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TWO NON-VANISHING RESULTS CONCERNING THE ANTI-CANONICAL BUNDLE

Part of: Birational geometry Surfaces and higher-dimensional varieties Special varieties

Published online by Cambridge University Press:  20 January 2025

NIKLAS MÜLLER Open the ORCID record for NIKLAS MÜLLER [Opens in a new window]
NIKLAS MÜLLER*
Affiliation:
Department of Mathematics Universität Duisburg-Essen Thea-Leymann-Strasse 9 45127 Essen Germany
*
[email protected]
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Abstract

Let $(X, \Delta )$ be a klt threefold pair with nef anti-log canonical divisor $-(K_X+\Delta )$. We show that $\kappa (X, -(K_X+\Delta ))\geq 0$. To do so, we prove a more general equivariant non-vanishing result for anti-log canonical bundles, which is valid in any dimension.

Keywords

non-vanishinganti-canonical bundleGIT

MSC classification

Primary: 14E30: Minimal model program (Mori theory, extremal rays)
Secondary: 14M25: Toric varieties, Newton polyhedra 14J30: $3$-folds
Type
Article
Information
Nagoya Mathematical Journal , First View , pp. 1 - 18
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Foundation Nagoya Mathematical Journal

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Footnotes

While working on this project, the author was supported by the DFG Research Training Group 2553 “Symmetries and Classifying Spaces: Analytic, Arithmetic and Derived”

References

Bauer, T. and Peternell, T., Nef reduction and anticanonical bundles , Asian J. Math. 8 (2004), no. 2, 315–352.CrossRefGoogle Scholar
Brion, M., Notes on automorphism groups of projective varieties, unpublished, 2018. Available from the author’s website https://www-fourier.univ-grenoble-alpes.fr/mbrion/autos_final.pdf.Google Scholar
Brion, M., Linearisations of algebraic group actions, unpublished, 2018. Available from the author’s website https://www-fourier.ujf-grenoble.fr/mbrion/lin.Google Scholar
Buczyński, J. and Wiśniewski, J., Algebraic torus actions on contact manifolds , J. Differ. Geom. 121 (2022), no. 2, 227–289.CrossRefGoogle Scholar
Campana, F., Cao, J. and Matsumura, S.-I., Projective klt pairs with nef anti-canonical divisor , Algebr. Geom. 8 (2021), no. 4, 430–464.CrossRefGoogle Scholar
Cao, J., Vanishing theorems and structure theorems of compact Kähler manifolds. Thesis, Institut Fourier, France, 2013. https://tel.archives-ouvertes.fr/tel-00919536.Google Scholar
Cao, J., Albanese maps of projective manifolds with nef anticanonical bundles , Ann. Sci. Éc. Norm. Supér. (4) 52 (2019), no. 5, 1137–1154.Google Scholar
Cao, J. and Höring, A., Manifolds with nef anticanonical bundle , J. Reine Angew. Math. 724 (2017), 203–244.CrossRefGoogle Scholar
Cao, J. and Höring, A., A decomposition theorem for projective manifolds with nef anticanonical bundle , J. Algebraic Geom. 28 (2019), no. 3, 567–597.CrossRefGoogle Scholar
Debarre, O., Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001.CrossRefGoogle Scholar
Ejiri, S. and Gongyo, Y., Nef anti-canonical divisors and rationally connected fibrations , Compos. Math. 155 (2019), no. 7, 1444–1456.CrossRefGoogle Scholar
Greb, D., Guenancia, H. and Kebekus, S., Klt varieties with trivial canonical class: holonomy , Geom. Topol. 23 (2019), no. 4, 2051–2124.CrossRefGoogle Scholar
Guillemin, V. and Sternberg, S., Convexity properties of the moment mapping , Invent. Math. 67 (1982), no. 3, 491–513.CrossRefGoogle Scholar
Han, J. and Liu, W., On numerical nonvanishing for generalized log canonical pairs , Doc. Math. 25 (2020), 93–123.CrossRefGoogle Scholar
Hausen, J. and Süß, H., The Cox ring of an algebraic variety with torus action , Adv Math. 225 (2010), no. 2, 977–1012.CrossRefGoogle Scholar
Hu, Y. and Keel, S., Mori dream spaces and GIT , Michigan Math J. 48 (2000), no. 1, 331–348.CrossRefGoogle Scholar
Koike, T., Ueda theory for compact curves with nodes , Indiana Univ. Math. J. 66 (2017), no. 3, 845–876.CrossRefGoogle Scholar
Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, Vol. 134, Cambridge University Press,Cambridge, 1998.CrossRefGoogle Scholar
Kollár, J., Lectures on resolution of singularities, Annals of Mathematics Studies, Vol. 166, Princeton University Press, Princeton, NJ, 2007.Google Scholar
Lazić, V. and Peternell, T., On generalised abundance, I , Publ. Res. Inst. Math. Sci. 56 (2020), no. 2, 353–389.CrossRefGoogle Scholar
Lazić, V., Matsumura, S.-I., Peternell, T., Tsakanikas, N. and Xie, Z., The nonvanishing problem for varieties with nef anticanonical bundle , Doc. Math. 28 (2023), no. 6, 1393–1440.CrossRefGoogle Scholar
Lu, S., Tu, Y., Zhang, Q. and Zheng, Q., On semistability of Albanese maps , Manuscripta Math. 131 (2010), no. 3–4, 531–535.CrossRefGoogle Scholar
Matsumura, S.-I. and Wang, J., Structure theorem for projective klt pairs with nef anti-canonical divisor, preprint, 2021, arXiv.2105.14308.Google Scholar
Milne, J., Algebraic groups: The theory of group schemes of finite type over a field, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2017.CrossRefGoogle Scholar
Müller, N., Locally constant fibrations and positivity of curvature, preprint, 2022, arXiv:2212.11530v1.Google Scholar
Mumford, D., Fogarty, D. and Kirwan, F., Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), Vol. 34, Springer-Verlag, Berlin, 1994.CrossRefGoogle Scholar
Păun, M., Sur le groupe fondamental des variétés kählériennes compactes à classe de Ricci numériquement effective , C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 11, 1249–1254.CrossRefGoogle Scholar
Păun, M., On the Albanese map of compact Kähler manifolds with numerically effective Ricci curvature , Comm. Anal. Geom. 9 (2001), no. 1, 35–60.CrossRefGoogle Scholar
Ueno, K., Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Heidelberg, 1975.CrossRefGoogle Scholar
Wang, J., Structure of projective varieties with nef anticanonical divisor: the case of log terminal singularities , Math. Ann. 384 (2022), no. 1–2, 47–100.CrossRefGoogle Scholar
Xie, Z., Rationally connected threefolds with nef and bad anticanonical divisor, Ann. Inst. Fourier 74 (2024), no. 5, 1819–1850.Google Scholar
Xie, Z., Rationally connected threefolds with nef and bad anticanonical divisor, II, preprint, 2023, arXiv.2301.09466.CrossRefGoogle Scholar
Zhang, Q., On projective varieties with nef anticanonical divisors , Math. Ann. 332 (2005), no. 3, 697–703.CrossRefGoogle Scholar