Classification of Mukai pairs with dimension 4 and rank 2

Kanemitsu, Akihiro

doi:10.1090/tran/7824

Classification of Mukai pairs with dimension $4$ and rank $2$
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Abstract:

We give the complete classification of Mukai pairs of dimension $4$ and rank $2$ with Picard number one, that is, pairs $(X,\mathcal {E})$ where $X$ is a Fano $4$-fold with Picard number one, and $\mathcal {E}$ is an ample vector bundle of rank 2 on $X$ with $c_1(X) = c_1(\mathcal {E})$. Equivalently, the present paper completes the classification of ruled Fano $5$-folds with index two, which was partially done by C. Novelli and G. Occhetta in 2007.

References

M. Andreatta, E. Ballico, and J. A. Wiśniewski, Two theorems on elementary contractions, Math. Ann. 297 (1993), no. 2, 191–198. MR 1241801, DOI 10.1007/BF01459496
Marco Andreatta, Elena Chierici, and Gianluca Occhetta, Generalized Mukai conjecture for special Fano varieties, Cent. Eur. J. Math. 2 (2004), no. 2, 272–293. MR 2113552, DOI 10.2478/BF02476544
F. Ambro, Ladders on Fano varieties, J. Math. Sci. (New York) 94 (1999), no. 1, 1126–1135. Algebraic geometry, 9. MR 1703912, DOI 10.1007/BF02367253
Vincenzo Ancona, Thomas Peternell, and Jarosław A. Wiśniewski, Fano bundles and splitting theorems on projective spaces and quadrics, Pacific J. Math. 163 (1994), no. 1, 17–42. MR 1256175
M. Andreatta and J. A. Wiśniewski, A note on nonvanishing and applications, Duke Math. J. 72 (1993), no. 3, 739–755. MR 1253623, DOI 10.1215/S0012-7094-93-07228-6
Edoardo Ballico and Jarosław A. Wiśniewski, On Bǎnicǎ sheaves and Fano manifolds, Compositio Math. 102 (1996), no. 3, 313–335. MR 1401426
Koji Cho, Yoichi Miyaoka, and N. I. Shepherd-Barron, Characterizations of projective space and applications to complex symplectic manifolds, Higher dimensional birational geometry (Kyoto, 1997) Adv. Stud. Pure Math., vol. 35, Math. Soc. Japan, Tokyo, 2002, pp. 1–88. MR 1929792, DOI 10.2969/aspm/03510001
Elena Chierici and Gianluca Occhetta, The cone of curves of Fano varieties of coindex four, Internat. J. Math. 17 (2006), no. 10, 1195–1221. MR 2287674, DOI 10.1142/S0129167X06003850
Elena Chierici and Gianluca Occhetta, Fano fivefolds of index two with blow-up structure, Tohoku Math. J. (2) 60 (2008), no. 4, 471–498. MR 2487822, DOI 10.2748/tmj/1232376163
Thomas Dedieu and Andreas Höring, Numerical characterisation of quadrics, Algebr. Geom. 4 (2017), no. 1, 120–135. MR 3592468, DOI 10.14231/AG-2017-006
Takao Fujita, Classification of projective varieties of $\Delta$-genus one, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 3, 113–116. MR 664549
Takao Fujita, On polarized varieties of small $\Delta$-genera, Tohoku Math. J. (2) 34 (1982), no. 3, 319–341. MR 676113, DOI 10.2748/tmj/1178229197
Takao Fujita, On polarized manifolds whose adjoint bundles are not semipositive, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 167–178. MR 946238, DOI 10.2969/aspm/01010167
Takao Fujita, On adjoint bundles of ample vector bundles, Complex algebraic varieties (Bayreuth, 1990) Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992, pp. 105–112. MR 1178722, DOI 10.1007/BFb0094513
Robin Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121–176. MR 597077, DOI 10.1007/BF01467074
Paltin Ionescu, Generalized adjunction and applications, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 3, 457–472. MR 830359, DOI 10.1017/S0305004100064409
V. A. Iskovskikh and Yu. G. Prokhorov, Fano varieties, Algebraic geometry, V, Encyclopaedia Math. Sci., vol. 47, Springer, Berlin, 1999, pp. 1–247. MR 1668579
Akihiro Kanemitsu, Classification of Mukai pairs with corank $3$, Annales de l’Institut Fourier, 69 (2019), no. 1, 231–282. DOI 10.5802/aif.3242.
