Termination of (many) 4-dimensional log flips

By Valery Alexeev, Christopher D. Hacon, Y. Kawamata. 2006

50 citations recorded by the source corpus.

We prove that any sequence of 4-dimensional log flips that begins with a klt pair (X,D) such that -(KX+D) is numerically equivalent to an effective divisor, terminates. This implies termination of flips that begin with a log Fano pair and termination of flips in a relative birational setting. We also prove termination of directed flips with big KX+D. As a consequence, we prove existence of minimal models of 4-dimensional dlt pairs of general type, existence of 5-dimensional log flips, and rationality of Kodaira energy in dimension 4.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/73f2275722c254010a1db06403e01e61590c189d

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In Computational Theory and Mathematics

1. G. Bellamy
2. C. Birkar

In Geometry and Topology

1. G. Bellamy
2. C. Birkar
3. Y. Odaka
4. Christopher D. Hacon
5. H. Blum
6. O. Fujino
7. F. Catanese
8. Ingrid Bauer

Citations in the corpus

• Existence of log canonical flips and a special LMMP (2011, background): “Alexeev-Hacon-Kawamata [1] and Birkar [5] proved it in dimension 5 in the klt case.”
• On existence of log minimal models (2007, methodology): “A pair (X/Z, B) consists of normal quasi-projective varieties X, Z over k, an R-divisor B on X with coefficients in [0, 1 ] such that KX + B is R-Cartier, and a projective morphism X → Z. For a prime divisor D on some birational model of X with a nonempty centre on X, a(D, X, B) denotes the log dis”
• On singular Calogero–Moser spaces (2007, methodology): “Therefore the pair (X, ∅) is a Kawamata log terminal pair (as defined in [ AHK ]) and we can apply [AHK, Lemma 2.1] to conclude that there exists an effective Q-factorial terminal pair (Y, B) together with a birational morphism f : Y → X such that”
• On existence of log minimal models II (2009, methodology): “4 in dimension four we directly use termination of log flips with scaling in dimension four which we prove that it follows from results of Shokurov [22] and AlexeevHacon-Kawamata [1].”