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4-folds with numerically effective tangent bundles and second Betti numbers greater than one. (English) Zbl 0799.14022
The authors study the four-dimensional manifolds \(X\) whose tangent bundles \(T_ X\) are numerically effective and they give a complete classification, except for the case of Fano 4-folds with \(b_ 2 = 1\) and index 1. – The main tools for the proof are the Albanese map, the “Mori contraction” of extremal rays and the results obtained by the same authors in their previous paper: Math. Ann. 289, No. 1, 169-187 (1991; Zbl 0729.14032).

14J35 \(4\)-folds
14C20 Divisors, linear systems, invertible sheaves
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[1] [APW] Ancona, V., Peternell, Th., Wisniewski., J.: Fano bundles and splitting theorems on projective spaces and quadrics. To appear in Pacific Journal of Math · Zbl 0808.14013
[2] Campana, F.; Peternell, Th., Projective manifolds whose tangent bundles are numerically effective, Math. Ann., 289, 169-187, (1991) · Zbl 0729.14032
[3] [CP2] Campana, F.; Peternell, Th.: On the second exterior power of tangent bundles of threefolds. Preprint, Bayreuth 1990. To appear in Comp. Math.
[4] [DPS] Demailly, J.P.; Peternell, Th.; Schneider, M.: Compact Kähler manifolds with numerically effective tangent bundles. Preprint (1991)
[5] [Fj] Fujita, T.: Classification theories of polarized varieties. London Math. Soc. Lecture Notes Series 155 Cambridge, 1990 · Zbl 0743.14004
[6] [Fu] Fulton, W.: Intersection Theory. Erg. Math. vol2. Springer 1984
[7] Iskovskih, V. A., Fano 3-folds I, II, Math. USSR., 11, 485-527, (1977) · Zbl 0382.14013
[8] Kawamata, Y.; Matsuda, K.; Matsuki, K., Introduction to the minimal model problem, Adv. Stud. Pure Math., 10, 283-360, (1987) · Zbl 0672.14006
[9] Kawamata, Y., Moderate degenerations of algebraic surfaces, Lecture Notes in Math., 1507, 113-132, (1992) · Zbl 0774.14032
[10] [Mu] Mukai, S.: New classification of Fano 3-folds and Fano manifolds of coindex 3. Preprint 1988
[11] [OSS] Okonek, C.; Schneider, M.; Spindler, H.: Vector bundles on complex projective spaces. Birkhäuser 1980 · Zbl 0438.32016
[12] Sols, I.; Szurek, M.; Wiśniewski, J., Fibering Fano 4-folds over a smooth quadric Q_{3}, Pacific J. Math., 148, 153-159, (1991) · Zbl 0733.14006
[13] Szurek, M.; Wiśniewski, J., Fano bundles over ℙ_{k} and Q_{3}, Pacific Journal of Math., 141, 197-208, (1990) · Zbl 0705.14016
[14] Szurek, M.; Wiśniewski, J., On Fano manifolds which are ℙ_{k} over ℙ_{2}, Nagoya J. Math., 120, 89-101, (1990) · Zbl 0728.14037
[15] [Wi] Wilson, P.M.H.: Fano fourfolds of index greater than one. Crelle’s Journal 389, 172-181 · Zbl 0611.14034
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