Classification of Mukai pairs with corank $3$

Kanemitsu, Akihiro

doi:10.5802/aif.3242

Classification of Mukai pairs with corank

3

[Classification des paires de Mukai de corang

3

]

Kanemitsu, Akihiro ¹

¹ Graduate School of Mathematical Sciences The University of Tokyo 3-8-1 Komaba Meguro-ku Tokyo 153-8914 (Japan)

Annales de l'Institut Fourier, Tome 69 (2019) no. 1, pp. 231-282.

Résumé
Abstract

On classifie les paires $(X, ℰ)$ où $X$ est une variété de Fano lisse de dimension $n \geq 5$ et $ℰ$ est un fibré vectoriel ample de rang $n - 2$ sur $X$ tel que $c_{1} (ℰ) = c_{1} (X)$ .

We classify the pairs $(X, ℰ)$ where $X$ is a smooth Fano manifold of dimension $n \geq 5$ and $ℰ$ is an ample vector bundle of rank $n - 2$ with $c_{1} (ℰ) = c_{1} (X)$ .

Reçu le : 2017-06-22
Révisé le : 2018-01-12
Accepté le : 2018-02-02
Publié le : 2019-06-03

DOI : 10.5802/aif.3242

Classification : 14J45, 14J40, 14J60
Keywords: Fano manifold, vector bundle
Mot clés : variété de Fano, fibré vectoriel

Affiliations des auteurs :

Kanemitsu, Akihiro ¹

¹ Graduate School of Mathematical Sciences The University of Tokyo 3-8-1 Komaba Meguro-ku Tokyo 153-8914 (Japan)

@article{AIF_2019__69_1_231_0,
     author = {Kanemitsu, Akihiro},
     title = {Classification of {Mukai} pairs with corank $3$},
     journal = {Annales de l'Institut Fourier},
     pages = {231--282},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {1},
     year = {2019},
     doi = {10.5802/aif.3242},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3242/}
}

TY  - JOUR
AU  - Kanemitsu, Akihiro
TI  - Classification of Mukai pairs with corank $3$
JO  - Annales de l'Institut Fourier
PY  - 2019
SP  - 231
EP  - 282
VL  - 69
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.3242/
DO  - 10.5802/aif.3242
LA  - en
ID  - AIF_2019__69_1_231_0
ER  -

%0 Journal Article
%A Kanemitsu, Akihiro
%T Classification of Mukai pairs with corank $3$
%J Annales de l'Institut Fourier
%D 2019
%P 231-282
%V 69
%N 1
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.3242/
%R 10.5802/aif.3242
%G en
%F AIF_2019__69_1_231_0

Kanemitsu, Akihiro. Classification of Mukai pairs with corank $3$. Annales de l'Institut Fourier, Tome 69 (2019) no. 1, pp. 231-282. doi : 10.5802/aif.3242. http://archive.numdam.org/articles/10.5802/aif.3242/

Bibliographie
Cité par

[1] Ambro, Florin Ladders on Fano varieties, J. Math. Sci., New York, Volume 94 (1999) no. 1, pp. 1126-1135 | DOI | MR | Zbl

[2] Ancona, Vincenzo; Peternell, Thomas; Wiśniewski, Jarosław A. Fano bundles and splitting theorems on projective spaces and quadrics, Pac. J. Math., Volume 163 (1994) no. 1, pp. 17-42 http://projecteuclid.org/euclid.pjm/1102622627 | DOI | MR | Zbl

[3] Andreatta, Marco Some remarks on the study of good contractions, Manuscr. Math., Volume 87 (1995) no. 3, pp. 359-367 | DOI | MR | Zbl

[4] Andreatta, Marco; Ballico, Edoardo; Wiśniewski, Jarosław A. Vector bundles and adjunction, Int. J. Math., Volume 3 (1992) no. 3, pp. 331-340 | DOI | MR | Zbl

[5] Andreatta, Marco; Ballico, Edoardo; Wiśniewski, Jarosław A. Two theorems on elementary contractions, Math. Ann., Volume 297 (1993) no. 2, pp. 191-198 | DOI | MR | Zbl

[6] Andreatta, Marco; Chierici, Elena; Occhetta, Gianluca Generalized Mukai conjecture for special Fano varieties, Cent. Eur. J. Math., Volume 2 (2004) no. 2, pp. 272-293 | DOI | MR | Zbl

[7] Andreatta, Marco; Mella, Massimiliano Contractions on a manifold polarized by an ample vector bundle, Trans. Am. Math. Soc., Volume 349 (1997) no. 11, pp. 4669-4683 | DOI | MR | Zbl

[8] Andreatta, Marco; Wiśniewski, Jarosław A. On manifolds whose tangent bundle contains an ample subbundle, Invent. Math., Volume 146 (2001) no. 1, pp. 209-217 | DOI | MR | Zbl

