On the structure of a log smooth pair in the equality case of the Bogomolov–Gieseker inequality
[Sur la structure d’une paire lisse logarithmique dans le cas d’égalité de l’inégalité de Bogomolov–Gieseker]
Iwai, Masataka1
1 Department of Mathematics, Graduate School of Science, Osaka University, 1-1, Machikaneyama-cho, Toyonaka, Osaka 560-0043 (Japan)
Annales de l'Institut Fourier, Online first, 17 p.

Nous étudions la structure d’une paire lisse logarithmique lorsque l’égalité tient dans l’inégalité de Bogomolov–Gieseker pour le faisceau tangent logarithmique et que ce faisceau est semistable par rapport à un certain diviseur ample. Nous étudions également le cas du sheaf d’extension canonique.

We study the structure of a log smooth pair when the equality holds in the Bogomolov–Gieseker inequality for the logarithmic tangent bundle and this bundle is semistable with respect to some ample divisor. We also study the case of the canonical extension sheaf.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3651
Classification : 32Q30, 14M22, 14E30, 32Q26
Keywords: Bogomolov–Gieseker inequality, Logarithmic tangent bundle, Log smooth pair, Projectively flat, Numerically projectively flat, Stability, Uniformization, Rational curve, MRC fibration, Rationally connected.
Mot clés : Inégalité de Bogomolov–Gieseker, fibré tangent logarithmique, paire lisse logarithmique, projectivement plat, numériquement projectivement plat, stabilité, uniformisation, courbe rationnelle, fibration MRC, rationnellement connexe.
Iwai, Masataka 1

1 Department of Mathematics, Graduate School of Science, Osaka University, 1-1, Machikaneyama-cho, Toyonaka, Osaka 560-0043 (Japan) 
@unpublished{AIF_0__0_0_A100_0,
     author = {Iwai, Masataka},
     title = {On the structure of a log smooth pair in the equality case of the {Bogomolov{\textendash}Gieseker} inequality},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3651},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Iwai, Masataka
TI  - On the structure of a log smooth pair in the equality case of the Bogomolov–Gieseker inequality
JO  - Annales de l'Institut Fourier
PY  - 2024
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3651
LA  - en
ID  - AIF_0__0_0_A100_0
ER  - 
%0 Unpublished Work
%A Iwai, Masataka
%T On the structure of a log smooth pair in the equality case of the Bogomolov–Gieseker inequality
%J Annales de l'Institut Fourier
%D 2024
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3651
%G en
%F AIF_0__0_0_A100_0
Iwai, Masataka. On the structure of a log smooth pair in the equality case of the Bogomolov–Gieseker inequality. Annales de l'Institut Fourier, Online first, 17 p.

[1] 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 | MR | Zbl

[2] Brown, Morgan V.; McKernan, James; Svaldi, Roberto; Zong, Hong R. A geometric characterization of toric varieties, Duke Math. J., Volume 167 (2018) no. 5, pp. 923-968 | DOI | MR | Zbl

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

[4] 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

[5] Campana, Frédéric; Peternell, Thomas Projective manifolds whose tangent bundles are numerically effective, Math. Ann., Volume 289 (1991) no. 1, pp. 169-187 | DOI | MR | Zbl

[6] Chintapalli, Seshadri; Iyer, Jaya NN Semistability of logarithmic cotangent bundle on some projective manifolds, Commun. Algebra, Volume 42 (2014) no. 4, pp. 1732-1746 | DOI | MR | Zbl

[7] Debarre, Olivier Higher-dimensional algebraic geometry, Universitext, Springer, 2001, xiv+233 pages | DOI | MR | Zbl

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

[9] Deng, Ya A characterization of complex quasi-projective manifolds uniformized by unit balls (2020) (to appear in Math. Ann., https://arxiv.org/abs/2006.16178v2)

[10] Druel, Stéphane Projectively flat log smooth pairs (2021) (https://arxiv.org/abs/2112.05449)

[11] Druel, Stéphane; Bianco, Federico Lo Numerical characterization of some toric fiber bundles, Math. Z., Volume 300 (2022) no. 4, pp. 3357-3382 | DOI | MR | Zbl

[12] Esnault, Hélène; Viehweg, Eckart Lectures on vanishing theorems, DMV Seminar, 20, Birkhäuser, 1992, vi+164 pages | DOI | MR | Zbl

