×

On uniruled degenerations of algebraic varieties with trivial canonical divisor. (English) Zbl 1138.14008

Let \(\pi : X\to C\) be a projective morphism (surjective with connected fibers) from a normal variety with canonical singularities to a smooth curve \(C\). Let \(0\in C\) be a point and \(X_0=\sum _{i\in I} m_i F_i\) be the corresponding fiber. If the general fiber has numerically trivial canonical divisor \(K_{X_t}\equiv 0\), then there is a unique way to write \(K_{X/C}=\sum _{j\in J}r_jF_j\) where \(r_j\in \mathbb Q ^{>0}\) and \(J\neq I\). In the paper under review, the author shows that:
{(0)}All components \(F_i\) are uniruled (i.e. covered by rational curves) except possibly one component \(F_j\) with \(\kappa (F_j )=0\) (i.e. \(\text{ max} \{ \dim H^0(\mathcal O (mK_{F_j}))| m>0 \}=1\)).
(1) If \(| I| =1\) so that \(X_0=m_1F_1\), then if \(F_1\) is not normal or it contains a \(2\) dimensional component of the singular locus of \(X\), then \(F_1\) is uniruled; if instead \(F_1\) is normal and it does not contain any \(2\) dimensional component of the singular locus of \(X\), then \(F_1\) is uniruled iff it is not canonical.
(2) If \(| I| \geq 2\) and \(J\cup \{ i\}\neq I\), then \(F_i\) is uniruled. Therefore: \(K_{X/C}\) is supported by uniruled divisors; if \(F_1\) is not uniruled, then \(I=J\cup \{1 \}\); and if \(K_{X/C}\) is \(\pi\)-nef, then all \(F_i\) are uniruled.
The proof is based on the technique of extensions of pluricanonical forms due to Y.-T. Siu [Invent. Math. 134, 661-673 (1998; Zbl 0955.32017); in: Finite or infinite dimensional complex analysis and applications. Proc. int. conf. Hanoi, Vietnam, August 8–12, 2001. Adv. Complex Anal. Appl. 2, 45–83 (2004; Zbl 1044.32016)].

MSC:

