X})\), such that \[ ( \bar{M}_{1}, \bar{g}_{1,s}) \longrightarrow (X, d_{X}) \longleftarrow ( \bar{M}_{2}, \bar{g}_{2,s}), \quad s\rightarrow 0. \] (The convergence is in the sense of Gromov-Hausdorff.) The paper is devoted to the proof of these versions of the conjecture. The following are two main results. (Recall that a Calabi-Yau variety is a simply connected projective normal variety \(M_{0}\) with trivial canonical sheaf and only canonical singularities.)

Theorem 1.1. Let \(M_{0}\) be a Calabi-Yau \(n\)-variety with singular set \(S\). Assume that \(M_{0}\) admits a smoothing \(\pi: \mathcal{M}\rightarrow \Delta\), such that the relative canonical bundle \(\mathcal{K}_{\mathcal{M}/\Delta}\) is trivial and \( \mathcal{M}\) admits an ample line bundle \(\mathcal{L}\). For any \(t\in \Delta\backslash \{0\}\), let \(\tilde{g}_{t}\) be the unique Ricci-flat Kähler metric on \(M_{t}=\pi^{-1}(t) \) with Kähler form \(\tilde{\omega}_{t}\in c_{1}(\mathcal{L})|_{M_{t}} \). Assume that \(M_{0}\) admits also a crepant resolution \((\bar{M}, \bar{\pi})\). Let \(\{\bar{g}_{s}\}_{s\in (0, 1]}\) be a family of Ricci-flat Kähler metrics with Kähler forms \(\bar{\omega}_{s}\), such that \(\lim\limits_{s\rightarrow 0}[\bar{\omega}_{s}]=\bar{\pi}^{*} c_{1}(\mathcal{L})|_{M_{0}} \) in \(H^{1,1}(\bar{M}, \mathbb{R} )\). Then there exists a compact length metric space \((X, d_{X})\) such that \[ \lim_{t\rightarrow 0}d_{GH}((M_{t}, \tilde{g}_{t}),(X, d_{X}))= \lim_{s\rightarrow 0}d_{GH}((\bar{M}, \bar{g}_{s}),(X, d_{X}))=0. \] Furthermore, \((X, d_{X})\) is isometric to the metric completion \(\overline{(M_{0}\backslash S,d_{g})} \) where \(g\) is a Ricci-flat Kähler metric on \(M_{0}\backslash S\), and \(d_{g}\) is Riemannian distance function of \(g\).

Theorem 1.2. Let \(M_{0}\) be an \(n\)-dimensional Calabi-Yau variety with singular set \(S\), and let \(\mathcal{L}\) be an ample line bundle on \(M_0\). Assume that \(M_{0}\) admits two crepant resolutions \((\bar{M}_{1}, \bar{\pi}_{1})\) and \((\bar{M}_{2}, \bar{\pi}_{2})\). Let \(\{\bar{g}_{\alpha,s}\}_{s\in (0,1]}\) be a family of Ricci-flat Kähler metrics on \(\bar{M}_{\alpha} \) with Kähler classes \(\lim\limits_{s\rightarrow 0}[\bar{\omega}_{\alpha,s}]=\bar{\pi}_{\alpha}^{*} c_{1}(\mathcal{L}) \), \(\alpha=1,2\). Then there exists a compact length metric space \((X, d_{X})\) such that \[ \lim_{s\rightarrow 0}d_{GH}((\bar{M}_{1}, \bar{g}_{1,s}),(X, d_{X}))= \lim_{s\rightarrow 0}d_{GH}((\bar{M}_{2}, \bar{g}_{2,s}),(X, d_{X}))=0. \] Furthermore, \((X, d_{X})\) is isometric to the metric completion \(\overline{(M_{0}\backslash S,d_{g})} \) where \(g\) is a Ricci-flat Kähler metric on \(M_{0}\backslash S\), and \(d_{g}\) is the Riemannian distance function of \(g\).

Reviewer: Alessandro Ghigi (Milano)