Open problems on structure of positively curved projective varieties

Matsumura, Shin-ichi

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 31 (2022) no. 3, pp. 1011-1029.

Nous fournissons des suppléments et des problèmes ouverts liés aux théorèmes de structure pour les fibrations maximales rationnellement connectées de certaines variétés projectives à courbure positive, y compris les variétés projectives lisses avec une courbure de section holomorphe semi-positive, un faisceau tangent pseudo-efficace et un diviseur anticanonique nef.

We provide supplements and open problems related to structure theorems for maximal rationally connected fibrations of certain positively curved projective varieties, including smooth projective varieties with semi-positive holomorphic sectional curvature, pseudo-effective tangent bundle, and nef anti-canonical divisor.

Classification : 32J25, 53C25, 14E30
Mots clés : Rational curves, Maximal rationally connected fibrations, Albanese maps, Structure theorems, Holomorphic sectional curvatures, Pseudo-effective tangent bundles, Nef anti-canonical divisors, klt pairs.

[1] Alvarez, Angelynn; Heier, Gordon; Zheng, Fangyang On projectivized vector bundles and positive holomorphic sectional curvature, Proc. Am. Math. Soc., Volume 146 (2018) no. 7, pp. 2877-2882 | DOI | MR | Zbl

[2] Bauer, Thomas; Kovács, Sándor J.; Küronya, Alex; Mistretta, Ernesto C.; Szemberg, Tomasz; Urbinati, Stefano On positivity and base loci of vector bundles, Eur. J. Math., Volume 1 (2015) no. 2, pp. 229-249 | DOI | MR | Zbl

[3] Berndtsson, Bo Curvature of vector bundles associated to holomorphic fibrations, Ann. Math., Volume 169 (2009) no. 2, pp. 531-560 | DOI | MR | Zbl

[4] Berndtsson, Bo; Păun, Mihai Bergman kernels and the pseudoeffectivity of relative canonical divisors, Duke Math. J., Volume 145 (2008) no. 2, pp. 341-378 | DOI | Zbl

[5] Boucksom, Sébastien Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. Éc. Norm. Supér., Volume 37 (2004) no. 1, pp. 45-76 | DOI | Numdam | MR | Zbl

[6] 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

[7] Campana, Frédéric Connexité rationnelle des variétés de Fano, Ann. Sci. Éc. Norm. Supér., Volume 25 (1992) no. 5, pp. 539-545 | DOI | Zbl

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

[9] Campana, Frédéric Orbifold slope rational connectedness (2017) (https://arxiv.org/abs/1607.07829v2)

[10] Campana, Frédéric; Cao, Junyan; Matsumura, Shin-ichi Projective klt pairs with nef anti-canonical divisor, Algebr. Geom., Volume 8 (2021) no. 4, pp. 430-464 | DOI | MR | Zbl

[11] Campana, Frédéric; Demailly, Jean-Pierre; Peternell, Thomas Rationally connected manifolds and semipositivity of the Ricci curvature, Recent advances in algebraic geometry (London Mathematical Society Lecture Note Series), Volume 417, Cambridge University Press, 2014, pp. 71-91 | Zbl

[12] Cao, Junyan Albanese maps of projective manifolds with nef anticanonical divisors, Ann. Sci. Éc. Norm. Supér., Volume 52 (2019) no. 5, pp. 1137-1154 | DOI | MR | Zbl

[13] Cao, Junyan; Demailly, Jean-Pierre; Matsumura, Shin-ichi A general extension theorem for cohomology classes on non reduced analytic subspaces, Sci. China, Math., Volume 60 (2017) no. 6, pp. 949-962 | MR | Zbl

[14] Cao, Junyan; Höring, Andreas A decomposition theorem for projective manifolds with nef anticanonical bundle, J. Algebr. Geom., Volume 28 (2019) no. 3, pp. 567-597 | MR | Zbl

[15] Cheeger, Jeff; Colding, Tobias H. Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. Math., Volume 144 (1996) no. 1, pp. 189-237 | DOI | MR | Zbl

[16] Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael Compact complex manifolds with numerically effective tangent bundles, Complex Manifolds, Volume 3 (1994) no. 2, pp. 295-345 | MR | Zbl

