On Kodaira energy and adjoint reduction of polarized manifolds. (English) Zbl 0766.14027
The author classifies polarized complex \(n\)-folds \((M,L)\) (respectively their Sommese second reduction \((M'',A))\) when \(n\geq 5\) (respectively \(n=4)\) and \(-\kappa\varepsilon(M,L)\geq n=3\). Here, \(\kappa\varepsilon(M,L)\) denotes the Kodaira energy, defined by: \(- \kappa\varepsilon(M,L)=\text{Inf}\{t\in\mathbb{Q}\mid \kappa(M,K_ M+tL)\geq 0\}\).
The case \(n\geq 6\) was settled by Beltrami and Sommese (the author precizes their result), but the cases \(n=4\) and 5 need a special analysis, since many new possibilities appear. This classification allows the author to check the spectrum and the fibration conjectures in that range. For example, when \(n=5\) and \(-\kappa\varepsilon(M,L)\geq 2\), then \(-\kappa\varepsilon(M,L)\) is one of the following: \(6,5,4,3,{5\over 2},{7\over 3},2\).
The case \(n\geq 6\) was settled by Beltrami and Sommese (the author precizes their result), but the cases \(n=4\) and 5 need a special analysis, since many new possibilities appear. This classification allows the author to check the spectrum and the fibration conjectures in that range. For example, when \(n=5\) and \(-\kappa\varepsilon(M,L)\geq 2\), then \(-\kappa\varepsilon(M,L)\) is one of the following: \(6,5,4,3,{5\over 2},{7\over 3},2\).
Reviewer: F.Campana (Vandœuvre-les-Nancy)
MSC:
| 14J10 | Families, moduli, classification: algebraic theory |
| 32C15 | Complex spaces |
| 14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |
| 14J15 | Moduli, classification: analytic theory; relations with modular forms |
Keywords:
adjunction theory; elementary contraction; classification of polarized complex \(n\)-folds; Kodaira energyReferences:
| [1] | [B] L. Badescu,On ample divisors, Nagoya Math. J.86 (1982), 155–171 · Zbl 0445.14002 |
| [2] | [BFS] M. C. Beltrametti, M. L. Fania and A. J. Sommese,On the adjunction theoretic classification of projective varieties, Math. Ann.290 (1991), 31–62 · Zbl 0712.14029 · doi:10.1007/BF01459237 |
| [3] | [BS] M. C. Beltrametti and A. J. Sommese,On the adjunction theoretic classification of polarized varieties, preprint · Zbl 0762.14005 |
| [4] | [BSW] M. C. Beltrametti, A. J. Sommese and J. A. Wiśniewski,Results on varieties with many lines and their application to adjunction theory, preprint |
| [5] | [F1] T. Fujita,On polarized manifolds whose adjoint bundles are not semipositive, in ”Algebraic Geometry; Sendai 1985,” Advanced Studies in Pure Math.,10, 1987, pp. 167–178 |
| [6] | [F2]–,Remarks on quasi-polarized varieties, Nagoya Math. J.115 (1989), 105–123 · Zbl 0699.14002 |
| [7] | [F3]–,On Del Pezzo fibrations over curves, Osaka J. Math.27 (1990) 229–245 · Zbl 0715.14030 |
| [8] | [F4] T. Fujita,On singular Del Pezzo varieties, in ”Algebraic Geometry; Proc. L’Aquila 1988,” Lecture Notes in Math. (Springer), 1417, 1990, pp. 117–128 |
| [9] | [F5] T. Fujita, ”Classification Theories of Polarized Varieties,” London Math. Soc. Lecture Note Series 155, Cambridge Univ. Press, 1990 · Zbl 0743.14004 |
| [10] | [F6] T. Fujita,Remarks on Kodaira energy of quasi-polarized varieties, in preparation |
| [11] | [G] R. Goren,Characterization and algebraic deformations of projective space, J. Math. Kyoto Univ.8 (1968), 41–47 · Zbl 0213.47802 |
| [12] | [I] P. Ionescu,Generalized adjunction and applications, Math. Proc. Cambridge Phil. Soc.99 (1986), 457–472 · Zbl 0619.14004 · doi:10.1017/S0305004100064409 |
| [13] | [KMM] Y. Kawamata, K. Matsuda and K. Matsuki,Introduction to the minimal model problem, in ”Algebraic Geometry; Sendai 1985,” Advanced Studies in Pure Math.,10, 1987, pp. 283–360 |
| [14] | [KO] S. Kobayashi and T. Ochiai,Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ.13 (1973), 31–47 · Zbl 0261.32013 |
| [15] | [M1] S. Mori,Threefolds whose canonical bundles are not numerically effective, Ann. of Math.116 (1982), 133–176 · Zbl 0557.14021 · doi:10.2307/2007050 |
| [16] | [M2]–,Flip theorem and the existence of minimal models for 3-folds, Journal of AMS1 (1988), 117–253 · Zbl 0649.14023 |
| [17] | [S] A. J. Sommese,On the adjunction theoretic structure of projective varieties, in ”Complex Analysis and Algebraic Geometry; Proc. Göttingen 1985”, Lecture Notes in Math. (Springer),1194, 1986, pp. 175–213 |
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