[9] Anghel, Cristian; Manolache, Nicolae Globally generated vector bundles on $ℙ^{n}$ with $c_{1} = 3$ , Math. Nachr., Volume 286 (2013) no. 14-15, pp. 1407-1423 | MR | Zbl

[10] 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

[11] Chierici, Elena; Occhetta, Gianluca The cone of curves of Fano varieties of coindex four, Int. J. Math., Volume 17 (2006) no. 10, pp. 1195-1221 | DOI | MR | Zbl

[12] Cho, Koji; Miyaoka, Yoichi; Shepherd-Barron, Nicholas I. Characterizations of projective space and applications to complex symplectic manifolds, Higher dimensional birational geometry (Kyoto, 1997) (Advanced Studies in Pure Mathematics), Volume 35, Mathematical Society of Japan, 2002, pp. 1-88 | MR | Zbl

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

[14] Dedieu, Thomas; Höring, Andreas Numerical characterisation of quadrics, Algebr. Geom., Volume 4 (2017) no. 1, pp. 120-135 | MR | Zbl

[15] Fujita, Kento Around the Mukai conjecture for Fano manifolds, Eur. J. Math., Volume 2 (2016) no. 1, pp. 120-139 | DOI | MR | Zbl

[16] Fujita, Takao On the structure of polarized varieties with $Δ$ -genera zero, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 22 (1975), pp. 103-115 | MR | Zbl

[17] Fujita, Takao Classification of projective varieties of $Δ$ -genus one, Proc. Japan Acad., Ser. A, Volume 58 (1982) no. 3, pp. 113-116 http://projecteuclid.org/euclid.pja/1195516108 | DOI | MR | Zbl

[18] Fujita, Takao On polarized varieties of small $Δ$ -genera, Tôhoku Math. J., Volume 34 (1982) no. 3, pp. 319-341 | DOI | MR | Zbl

[19] Fujita, Takao On polarized manifolds whose adjoint bundles are not semipositive, Algebraic geometry (Sendai, 1985) (Advanced Studies in Pure Mathematics), Volume 10, North-Holland, 1987, pp. 167-178 | DOI | MR | Zbl

[20] Fujita, Takao On adjoint bundles of ample vector bundles, Complex algebraic varieties (Bayreuth, 1990) (Lecture Notes in Mathematics), Volume 1507, Springer, 1992, pp. 105-112 | DOI | MR | Zbl

[21] Horrocks, Goeffrey Vector bundles on the punctured spectrum of a local ring, Proc. Lond. Math. Soc., Volume 14 (1964), pp. 689-713 | DOI | MR | Zbl

[22] Ionescu, Paltin Generalized adjunction and applications, Math. Proc. Camb. Philos. Soc., Volume 99 (1986) no. 3, pp. 457-472 | DOI | MR | Zbl

[23] Iskovskikh, Vasily A.; Prokhorov, Yuri G. Fano varieties, Algebraic geometry, V (Encyclopaedia of Mathematical Sciences), Volume 47, Springer, 1999, pp. 1-247 | MR | Zbl

[24] Kachi, Yasuyuki; Sato, Eiichi Polarized varieties whose points are joined by rational curves of small degrees, Ill. J. Math., Volume 43 (1999) no. 2, pp. 350-390 http://projecteuclid.org/euclid.ijm/1255985220 | DOI | MR | Zbl

[25] Kachi, Yasuyuki; Sato, Eiichi Segre’s reflexivity and an inductive characterization of hyperquadrics, Mem. Am. Math. Soc., Volume 160 (2002) no. 763, x+116 pages | DOI | MR | Zbl

[26] Kanemitsu, Akihiro Extremal rays and nefness of tangent bundles (2016) (https://arxiv.org/abs/1605.04680, to appear in Mich. Math. J.)

[27] Kanemitsu, Akihiro Fano $5$ -folds with nef tangent bundles, Math. Res. Lett., Volume 24 (2017) no. 5, pp. 1453-1475 | DOI | MR | Zbl

[28] Kawamata, Yujiro; Matsuda, Katsumi; Matsuki, Kenji Introduction to the minimal model problem, Algebraic geometry (Sendai, 1985) (Advanced Studies in Pure Mathematics), Volume 10, North-Holland, 1987, pp. 283-360 | DOI | MR | Zbl

[29] Kebekus, Stefan Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron, Complex geometry (Göttingen, 2000), Springer, 2002, pp. 147-155 | DOI | MR | Zbl

[30] Kobayashi, Shoshichi; Ochiai, Takushiro Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ., Volume 13 (1973), pp. 31-47 | DOI | MR | Zbl

[31] Kollár, János Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 32, Springer, 1996, viii+320 pages | DOI | MR | Zbl

