Maps from K-trivial varieties and connectedness problems

Lazić, Vladimir; Peternell, Thomas

Annales Henri Lebesgue, Tome 3 (2020), pp. 473-500.

Dans cet article, nous étudions les variétés couvertes par des courbes rationnelles ou elliptiques. Nous montrons tout d’abord que les images des variétés de Calabi–Yau ou irréductibles symplectiques par des applications rationnelles sont presque toujours rationnellement connexes. Nous étudions ensuite les variétés elliptiquement connexes, elliptiquement connexes par chaînes ainsi que les variétés balayées par une famille de courbes elliptiques. Entre autres choses, nous montrons que les variétés de Calabi–Yau ou hyperkählériennes qui sont couvertes par une famille de courbes elliptiques contiennent des diviseurs uniréglés, et que les variétés elliptiquement connexes par chaînes de dimension au moins $2$ contiennent une courbe rationnelle, tout comme les variétés à fibré canonique trivial de groupe fondamental fini qui sont couvertes par des courbes elliptiques.

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.

Classification : 14J32, 14H10
Mots clés : Calabi–Yau varieties, families of elliptic curves, elliptically (chain) connected varieties

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

[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

[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

[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 | DOI | Zbl

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

[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, 4, Springer, 2004 | MR | Zbl

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

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

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

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

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

[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

[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 | Zbl

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

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

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

[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 | DOI | MR | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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

[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 | DOI | MR | Zbl

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

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

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

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

[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 | DOI | MR | Zbl

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

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

[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 | Zbl

[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) | DOI | MR | Zbl

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

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

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