[17] Diverio, Simone; Trapani, Stefano Quasi-negative holomorphic sectional curvature and positivity of the canonical divisor, J. Differ. Geom., Volume 111 (2019) no. 2, pp. 303-314 | Zbl

[18] Druel, Stéphane A decomposition theorem for singular spaces with trivial canonical class of dimension at most five, Invent. Math., Volume 211 (2018) no. 1, pp. 245-296 | DOI | MR | Zbl

[19] Ein, Lawrence; Lazarsfeld, Robert; Mustaţă, Mircea; Nakamaye, Michael; Popa, Mihnea Asymptotic invariants of base loci, Ann. Inst. Fourier, Volume 56 (2006) no. 6, pp. 1701-1734 | Numdam | MR | Zbl

[20] Ein, Lawrence; Lazarsfeld, Robert; Mustaţă, Mircea; Nakamaye, Michael; Popa, Mihnea Restricted volumes and base loci of linear series, Am. J. Math., Volume 131 (2009) no. 3, pp. 607-651 | DOI | MR | Zbl

[21] Ejiri, Sho; Gongyo, Yoshinori Nef anti-canonical divisors and rationally connected fibrations, Compos. Math., Volume 155 (2019) no. 7, pp. 1444-1456 | DOI | MR | Zbl

[22] Ejiri, Sho; Iwai, Masataka; Matsumura, Shin-ichi On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces (2005) (https://arxiv.org/abs/2005.04566v1)

[23] Fulger, Mihai; Murayama, Takumi Seshadri constants for vector bundles, J. Pure Appl. Algebra, Volume 225 (2021) no. 4, 106559, 35 pages | MR | Zbl

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

[25] Greb, Daniel; Guenancia, Henri; Kebekus, Stefan Klt varieties with trivial canonical class: holonomy, differential forms, and fundamental groups, Geom. Topol., Volume 23 (2019) no. 4, pp. 2051-2124 | DOI | MR | Zbl

[26] Greb, Daniel; Kebekus, Stefan; Peternell, Thomas Singular spaces with trivial canonical class, Minimal models and extremal rays (Kyoto, 2011) (Advanced Studies in Pure Mathematics), Volume 70, Mathematical Society of Japan, 2016, pp. 67-113 | DOI | MR | Zbl

[27] Hacon, Christopher; McKernan, James On Shokurov’s rational connectedness conjecture, Duke Math. J., Volume 138 (2007) no. 1, pp. 119-136 | MR | Zbl

[28] Hacon, Christopher; Popa, Mihnea; Schnell, Christian Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Pǎun, Local and global methods in algebraic geometry (Contemporary Mathematics), Volume 712, American Mathematical Society, 2018, pp. 143-195 | Zbl

[29] Heier, Gordon; Lu, Steven S. Y.; Wong, Bun; Zheng, Fangyang Reduction of manifolds with semi-negative holomorphic sectional curvature, Math. Ann., Volume 372 (2018) no. 3-4, pp. 951-962 | DOI | MR | Zbl

[30] Heier, Gordon; Wong, Bun On projective Kähler manifolds of partially positive curvature and rational connectedness, Doc. Math., Volume 25 (2020), pp. 219-238 | Zbl

[31] Hitchin, Nigel J. On the curvature of rational surfaces, Differential geometry (Stanford Univ., Stanford, 1973) (Proceedings of Symposia in Pure Mathematics), Volume 27 part 2, American Mathematical Society, 1973, pp. 65-80 | Zbl

[32] Höring, Andreas Uniruled varieties with split tangent bundle, Math. Z., Volume 256 (2007) no. 3, pp. 465-479 | DOI | MR | Zbl

[33] Höring, Andreas; Liu, Jie; Shao, Feng Examples of Fano manifolds with non-pseudoeffective tangent bundle (2020) (https://arxiv.org/abs/2003.09476v1)

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

[35] Hosono, Genki; Iwai, Masataka; Matsumura, Shin-ichi On projective manifolds with pseudo-effective tangent bundle (2021) (to appear in J. Inst. Math. Jussieu, https://doi.org/10.1017/S1474748020000754) | DOI

