# zbMATH — the first resource for mathematics

On ample vector bundles whose adjunction bundles are not numerically effective. (English) Zbl 0709.14011
By a generalized polarized variety the authors mean an algebraic variety X of dimension n over an algebraically closed field of characteristic zero together with an ample vector bundle E on it, i.e. a pair (X,E). Then $$K_ X+c_ 1(E)$$ is its associated adjunction bundle. The paper studies the numerical effectiveness of this bundle. Most of the results from T. Fujita’s paper [Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 167-178 (1987; Zbl 0659.14002)] about adjunction bundles of polarized varieties are generalized to those of generalized varieties. Main results:
Theorem 1. If the rank of E is $$n+1$$ then $$K_ X+c_ 1(E)$$ is always numerically effective and $$K_ X+c_ 1(E)$$ is equal to zero if and only if $$(X,E)=(P^ n,\oplus^{n+1}{\mathcal O}_{P^ n}(1)).$$
Theorem 2. If the rank of E is n then $$K_ X+c_ 1(E)$$ is numerically effective unless $$(X,E)=(P^ n,\oplus^ nO_ Pn(1)).$$
Theorem 3. If the rank of E is $$n-1$$ then $$K_ X+c_ 1(E)$$ is numerically effective unless (X,E) is one of the following:
1. X is a scroll over a smooth curve and $$E|_ F=\oplus^{n-1}O_ Pn-1(1)$$, where $$F=P^{n-1}$$ is any fiber of this scroll.
2. $$(P^ n,\oplus^{n-1}{\mathcal O}_ Pn(1))$$. 3. $$(P^ n,\oplus^{n- 2}{\mathcal O}_ Pn(1)\oplus {\mathcal O}_ P^ n(2))$$. 4. $$(Q^ n,\oplus^{n-1}{\mathcal O}_ Qn(1))$$. Here $$Q^ n$$ is a smooth hyperquadric.
Theorem 4. If the rank of E is n and $$c_ 1(E)=c_ 1(X)$$, then P(E) is a Fano-$$(2n-2)$$-fold with index n and Picard number two. Moreover P(E) has exactly two extremal rays, and the contraction maps associated with these extremal rays are both of fiber type and equidimensional unless $$(X,E)=(P^ n,\oplus^{n-1}O_ Pn(1)\oplus O_ Pn(2))$$ or $$(Q^ 2,O_ Q2(1)\oplus O_ Q2(1))$$.
Reviewer: V.K.Vedernikov

##### MSC:
 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14C20 Divisors, linear systems, invertible sheaves 14J40 $$n$$-folds ($$n>4$$)
Full Text:
##### References:
 [1] T. Fujita, On the hyperplane section principle of Lefschetz , J. Math. Soc. Japan 32 (1980), no. 1, 153-169. · Zbl 0414.14007 · doi:10.2969/jmsj/03210153 [2] T. Fujita, On the structure of polarized varieties with $$\Delta$$-genera zero , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 103-115. · Zbl 0333.14004 [3] T. Fujita, On polarized manifolds whose adjoint bundles are not semipositive , Algebraic Geometry, Sendai, 1985, Advances Studies in Pure Mathematics, vol. 10, North-Holland, Amsterdam, 1987, pp. 167-178. · Zbl 0659.14002 [4] R. Hartshorne, Algebraic Geometry , Springer-Verlag, New York, 1977. · Zbl 0367.14001 [5] R. Hartshorne, Ample vector bundles , Inst. Hautes Études Sci. Publ. Math. (1966), no. 29, 63-94. · Zbl 0173.49003 · numdam:PMIHES_1966__29__63_0 · eudml:103864 [6] P. Ionescu, Generalized adjunction and applications , Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 3, 457-472. · Zbl 0619.14004 · doi:10.1017/S0305004100064409 [7] Y. Kawamata, A generalization of Kodaira-Ramanujam’s vanishing theorem , Math. Ann. 261 (1982), no. 1, 43-46. · Zbl 0476.14007 · doi:10.1007/BF01456407 · eudml:182862 [8] Y. Kawamata, The cone of curves of algebraic varieties , Ann. of Math. (2) 119 (1984), no. 3, 603-633. JSTOR: · Zbl 0544.14009 · doi:10.2307/2007087 · links.jstor.org [9] S. Kobayashi and T. Ochiai, Characterizations of complex projective spaces and hyperquadrics , J. Math. Kyoto Univ. 13 (1973), 31-47. · Zbl 0261.32013 [10] J. Kollár, Higher direct images of dualizing sheaves I , Ann. of Math. (2) 123 (1986), no. 1, 11-42. JSTOR: · Zbl 0598.14015 · doi:10.2307/1971351 · links.jstor.org [11] A. Lanteri and A. J. Sommese, A vector bundle characterization of $$p^n$$ , · Zbl 0701.14050 [12] R. Lazarsfeld, Some applications of the theory of positive vector bundles , Complete intersections (Acireale, 1983), Lecture Notes in Mathematics, vol. 1092, Springer-Verlag, Berlin, 1984, pp. 29-61. · Zbl 0547.14009 · doi:10.1007/BFb0099356 [13] S. Mori, Projective manifolds with ample tangent bundles , Ann. of Math. (2) 110 (1979), no. 3, 593-606. JSTOR: · Zbl 0423.14006 · doi:10.2307/1971241 · links.jstor.org [14] S. Mori, Threefolds whose canonical bundles are not numerically effective , Ann. of Math. (2) 116 (1982), no. 1, 133-176. JSTOR: · Zbl 0557.14021 · doi:10.2307/2007050 · links.jstor.org [15] S. Mori and H. Sumihiro, On Hartshorne’s conjecture , J. Math. Kyoto Univ. 18 (1978), no. 3, 523-533. · Zbl 0422.14030 [16] S. Mukai, Problems on characterizations of the complex projective space , Birational Geometry of Algebraic Varieties-Open Problems, Katata, Japan, 1988, The 23th International Symposium Division of Mathematics, The Taniguchi Foundation, pp. 57-60. [17] C. Okonek, M. Schneider, and H. Spindler, Vector bundles on complex projective spaces , Progress in Mathematics, vol. 3, Birkhauser, Boston, 1980. · Zbl 0438.32016 [18] A. J. Sommese, On manifolds that cannot be ample divisors , Math. Ann. 221 (1976), no. 1, 55-72. · Zbl 0306.14006 · doi:10.1007/BF01434964 · eudml:162833 [19] T. Peternell, A Characterization of $$p^n$$ by Vector Bundles ,
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.