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Remarks on the collapsing of torus fibered Calabi-Yau manifolds. (English) Zbl 1345.32018

Let \(f: M\rightarrow N\) be a surjective holomorphic map between compact K\({\ddot{\text{a}}}\)hler manifolds and \(S\) be the set of critical points of \(f\). For all \(t\in (0, 1]\), let \(\tilde{w}_t\) be the unique Ricci-flat K\({\ddot{\text{a}}}\)hler metric on \(M\) cohomologous to \(w_0+tw_M\), where \(w_M\) and \(w_N\) are two fixed K\({\ddot{\text{a}}}\)hler metrics on \(M\) and \(N\) respectively, and \(w_0=f^*w_N\).
In this paper, the authors prove the main theorem:
Theorem 1.1. If \(M\) is Calabi-Yau and all smooth fibers of \(f\) are complex \(n\)-tori, given any compact set \(K\subset M \setminus f^{-1}(f(S))\) and any \(k\in \mathbb{N}_0\), there exists a constant \(C_{K, k}< \infty \), which does not depend on \(t\), such that \[ \|\tilde{w}_t \|_{C^k(K, w_M)} \leq C_{K, k} \] holds uniformly for all \(t\in (0, 1]\).
This theorem was proved by M. Gross et al. [Duke Math. J. 162, No. 3, 517–551 (2013; Zbl 1276.32020)] when \(M\) is projective. To remove the projectivity assumption, the authors prove that the above theorem is an application of the following theorem, using an idea applied in papers of Fujiki and Schumacher: the Siegel upper half-space more generally classifies polarised complex tori.
Theorem 1.2. Let \(B\subset N\setminus f(S) \) be a small coordinate ball, \(U= f^{-1}(B)\), and \(p: B \times {\mathbb{C}}^n\rightarrow U\) be the universal holomorphic cover of \(U\). Then there is a semi-flat form \(w_{SF}\geq 0\) on \(U\) such that \(p^*w_{SF}= i \partial\bar{\partial}\eta\) for a smooth real-valued function \(\eta\) on \(B \times {\mathbb{C}}^n\) with the scaling property \[ \eta(y, \lambda z)= \lambda^2\eta (y, z), \quad \lambda\in \mathbb{R}. \]

MSC:

32J27 Compact Kähler manifolds: generalizations, classification
32Q25 Calabi-Yau theory (complex-analytic aspects)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions

Citations:

Zbl 1276.32020
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References:

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