The dual complex of Calabi-Yau pairs. (English) Zbl 1360.14056
A log Calabi-Yau variety is a pair \((X, \Delta)\) such that \(X\) is a proper variety and \(\Delta\) an effective \(\mathbb Q\)-divisor on \(X\) such that \(K_X + \Delta\) is \(\mathbb Q\)-linearly equivalent to \(0\) and the pair \((X, \Delta)\) has at most log-canonical singularities. Let \(g: Y \rightarrow X\) be a log-resolution of the pair \((X, \Delta)\), then there is a unique divisor \(\Delta_Y\) such that \(g_* \Delta_Y = \Delta\) and \(g^*(K_X+\Delta) \sim_{\mathbb Q} K_Y + \Delta_Y\). Since \((X, \Delta)\) is log-canonical, the coefficients in \(\Delta_Y\) are at most one and the divisor \(\Delta_Y^{=1}\) formed by the components with coefficient equal to one is of special interest. In fact de Fernex and the authors showed in a recent paper [“The dual complex of singularities”, arXiv:1212.1675] that the dual complex \(\mathcal{DMR}(X, \Delta)\) of \(\Delta_Y^{=1}\) is independent of the choice of \((Y, \Delta_Y)\), up-to PL-homeomorphism.
In this paper the authors study the topological invariants of the dual complex. They prove that if \(\mathcal{DMR}(X, \Delta)\) has (real) dimension \(d \geq 2\), then the rational cohomology \(H^i(\mathcal{DMR}(X, \Delta), \mathbb Q)\) vanishes for all \(0<i<d\). Moreover they prove that there is a natural surjection \[ \pi_1(X^{\text{sm}}) \twoheadrightarrow \pi_1(\mathcal{DMR}(X, \Delta)). \] In particular, by earlier results of Greb-Kebekus-Peternell [D. Greb et al., Duke Math. J. 165, No. 10, 1965–2004 (2016; Zbl 1360.14094)] and the second author C. Xu [Compos. Math. 150, No. 3, 409–414 (2014; Zbl 1291.14057)], the profinite completion of the fundamental group is finite. In low dimension additional arguments lead to a more precise description: if \(\dim X \leq 4\), then the dual complex \(\mathcal{DMR}(X, \Delta)\) is \(PL\)-homeomorphic to the quotient of a sphere by a finite group. A second application concerns degenerations of Calabi-Yau manifolds, i.e. proper families \(Y \rightarrow \Delta\) such that \(K_Y\) is trivial and for the central fibre \(Y_0\) the pair \((\mathcal Y, \mathcal Y_0)\) is dlt. If the general fibres are Calabi-Yau in the strict sense and the dual complex \(\mathcal D(Y_0)\) of the central fibre has maximal dimension, the authors ask if \(\mathcal D(Y_0)\) is \(PL\)-homeomorphic to a sphere. They give a positive answer if \(\dim X \leq 3\) or \(\dim X=4\) and the central fibre is a SNC divisor.
In this paper the authors study the topological invariants of the dual complex. They prove that if \(\mathcal{DMR}(X, \Delta)\) has (real) dimension \(d \geq 2\), then the rational cohomology \(H^i(\mathcal{DMR}(X, \Delta), \mathbb Q)\) vanishes for all \(0<i<d\). Moreover they prove that there is a natural surjection \[ \pi_1(X^{\text{sm}}) \twoheadrightarrow \pi_1(\mathcal{DMR}(X, \Delta)). \] In particular, by earlier results of Greb-Kebekus-Peternell [D. Greb et al., Duke Math. J. 165, No. 10, 1965–2004 (2016; Zbl 1360.14094)] and the second author C. Xu [Compos. Math. 150, No. 3, 409–414 (2014; Zbl 1291.14057)], the profinite completion of the fundamental group is finite. In low dimension additional arguments lead to a more precise description: if \(\dim X \leq 4\), then the dual complex \(\mathcal{DMR}(X, \Delta)\) is \(PL\)-homeomorphic to the quotient of a sphere by a finite group. A second application concerns degenerations of Calabi-Yau manifolds, i.e. proper families \(Y \rightarrow \Delta\) such that \(K_Y\) is trivial and for the central fibre \(Y_0\) the pair \((\mathcal Y, \mathcal Y_0)\) is dlt. If the general fibres are Calabi-Yau in the strict sense and the dual complex \(\mathcal D(Y_0)\) of the central fibre has maximal dimension, the authors ask if \(\mathcal D(Y_0)\) is \(PL\)-homeomorphic to a sphere. They give a positive answer if \(\dim X \leq 3\) or \(\dim X=4\) and the central fibre is a SNC divisor.
Reviewer: Andreas Höring (Nice)
MSC:
| 14E30 | Minimal model program (Mori theory, extremal rays) |
| 14J17 | Singularities of surfaces or higher-dimensional varieties |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
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