Stefan Kebekus, Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron, Complex geometry (Göttingen, 2000) Springer, Berlin, 2002, pp. 147–155. MR 1922103
János Kollár, Yoichi Miyaoka, and Shigefumi Mori, Rational connectedness and boundedness of Fano manifolds, J. Differential Geom. 36 (1992), no. 3, 765–779. MR 1189503
Shoshichi Kobayashi and Takushiro Ochiai, Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ. 13 (1973), 31–47. MR 316745, DOI 10.1215/kjm/1250523432
János Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 32, Springer-Verlag, Berlin, 1996. MR 1440180, DOI 10.1007/978-3-662-03276-3
Massimiliano Mella, Existence of good divisors on Mukai varieties, J. Algebraic Geom. 8 (1999), no. 2, 197–206. MR 1675146
Yoichi Miyaoka, Numerical characterisations of hyperquadrics, Complex analysis in several variables—Memorial Conference of Kiyoshi Oka’s Centennial Birthday, Adv. Stud. Pure Math., vol. 42, Math. Soc. Japan, Tokyo, 2004, pp. 209–235. MR 2087053, DOI 10.2969/aspm/04210209
Shigefumi Mori, Projective manifolds with ample tangent bundles, Ann. of Math. (2) 110 (1979), no. 3, 593–606. MR 554387, DOI 10.2307/1971241
Shigeru Mukai, 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.
Shigeru Mukai, Biregular classification of Fano $3$-folds and Fano manifolds of coindex $3$, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 9, 3000–3002. MR 995400, DOI 10.1073/pnas.86.9.3000
Carla Novelli and Gianluca Occhetta, Ruled Fano fivefolds of index two, Indiana Univ. Math. J. 56 (2007), no. 1, 207–241. MR 2305935, DOI 10.1512/iumj.2007.56.2905
Gianluca Occhetta, A note on the classification of Fano manifolds of middle index, Manuscripta Math. 117 (2005), no. 1, 43–49. MR 2142900, DOI 10.1007/s00229-005-0540-y
Gianluca Occhetta, A characterization of products of projective spaces, Canad. Math. Bull. 49 (2006), no. 2, 270–280. MR 2226250, DOI 10.4153/CMB-2006-028-3
Gianluca Occhetta and Jarosław A. Wiśniewski, On Euler-Jaczewski sequence and Remmert-van de Ven problem for toric varieties, Math. Z. 241 (2002), no. 1, 35–44. MR 1930984, DOI 10.1007/s002090100405
Thomas Peternell, A characterization of $\textbf {P}_n$ by vector bundles, Math. Z. 205 (1990), no. 3, 487–490. MR 1082869, DOI 10.1007/BF02571257
Thomas Peternell, Ample vector bundles on Fano manifolds, Internat. J. Math. 2 (1991), no. 3, 311–322. MR 1104121, DOI 10.1142/S0129167X91000181
Thomas Peternell, MichałSzurek, and Jarosław A. Wiśniewski, Fano manifolds and vector bundles, Math. Ann. 294 (1992), no. 1, 151–165. MR 1180456, DOI 10.1007/BF01934319
Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin-New York, 1973 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II); Dirigé par P. Deligne et N. Katz. MR 0354657
Kiwamu Watanabe, $\Bbb {P}^1$-bundles admitting another smooth morphism of relative dimension one, J. Algebra 414 (2014), 105–119. MR 3223392, DOI 10.1016/j.jalgebra.2014.05.026
Jarosław A. Wiśniewski, Ruled Fano $4$-folds of index $2$, Proc. Amer. Math. Soc. 105 (1989), no. 1, 55–61. MR 929433, DOI 10.1090/S0002-9939-1989-0929433-7
Jarosław A. Wiśniewski, On a conjecture of Mukai, Manuscripta Math. 68 (1990), no. 2, 135–141. MR 1063222, DOI 10.1007/BF02568756
Jarosław A. Wiśniewski, On contractions of extremal rays of Fano manifolds, J. Reine Angew. Math. 417 (1991), 141–157. MR 1103910, DOI 10.1515/crll.1991.417.141
Jarosław A. Wiśniewski, On Fano manifolds of large index, Manuscripta Math. 70 (1991), no. 2, 145–152. MR 1085628, DOI 10.1007/BF02568366
Jarosław A. Wiśniewski, Fano manifolds and quadric bundles, Math. Z. 214 (1993), no. 2, 261–271. MR 1240888, DOI 10.1007/BF02572403
Yun-Gang Ye and Qi Zhang, On ample vector bundles whose adjunction bundles are not numerically effective, Duke Math. J. 60 (1990), no. 3, 671–687. MR 1054530, DOI 10.1215/S0012-7094-90-06027-2