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Minimal models for Kähler threefolds

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Abstract

Let \(X\) be a compact Kähler threefold that is not uniruled. We prove that \(X\) has a minimal model.

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Notes

  1. If \(X\) is projective this is known in arbitrary dimension by [8].

  2. We “twist” the definition from [9] in order to get a group that injects in \(H^2(X, \mathbb {R})\) rather than \(H^2(X, i \mathbb {R}).\)

  3. Since \(X\) has rational singularities we have \(H^1(\hat{X}, \mathbb {R}) \simeq H^1(X, \mathbb {R})\) and \(H^1(\hat{X}, {{\mathcal O}}_{\hat{X}}) \simeq H^1(X, {{\mathcal O}}_X)\) where \(\hat{X} \rightarrow X\) is a resolution by a compact Kähler manifold. Now we apply the classical result from the Kähler case.

  4. The statement in [22, Prop.6.1.iii)] is for compact manifolds, but the proof works in the singular setting.

  5. The statements in [12] are for complex compact manifolds, but generalise immediately to our situation: take \(\mu :X' \rightarrow X\) a desingularisation, and let \(m \in \mathbb {N}\) be the Cartier index of \(K_X\). Then \(\mu ^* (mK_X)\) is a pseudoeffective line bundle with divisorial Zariski decomposition \(\mu ^* (mK_X)= \sum \eta _j S_j' + N(mK_X)'\). The decomposition of \(K_X\) is defined by the push-forwards \(\mu _* (\frac{1}{m} \sum \eta _j S_j')\) and \(\mu _* (\frac{1}{m} N(mK_X)')\). Since a prime divisor \(D \subset X\) is not contained in the singular locus of \(X\), the decomposition has the stated properties.

  6. If \(S_0\) is \(\mathbb {P}^2\) just take the same MMP but omit the last blow-up.

  7. Note that \([\hat{C}] \not \in \mathbb {R}^+ [F]\) since \(\hat{C}\) is not rational.

  8. The cohomology class of a curve \(C \subset X\) is contained in the integral lattice \(H^4(X, \mathbb {Z})\), so the cohomology classes of curves in \(X\) form a discrete set in \(\overline{{NE}}(X)\). Thus for every Kähler form \(\omega '\) there exists a real constant \(\lambda >0\) such that

    $$\begin{aligned} \omega ' \cdot C \ge \lambda \end{aligned}$$

    for every curve \(C \subset X\).

  9. The proof of Theorem 1.3 uses only Theorem 6.3.

  10. Proposition 2.3. in [12] is for compact complex manifolds, but the proof goes through without changes for a variety with isolated singularities. A different way to obtain the decomposition is to take a resolution of singularities \(\nu :X' \rightarrow X\) and consider the nef and big class \(\nu ^* \alpha \). The non-Kähler locus is the union of the \(\nu \)-exceptional locus and the strict transforms of the curves in the non-Kähler locus of \(\alpha \). By [11, Thm.3.1.24] there exists a modification \(\mu ':\tilde{X} \rightarrow X'\) and a Kähler class \(\tilde{\alpha }\) on \(\tilde{X}\) such that \((\mu ')^* \nu ^* \alpha = \tilde{\alpha }+ E\). Then \(\mu :=\nu \circ \mu '\) has the stated properties.

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Acknowledgments

We thank the Forschergruppe 790 “Classification of algebraic surfaces and compact complex manifolds” of the Deutsche Forschungsgemeinschaft for financial support. A. Höring was partially also supported by the A.N.R. project CLASS (ANR-10-JCJC-0111). This work was done when A. Höring was member of the Institut de Mathématiques de Jussieu (UPMC). Last but not least we thank the referees for very helpful comments.

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  1. Laboratoire de Mathématiques J.A. Dieudonné, UMR 7351 CNRS, Université de Nice Sophia-Antipolis, 06108, Nice Cedex 02, France

    Andreas Höring

  2. Mathematisches Institut, Universität Bayreuth, 95440, Bayreuth, Germany

    Thomas Peternell

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Höring, A., Peternell, T. Minimal models for Kähler threefolds. Invent. math. 203, 217–264 (2016). https://doi.org/10.1007/s00222-015-0592-x

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