[13] Fujino, Osamu Iitaka conjecture. An introduction, SpringerBriefs in Mathematics, Springer, 2020, xiv+128 pages | DOI | MR | Zbl

[14] Fujino, Osamu; Miyamoto, Keisuke A characterization of projective spaces from the Mori theoretic viewpoint, Osaka J. Math., Volume 58 (2021) no. 4, pp. 827-837 | MR | Zbl

[15] Fujita, Kento Simple normal crossing Fano varieties and log Fano manifolds, Nagoya Math. J., Volume 214 (2014), pp. 95-123 | DOI | MR | Zbl

[16] Greb, Daniel; Kebekus, Stefan; Peternell, Thomas Projectively flat klt varieties, J. Éc. Polytech., Math., Volume 8 (2021), pp. 1005-1036 | DOI | Numdam | MR | Zbl

[17] Greb, Daniel; Kebekus, Stefan; Peternell, Thomas Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor, J. Algebr. Geom., Volume 31 (2022) no. 3, pp. 467-496 | DOI | MR | Zbl

[18] Guenancia, Henri; Taji, Behrouz Orbifold stability and Miyaoka–Yau inequality for minimal pairs, Geom. Topol., Volume 26 (2022) no. 4, pp. 1435-1482 | DOI | MR | Zbl

[19] Hartshorne, Robin Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1977, xvi+496 pages | DOI | MR | Zbl

[20] Hosono, Genki; Iwai, Masataka; Matsumura, Shin-ichi On projective manifolds with pseudo-effective tangent bundle, J. Inst. Math. Jussieu, Volume 21 (2022) no. 5, pp. 1801-1830 | DOI | MR | Zbl

[21] Iitaka, Shigeru Algebraic geometry. An introduction to birational geometry of algebraic varieties, North-Holland Mathematical Library, 24, Springer, 1982, x+357 pages | MR

[22] Kato, Kazuya Logarithmic structures of Fontaine–Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, 1989, pp. 191-224 | MR | Zbl

[23] Kawamata, Yujiro On deformations of compactifiable complex manifolds, Math. Ann., Volume 235 (1978) no. 3, pp. 247-265 | DOI | MR | Zbl

[24] Kebekus, Stefan; Kovács, Sándor J. Families of canonically polarized varieties over surfaces, Invent. Math., Volume 172 (2008) no. 3, pp. 657-682 | DOI | MR | Zbl

[25] Kobayashi, Shoshichi Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15, Princeton University Press, 1987, xii+305 pages (Kanô Memorial Lectures, 5) | DOI | MR | Zbl

[26] Lazarsfeld, Robert Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 49, Springer, 2004, xviii+385 pages | DOI | MR | Zbl

[27] Li, Chi On the stability of extensions of tangent sheaves on Kähler-Einstein Fano/Calabi-Yau pairs, Math. Ann., Volume 381 (2021) no. 3-4, pp. 1943-1977 | DOI | MR | Zbl

[28] Liu, Jie; Ou, Wenhao; Yang, Xiaokui Projective manifolds whose tangent bundle contains a strictly nef subsheaf (2020) (to appear in J. Algebraic Geom., https://arxiv.org/abs/2004.08507)

[29] Meng, Sheng; Zhang, De-Qi Characterizations of toric varieties via polarized endomorphisms, Math. Z., Volume 292 (2019) no. 3-4, pp. 1223-1231 | DOI | MR | Zbl

[30] Nakayama, Noboru Zariski-decomposition and abundance, MSJ Memoirs, 14, Mathematical Society of Japan, 2004, xiv+277 pages | DOI | MR | Zbl

[31] Tian, Gang On stability of the tangent bundles of Fano varieties, Int. J. Math., Volume 3 (1992) no. 3, pp. 401-413 | DOI | MR | Zbl

[32] Tian, Gang; Yau, Shing-Tung Existence of Kähler–Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, Mathematical aspects of string theory (San Diego, Calif., 1986) (Advanced Series in Mathematical Physics), Volume 1, World Scientific, 1987, pp. 574-628 | DOI | MR | Zbl

[33] Tsuji, Hajime A characterization of ball quotients with smooth boundary, Duke Math. J., Volume 57 (1988) no. 2, pp. 537-553 | DOI | MR | Zbl

[34] Winkelmann, Jörg On manifolds with trivial logarithmic tangent bundle, Osaka J. Math., Volume 41 (2004) no. 2, pp. 473-484 | MR | Zbl

Cité par Sources :