[32] Kollár, János; Miyaoka, Yoichi; Mori, Shigefumi Rational connectedness and boundedness of Fano manifolds, J. Differ. Geom., Volume 36 (1992) no. 3, pp. 765-779 http://projecteuclid.org/euclid.jdg/1214453188 | MR | Zbl

[33] Kollár, János; Miyaoka, Yoichi; Mori, Shigefumi Rational curves on Fano varieties, Classification of irregular varieties (Trento, 1990) (Lecture Notes in Mathematics), Volume 1515, Springer, 1992, pp. 100-105 | DOI | MR | Zbl

[34] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, 1998, viii+254 pages (With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original) | DOI | MR | Zbl

[35] Lanteri, Antonio Ample vector bundles with sections vanishing on surfaces of Kodaira dimension zero, Matematiche, Volume 51 (1996) no. suppl., pp. 115-125 | MR | Zbl

[36] Lazarsfeld, Robert Some applications of the theory of positive vector bundles, Complete intersections (Acireale, 1983) (Lecture Notes in Mathematics), Volume 1092, Springer, 1984, pp. 29-61 | DOI | MR | Zbl

[37] Mella, Massimiliano Existence of good divisors on Mukai varieties, J. Algebr. Geom., Volume 8 (1999) no. 2, pp. 197-206 | MR | Zbl

[38] Miyaoka, Yoichi The Chern classes and Kodaira dimension of a minimal variety, Algebraic geometry (Sendai, 1985) (Advanced Studies in Pure Mathematics), Volume 10, North-Holland, 1987, pp. 449-476 | DOI | MR | Zbl

[39] Miyaoka, Yoichi Numerical characterisations of hyperquadrics, Complex analysis in several variables—Memorial Conference of Kiyoshi Oka’s Centennial Birthday (Advanced Studies in Pure Mathematics), Volume 42, Mathematical Society of Japan, 2004, pp. 209-235 | DOI | MR | Zbl

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

[41] Mukai, Shigeru Problems on characterization of the complex projective space, Birational Geometry of Algebraic Varieties, Open Problems, Katata, the 23rd Int’l Symp., Taniguchi Foundation, 1988, pp. 57-60

[42] Mukai, Shigeru Biregular classification of Fano $3$ -folds and Fano manifolds of coindex $3$ , Proc. Natl. Acad. Sci. USA, Volume 86 (1989) no. 9, pp. 3000-3002 | DOI | MR | Zbl

[43] Muñoz, Roberto; Occhetta, Gianluca; Solá Conde, Luis E. Uniform vector bundles on Fano manifolds and applications, J. Reine Angew. Math., Volume 664 (2012), pp. 141-162 | MR | Zbl

[44] Novelli, Carla; Occhetta, Gianluca Ruled Fano fivefolds of index two, Indiana Univ. Math. J., Volume 56 (2007) no. 1, pp. 207-241 | DOI | MR | Zbl

[45] Novelli, Carla; Occhetta, Gianluca Projective manifolds containing a large linear subspace with nef normal bundle, Mich. Math. J., Volume 60 (2011) no. 2, pp. 441-462 | DOI | MR | Zbl

[46] Occhetta, Gianluca A note on the classification of Fano manifolds of middle index, Manuscr. Math., Volume 117 (2005) no. 1, pp. 43-49 | DOI | MR | Zbl

[47] Occhetta, Gianluca A characterization of products of projective spaces, Can. Math. Bull., Volume 49 (2006) no. 2, pp. 270-280 | DOI | MR | Zbl

[48] Occhetta, Gianluca; Solá Conde, Luis E.; Watanabe, Kiwamu; Wiśniewski, Jarosław A. Fano manifolds whose elementary contractions are smooth $ℙ^{1}$ -fibrations: a geometric characterization of flag varieties, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 17 (2017) no. 2, pp. 573-607 | MR | Zbl

[49] Ohno, Masahiro Classification of generalized polarized manifolds by their nef values, Adv. Geom., Volume 6 (2006) no. 4, pp. 543-599 | DOI | MR | Zbl

[50] Ohno, Masahiro Nef vector bundles on a projective space or a hyperquadric with the first Chern class small (2016) (https://arxiv.org/abs/1409.4191v3)

[51] Ohno, Masahiro Nef vector bundles on a projective space with first Chern class 3 and second Chern class 8, Matematiche, Volume 72 (2017) no. 2, pp. 69-81 | MR | Zbl

[52] Ohno, Masahiro Nef vector bundles on a projective space with first Chern class 3 (2018) (https://arxiv.org/abs/1604.05847v7)

[53] Ohno, Masahiro; Terakawa, Hiroyuki A spectral sequence and nef vector bundles of the first Chern class two on hyperquadrics, Ann. Univ. Ferrara, Sez. VII, Sci. Mat., Volume 60 (2014) no. 2, pp. 397-406 | DOI | MR | Zbl