[36] Howard, Alan; Smyth, Brian; Wu, Hung-Hsi On compact Kähler manifolds of nonnegative bisectional curvature I and II, Acta Math., Volume 147 (1981) no. 1-2, pp. 51-70 | DOI | Zbl

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

[38] Matsumura, Shin-ichi Asymptotic cohomology vanishing and a converse to the Andreotti-Grauert theorem on surfaces, Ann. Inst. Fourier, Volume 63 (2013) no. 6, pp. 2199-2221 | DOI | Numdam | MR | Zbl

[39] Matsumura, Shin-ichi On projective manifolds with semi-positive holomorphic sectional curvature (2018) (https://arxiv.org/abs/1811.04182v1, to appear in Am. J. Math.)

[40] Matsumura, Shin-ichi On the image of MRC fibrations of projective manifolds with semi-positive holomorphic sectional curvature, Pure Appl. Math. Q., Volume 16 (2020) no. 5, pp. 1443-1463 | MR | Zbl

[41] Mok, Ngaiming The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, J. Differ. Geom., Volume 27 (1988) no. 2, pp. 179-214 | Zbl

[42] Mori, Shigefumi Projective manifolds with ample tangent bundles, Ann. Math., Volume 110 (1979) no. 3, pp. 593-606 | DOI | MR | Zbl

[43] Nakayama, Noboru Zariski-decomposition and abundance, MSJ Memoirs, 14, Mathematical Society of Japan, 2004, xiv+277 pages

[44] Paris, Mtthieu Quelques aspects de la positivité du fibré tangent des variétés projectives complexes, Ph. D. Thesis, Université Joseph-Fourier (France) (2010) (https://tel.archives-ouvertes.fr/tel-00552308)

[45] Paun, Mihai Sur le groupe fondamental des variétés kählériennes compactes à classe de Ricci numériquement effective, C. R. Acad. Sci. Paris, Volume 324 (1997) no. 11, pp. 1249-1254 | DOI | MR | Zbl

[46] Păun, Mihai; Takayama, Shigeharu Positivity of twisted relative pluricanonical bundles and their direct images, J. Algebr. Geom., Volume 27 (2018) no. 2, pp. 211-272 | DOI | MR | Zbl

[47] Siu, Yum-Tong; Yau, Shing-Tung Compact Kähler manifolds of positive bisectional curvature, Invent. Math., Volume 59 (1980) no. 2, pp. 189-204 | Zbl

[48] Tosatti, Valentino; Yang, Xiaokui An extension of a theorem of Wu-Yau, J. Differ. Geom., Volume 107 (2017) no. 3, pp. 573-579 | MR | Zbl

[49] Wang, Juanyong Structure of projective varieties with nef anticanonical divisor: the case of log terminal singularities (2020) (https://arxiv.org/abs/2005.05782v2)

[50] Wu, Damin; Yau, Shing-Tung Negative holomorphic curvature and positive canonical divisor, Invent. Math., Volume 204 (2016) no. 2, pp. 595-604 | Zbl

[51] Yang, Xiaokui RC-positivity, rational connectedness and Yau’s conjecture, Camb. J. Math., Volume 6 (2018) no. 2, pp. 183-212 | DOI | MR | Zbl

[52] Yang, Xiaokui A partial converse to the Andreotti-Grauert theorem, Compos. Math., Volume 155 (2019) no. 1, pp. 89-99 | DOI | MR | Zbl

[53] Yang, Xiaokui RC-positive metrics on rationally connected manifolds, Forum Math. Sigma, Volume 8 (2020), e53, 19 pages | MR | Zbl

[54] Yau, Shing-Tung Problem section, Seminar on Differential Geometry (Annals of Mathematics Studies), Volume 102, Princeton University Press, 1982, pp. 669-706 | MR | Zbl

[55] Zhang, Qi On projective manifolds with nef anticanonical bundles, J. Reine Angew. Math., Volume 478 (1996), pp. 57-60 | MR | Zbl

[56] Zhang, Qi On projective varieties with nef anticanonical divisors, Math. Ann., Volume 332 (2005) no. 3, pp. 697-703 | DOI | MR | Zbl