Maps from K-trivial varieties and connectedness problems
Annales Henri Lebesgue, Volume 3 (2020) , pp. 473-500.

KeywordsCalabi–Yau varieties, families of elliptic curves, elliptically (chain) connected varieties

### Abstract

In this paper we study varieties covered by rational or elliptic curves. First, we show that images of Calabi–Yau or irreducible symplectic varieties under rational maps are almost always rationally connected. Second, we investigate elliptically connected and elliptically chain connected varieties, and varieties swept out by a family of elliptic curves. Among other things, we show that Calabi–Yau or hyperkähler manifolds which are covered by a family of elliptic curves contain uniruled divisors and that elliptically chain connected varieties of dimension at least $2$ contain a rational curve, and so do $K$-trivial varieties with finite fundamental group which are covered by elliptic curves.

### References

[Amb05] Ambro, Florin The moduli $b$-divisor of an lc-trivial fibration, Compos. Math., Volume 141 (2005) no. 2, pp. 385-403 | Article | MR 2134273 | Zbl 1094.14025

[Ame04] Amerik, Ekaterina Variétés de Fano et morphismes: notes d’un mini-cours (2004) (https://arxiv.org/abs/math/0411511)

[Bar75] Barlet, Daniel Espace analytique réduit des cycles analytiques complexes compacts d’un espace analytique complexe de dimension finie, Fonctions de plusieurs variables complexes, II (Sém. François Norguet, 1974–1975) (Lecture Notes in Mathematics) Volume 482, Springer, 1975, pp. 1-158 | Zbl 0331.32008

[BCE + 02] Bauer, Thomas; Campana, Frédéric; Eckl, Thomas; Kebekus, Stefan; Peternell, Thomas; Rams, Sławomir; Szemberg, Tomasz; Wotzlaw, Lorenz A reduction map for nef line bundles, Complex geometry. Collection of papers dedicated to Hans Grauert on the occasion of his 70th birthday (Göttingen, 2000), Springer, 2002, pp. 27-36 | Zbl 1054.14019

[BDPP13] Boucksom, Sébastien; Demailly, Jean-Pierre; Păun, Mihai; Peternell, Thomas The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebr. Geom., Volume 22 (2013) no. 2, pp. 201-248 | Article | Zbl 1267.32017

[Bea96] Beauville, Arnaud Complex algebraic surfaces, London Mathematical Society Student Texts, Volume 34, Cambridge University Press, 1996, ix+132 pages | MR 1406314 | Zbl 0849.14014

[BHPVdV04] Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Volume 4, Springer, 2004 | MR 2030225 | Zbl 1036.14016

[Bir12] Birkar, Caucher Existence of log canonical flips and a special LMMP, Publ. Math., Inst. Hautes Étud. Sci., Volume 115 (2012), pp. 325-368 | Article | MR 2929730 | Zbl 1256.14012

[BM14] Barlet, Daniel; Magnússon, Jón Cycles analytiques complexes. I. Théorèmes de préparation des cycles, Contributions in Mathematical and Computational Sciences, Volume 22, Société Mathématique de France, 2014, 525 pages | Zbl 1318.32001

[Cam81] Campana, Frédéric Coréduction algébrique d’un espace analytique faiblement kählérien compact, Invent. Math., Volume 63 (1981) no. 2, pp. 187-223 | Article | Zbl 0436.32024

[Cam04a] Campana, Frédéric Orbifolds, special varieties and classification theory, Ann. Inst. Fourier, Volume 54 (2004) no. 3, pp. 499-630 | Article | Numdam | MR 2097416 | Zbl 1062.14014

[Cam04b] Campana, Frédéric Orbifolds, special varieties and classification theory: an appendix, Ann. Inst. Fourier, Volume 54 (2004) no. 3, pp. 631-665 | Article | MR 2097417 | Zbl 1062.14015

[CKM88] Clemens, Herbert; Kollár, János; Mori, Shigefumi Higher-dimensional complex geometry. A summer seminar at the University of Utah, Salt Lake City, 1987, Astérisque, Société Mathématique de France, 1988 no. 166 | Numdam | Zbl 0689.14016

[Cor81] Cornalba, Maurizio A remark on the topology of cyclic coverings of algebraic varieties, Boll. Unione Mat. Ital., Volume 18 (1981) no. 2, pp. 323-328 | MR 618353 | Zbl 0462.14007

[Cyn03] Cynk, Sławomir Cyclic coverings of Fano threefolds, Ann. Pol. Math., Volume 80 (2003), pp. 117-124 | Article | MR 1972838 | Zbl 1015.14008

[DFM19] Diverio, Simone; Fontanari, Claudio; Martinelli, Diletta Rational curves on fibered Calabi–Yau manifolds, Doc. Math. (2019), pp. 663-675 | MR 3960117 | Zbl 07058479

[GHS03] Graber, Tom; Harris, Joe; Starr, Jason Families of rationally connected varieties, J. Amer. Math. Soc., Volume 16 (2003) no. 1, pp. 57-67 | Article | MR 1937199 | Zbl 1092.14063

[GKKP11] Greb, Daniel; Kebekus, Stefan; Kovács, Sándor; Peternell, Thomas Differential forms on log canonical spaces, Publ. Math., Inst. Hautes Étud. Sci., Volume 114 (2011), pp. 87-169 | Article | MR 2854859 | Zbl 1258.14021

