On uniruled degenerations of algebraic varieties with trivial canonical divisor
type: Item
instance of: scholarly article
title: On uniruled degenerations of algebraic varieties with trivial canonical divisor
zbMATH Open document ID: 1138.14008
author: Shigeharu Takayama
published in: Mathematische Zeitschrift
publication date: 2008-05-27T00:00:00Z
review text: 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: \noindent {(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 \textit{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)].
Mathematics Subject Classification ID: 14D06
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 14J26
zbMATH DE Number: 5279019
zbMATH Keywords: degenerations
zbMATH Keywords: uniruled
zbMATH Keywords: Calabi-Yau
reviewed by: Christopher Hacon
MaRDI profile type: Publication
full work available at URL: s00209-007-0235-z
OpenAlex ID: W2045560746
cites work: Q4434669
cites work: Divisorial Zariski decompositions on compact complex manifolds
cites work: Q4379894
cites work: Asymptotic invariants of base loci
cites work: Zariski decomposition and canonical rings of elliptic threefolds
cites work: On Shokurov's rational connectedness conjecture
cites work: Compact hyperkähler manifolds: Basic results
cites work: Minimal models and the Kodaira dimension of algebraic fiber spaces.
cites work: On the length of an extremal rational curve
cites work: On compact analytic surfaces. II
cites work: Q4212603
cites work: DEGENERATIONS OFK3 SURFACES AND ENRIQUES SURFACES
cites work: Q4821063
cites work: On singular fibres of Lagrangian fibrations over holomorphic symplectic manifolds
cites work: A numerical criterion for uniruledness
cites work: Semistable degenerations of Enriques' and hyperelliptic surfaces
cites work: Zariski-decomposition and abundance
cites work: On degenerations of algebraic surfaces
cites work: Invariance of plurigenera
cites work: Q3147685
cites work: On the invariance and the lower semi-continuity of plurigenera of algebraic varieties
cites work: Pluricanonical systems on algebraic varieties of general type
cites work: On the uniruledness of stable base loci
cites work: Q5618471
cites work: Classification theory of algebraic varieties and compact complex spaces. Notes written in collaboration with P. Cherenack
cites work: On the incompleteness of the Weil-Petersson metric along degenerations of Calabi-Yau manifolds
DOI: 10.1007/S00209-007-0235-Z
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description: scientific article; zbMATH DE number 5279019