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3 results

Nef anti-canonical divisors and rationally connected fibrations
Part of
- Families, fibrations
- Special varieties
Sho Ejiri, Yoshinori Gongyo
Journal: Compositio Mathematica / Volume 155 / Issue 7 / July 2019
Published online by Cambridge University Press: 28 June 2019, pp. 1444-1456
Print publication: July 2019
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We study the Iitaka–Kodaira dimension of nef relative anti-canonical divisors. As a consequence, we prove that given a complex projective variety with klt singularities, if the anti-canonical divisor is nef, then the dimension of a general fibre of the maximal rationally connected fibration is at least the Iitaka–Kodaira dimension of the anti-canonical divisor.

Log pluricanonical representations and the abundance conjecture
Part of
- Birational geometry
Osamu Fujino, Yoshinori Gongyo
Journal: Compositio Mathematica / Volume 150 / Issue 4 / April 2014
Published online by Cambridge University Press: 10 March 2014, pp. 593-620
Print publication: April 2014
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We prove the finiteness of log pluricanonical representations for projective log canonical pairs with semi-ample log canonical divisor. As a corollary, we obtain that the log canonical divisor of a projective semi log canonical pair is semi-ample if and only if the log canonical divisor of its normalization is semi-ample. We also treat many other applications.

Reduction maps and minimal model theory
Part of
- Birational geometry
Yoshinori Gongyo, Brian Lehmann
Journal: Compositio Mathematica / Volume 149 / Issue 2 / February 2013
Published online by Cambridge University Press: 04 December 2012, pp. 295-308
Print publication: February 2013
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We use reduction maps to study the minimal model program. Our main result is that the existence of a good minimal model for a Kawamata log terminal pair (X,Δ) can be detected on a birational model of the base of the (KX+Δ)-trivial reduction map. We then interpret the main conjectures of the minimal model program as a natural statement about the existence of curves on X.