[GKP16] Greb, Daniel; Kebekus, Stefan; Peternell, Thomas Singular spaces with trivial canonical class, Minimal models and extremal rays. Proceedings of the conference, RIMS, Kyoto, Japan, June 20–24, 2011 (Advanced Studies in Pure Mathematics) Volume 70, Mathematical Society of Japan, 2016, pp. 67-113 | MR 3617779 | Zbl 1369.14052

[Gou16] Gounelas, Frank Free curves on varieties, Doc. Math., Volume 21 (2016), pp. 287-308 | MR 3505133 | Zbl 1369.14063

[HP19] Höring, Andreas; Peternell, Thomas Algebraic integrability of foliations with numerically trivial canonical bundle, Invent. Math., Volume 216 (2019) no. 2, pp. 395-419 | Article | MR 3953506 | Zbl 07061101

[Huy16] Huybrechts, Daniel Lectures on K3 surfaces, Cambridge Studies in Advanced Mathematics, Volume 158, Cambridge University Press, 2016 | MR 3586372 | Zbl 1360.14099

[HX13] Hacon, Christopher D.; Xu, Chenyang Existence of log canonical closures, Invent. Math., Volume 192 (2013) no. 1, pp. 161-195 | Article | MR 3032329 | Zbl 1282.14027

[IP99] Iskovskikh, Vasily A.; Prokhorov, Yuri G. Fano varieties, Algebraic geometry V: Fano varieties. Transl. from the Russian by Yuri G. Prokhorov and S. Tregub (Encyclopaedia of Mathematical Sciences) Volume 47, Springer, 1999, pp. 1-247 | MR 1668579 | Zbl 0912.14013

[Kaw85] Kawamata, Yujiro Minimal models and the Kodaira dimension of algebraic fiber spaces, J. Reine Angew. Math., Volume 363 (1985), pp. 1-46 | MR 814013 | Zbl 0589.14014

[KKMSD73] Kempf, Georges R.; Knudsen, Finn F.; Mumford, David B.; Saint-Donat, Bernard Toroidal embeddings. I, Lecture Notes in Mathematics, Volume 339, Springer, 1973, 209 pages | MR 335518 | Zbl 0271.14017

[KL09] Kollár, János; Larsen, Michael Quotients of Calabi–Yau varieties, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II (Tschinkel, Yuri; Zarhin, Yuri, eds.) (Progress in Mathematics) Volume 270, Birkhäuser, 2009, pp. 179-211 | Article | MR 2641190 | Zbl 1200.14074

[KM98] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, Volume 134, Cambridge University Press, 1998 | MR 1658959 | Zbl 0926.14003

[KMM92] Kollár, János; Miyaoka, Yoichi; Mori, Shigefumi Rationally connected varieties, J. Algebr. Geom., Volume 1 (1992) no. 3, pp. 429-448 | MR 1158625 | Zbl 0780.14026

[Kol96] Kollár, János Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Volume 32, Springer, 1996 | MR 1440180 | Zbl 0877.14012

[Kol13] Kollár, János Singularities of the minimal model program, Cambridge Tracts in Mathematics, Volume 200, Cambridge University Press, 2013 (with a collaboration of Sándor Kovács) | MR 3057950 | Zbl 1282.14028

[KPS18] Kuznetsov, Alexander G.; Prokhorov, Yuri G.; Shramov, Constantin A. Hilbert schemes of lines and conics and automorphism groups of Fano threefolds, Jpn. J. Math., Volume 13 (2018) no. 1, pp. 109-185 | Article | MR 3776469 | Zbl 1406.14031

[Lin20] Lin, Hsueh-Yung Lagrangian constant cycle subvarieties in Lagrangian fibrations, Int. Math. Res. Not. IMRN (2020) no. 1, pp. 14-24 | MR 4050560

[Nak04] Nakayama, Noboru Zariski–decomposition and abundance, MSJ Memoirs, Volume 14, Mathematical Society of Japan, 2004 | MR 2104208 | Zbl 1061.14018

[Pet11] Peternell, Thomas Generically nef vector bundles and geometric applications, Complex and differential geometry. Conference held at Leibniz Universität Hannover, Germany, September 14–18, 2009. Proceedings (Springer Monographs in Mathematics) Volume 8, Springer, 2011, pp. 345-368 | MR 2964482 | Zbl 1231.14009

[PS00] Peternell, Thomas; Sommese, Andrew J. Ample vector bundles and branched coverings, Commun. Algebra, Volume 28 (2000) no. 12, pp. 5573-5599 (with an appendix by Robert Lazarsfeld, Special issue in honor of Robin Hartshorne) | Article | MR 1808590 | Zbl 0982.14025

[Uen75] Ueno, Kenji Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, Volume 439, Springer, 1975 (notes written in collaboration with P. Cherenack) | MR 506253 | Zbl 0299.14007

[Vie95] Viehweg, Eckart Quasi-projective moduli for polarized manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Volume 30, Springer, 1995 | MR 1368632 | Zbl 0844.14004

[Voi03] Voisin, Claire On some problems of Kobayashi and Lang; algebraic approaches, Current developments in mathematics, 2003, International Press, 2003, pp. 53-125 | Article | Zbl 1215.32014