14D06 Fibrations, degenerations in algebraic geometry
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J26 Rational and ruled surfaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Barth, W., Hulek, K., Peters, C., Van de Ven, A.: Compact complex surfaces, 2nd edn. Ergebnisse der Math. und ihrer Grenzgebiete (3), vol 4. Springer-Verlag, Berlin (2004) · Zbl 1036.14016
[2] Boucksom S. (2004). Divisorial Zariski decompositions on compact complex manifolds. Ann. Sci. École Norm. Sup. 37: 209–233 · Zbl 1054.32010
[3] Boucksom, S., Demailly, J.-P., Paun, M., Peternell, Th.: The pseudo-effective cone of a compact Kähler manifold and varieties of non-negative Kodaira dimension. arXiv:math.AG/0405285
[4] Demailly, J.-P.: Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. Algebraic geometry–Santa Cruz 1995. Proc. Symp. Pure Math., vol 62, (2), pp. 285–360. Amer. Math. Soc., Providence (1997) · Zbl 0919.32014
[5] Ein L., Lazarsfeld R., Mustata M., Nakamaye M. and Popa M. (2006). Asymptotic invariants of base loci. Ann. Inst. Fourier 56: 1701–1734 · Zbl 1127.14010
[6] Friedman, R., Morrison, D.: The birational geometry of degenerations: an overview. The birational geometry of degenerations (Cambridge, Mass., 1981), pp. 1–32. Progr. Math., vol 29. Birkhäuser, Boston (1983)
[7] Fujita T. (1986). Zariski decompositions and canonical rings of elliptic threefolds. J. Math. Soc. Japan 38: 19–37 · Zbl 0627.14031 · doi:10.2969/jmsj/03810019
[8] Hacon C. and McKernan J. (2007). On Shokurov’s rational connectedness conjecture. Duke Math. J. 138: 119–136 · Zbl 1128.14028
[9] Huybrechts D. (1999). Compact hyperkähler manifolds: basic results. Invent. Math. 135: 63–113 · Zbl 0953.53031 · doi:10.1007/s002220050280
[10] Kawamata Y. (1985). Minimal models and the Kodaira dimension of algebraic fiber spaces. J. Reine Angew. Math. 363: 1–46 · Zbl 0589.14014 · doi:10.1515/crll.1985.363.1
[11] Kawamata Y. (1991). On the length of an extremal rational curve. Invent. Math. 105: 609–611 · Zbl 0751.14007 · doi:10.1007/BF01232281
[12] Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the minimal model problem. Algebraic Geometry (Sendai, 1985). Advanced Studies in Pure Math., vol 10, pp. 283–360. North-Holland, Amsterdam (1987)
[13] Kodaira K. (1963). On compact complex analytic surfaces II. Ann. Math. 77: 563–626 · Zbl 0118.15802 · doi:10.2307/1970131
[14] Kollár, J. (with 14 coauthors): Flips and abundance for algebaric threefolds. Astérisque 211 (1992)
[15] Kollár J. and Mori S. (1998). Birational geometry of algebraic varieties. Cambridge Tracts Math., vol 134. Cambridge University Press, Cambridge · Zbl 0926.14003
[16] Kulikov V. (1977). Degenerations of K3 surfaces and Enriques surfaces. Math. USSR Izvestija 11: 957–989 · Zbl 0387.14007 · doi:10.1070/IM1977v011n05ABEH001753
[17] Lazarsfeld, R.: Positivity in algebraic geometry I, II. Ergebnisse der Math. und ihrer Grenzgebiete (3), vol 48–49. Springer-Verlag, Berlin (2004) · Zbl 1066.14021
[18] Mastushita D. (2001). On singular fibres of Lagrangian fibrations over holomorphic symplectic manifolds. Math. Ann. 321: 755–773 · Zbl 1017.53069 · doi:10.1007/s002080100251
[19] Miyaoka Y. and Mori S. (1986). A numerical criterion of uniruledness. Ann. Math. 124: 65–69 · Zbl 0606.14030 · doi:10.2307/1971387
[20] Morrison D. (1981). Semistable degenerations of Enriques’ and hyperelliptic surfaces. Duke Math. J. 48: 197–249 · Zbl 0476.14015 · doi:10.1215/S0012-7094-81-04813-4
[21] Nakayama, N.: Zariski-decomposition and abundance. MSJ Memoirs, vol 14. Math. Soc. Japan (2004) · Zbl 1061.14018
[22] Persson, U.: On degenerations of algebraic surfaces. Memoirs AMS, vol 189. Amer. Math. Soc. (1977) · Zbl 0368.14008
[23] Persson U. and Pinkham H. (1981). Degeneration of surfaces with trivial canonical bundle. Ann. Math. 113: 45–66 · Zbl 0447.14007 · doi:10.2307/1971133
[24] Reid, M.: Young person’s guide to canonical singularities, in Algebraic geometry, Bowdoin 1985. In: Bloch, S., (ed.) Proc. of Symp. in Pure Math., vol 46, (1), pp. 345–414. Amer. Math.Soc. (1987)
[25] Siu Y.-T. (1998). Invariance of plurigenera. Invent. Math. 134: 661–673 · Zbl 0955.32017 · doi:10.1007/s002220050276
[26] Siu Y.-T. (2002). Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semi-positively twisted plurigenera for manifolds not necessarily of general type. In: Bauer, I. (eds) Complex Geometry (Göttingen, 2000), pp 223–277. Springer, Berlin · Zbl 1007.32010
[27] Takayama S. (2006). On the invariance and the lower semi-continuity of plurigenera of algebraic varieties. J. Algebraic Geom. 16: 1–18 · Zbl 1113.14028
[28] Takayama S. (2006). Pluricanonical systems on algebraic varieties of general type. Invent. Math. 165: 551–587 · Zbl 1108.14031 · doi:10.1007/s00222-006-0503-2
[29] Takayama, S.: On the uniruledness of stable base loci, to appear in J. Differential Geom · Zbl 1136.14005
[30] Tsunoda, S.: Degenerations of surfaces. Algebraic Geometry (Sendai, 1985), Advanced Studies in Pure Math., vol 10, pp 755–764. North-Holland, Amsterdam (1987)
[31] Ueno K. (1971). On fibre spaces of normally polarized abelian varieties of dimesion 2. I. Singular fibres of the first kind. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18: 37–95 · Zbl 0215.37303
[32] Ueno K. (1975). Classification theory of algebraic varieties and compact complex spaces. Lecture Notes in Math., vol 439. Springer-Verlag, Berlin · Zbl 0299.14007
[33] Wang C.-L. (1997). On the incompleteness of the Weil-Petersson metric along degenerations of Calabi–Yau manifolds. Math. Res. Lett. 4: 157–171 · Zbl 0881.32017
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.