[54] Okonek, Christian; Schneider, Michael; Spindler, Heinz Vector bundles on complex projective spaces, Progress in Mathematics, 3, Birkhäuser, 1980, vii+389 pages | MR | Zbl

[55] Ottaviani, Giorgio Spinor bundles on quadrics, Trans. Am. Math. Soc., Volume 307 (1988) no. 1, pp. 301-316 | DOI | MR | Zbl

[56] Ottaviani, Giorgio On Cayley bundles on the five-dimensional quadric, Boll. Unione Mat. Ital., VII. Ser., A, Volume 4 (1990) no. 1, pp. 87-100 | MR | Zbl

[57] Peternell, Thomas A characterization of $P_{n}$ by vector bundles, Math. Z., Volume 205 (1990) no. 3, pp. 487-490 | DOI | MR | Zbl

[58] Peternell, Thomas Ample vector bundles on Fano manifolds, Int. J. Math., Volume 2 (1991) no. 3, pp. 311-322 | DOI | MR | Zbl

[59] Peternell, Thomas; Szurek, Michał; Wiśniewski, Jarosław A. Fano manifolds and vector bundles, Math. Ann., Volume 294 (1992) no. 1, pp. 151-165 | DOI | MR | Zbl

[60] Peternell, Thomas; Szurek, Michał; Wiśniewski, Jarosław A. Numerically effective vector bundles with small Chern classes, Complex algebraic varieties (Bayreuth, 1990) (Lecture Notes in Mathematics), Volume 1507, Springer, 1992, pp. 145-156 | DOI | MR | Zbl

[61] Sato, Eiichi Uniform vector bundles on a projective space, J. Math. Soc. Japan, Volume 28 (1976) no. 1, pp. 123-132 | DOI | MR | Zbl

[62] Sierra, José Carlos; Ugaglia, Luca On globally generated vector bundles on projective spaces II, J. Pure Appl. Algebra, Volume 218 (2014) no. 1, pp. 174-180 | DOI | MR | Zbl

[63] Szurek, Michał; Wiśniewski, Jarosław A. Fano bundles over $ℙ^{3}$ and $Q_{3}$ , Pac. J. Math., Volume 141 (1990) no. 1, pp. 197-208 http://projecteuclid.org/euclid.pjm/1102646779 | DOI | MR | Zbl

[64] Szurek, Michał; Wiśniewski, Jarosław A. On Fano manifolds, which are $P^{k}$ -bundles over $P^{2}$ , Nagoya Math. J., Volume 120 (1990), pp. 89-101 http://projecteuclid.org/euclid.nmj/1118782199 | DOI | MR | Zbl

[65] Tironi, Andrea L. Nefness of adjoint bundles for ample vector bundles of corank 3, Math. Nachr., Volume 286 (2013) no. 14-15, pp. 1548-1570 | MR | Zbl

[66] Watanabe, Kiwamu Lengths of chains of minimal rational curves on Fano manifolds, J. Algebra, Volume 325 (2011), pp. 163-176 | DOI | MR | Zbl

[67] Wiśniewski, Jarosław A. Length of extremal rays and generalized adjunction, Math. Z., Volume 200 (1989) no. 3, pp. 409-427 | DOI | MR | Zbl

[68] Wiśniewski, Jarosław A. Ruled Fano $4$ -folds of index $2$ , Proc. Am. Math. Soc., Volume 105 (1989) no. 1, pp. 55-61 | DOI | MR | Zbl

[69] Wiśniewski, Jarosław A. On a conjecture of Mukai, Manuscr. Math., Volume 68 (1990) no. 2, pp. 135-141 | DOI | MR | Zbl

[70] Wiśniewski, Jarosław A. On contractions of extremal rays of Fano manifolds, J. Reine Angew. Math., Volume 417 (1991), pp. 141-157 | DOI | MR | Zbl

[71] Wiśniewski, Jarosław A. A report on Fano manifolds of middle index and $b_{2} \geq 2$ , Projective geometry with applications (Lecture Notes in Pure and Applied Mathematics), Volume 166, Dekker, New York, 1994, pp. 19-26 | MR | Zbl

[72] Ye, Yun-Gang; Zhang, Qi On ample vector bundles whose adjunction bundles are not numerically effective, Duke Math. J., Volume 60 (1990) no. 3, pp. 671-687 | DOI | MR | Zbl

[73] Zhang, Qi A theorem on the adjoint system for vector bundles, Manuscr. Math., Volume 70 (1991) no. 2, pp. 189-201 | DOI | MR | Zbl

[74] Zhang, Qi On the spannedness of adjunction of vector bundles, Math. Z., Volume 223 (1996) no. 4, pp. 725-729 | DOI | MR | Zbl

Cité par Sources :