On existence of log minimal models

By C. Birkar. Compositio Mathematica, 2007

75 citations recorded by the source corpus.

Abstract In this paper, we prove that the log minimal model program in dimension d−1 implies the existence of log minimal models for effective lc pairs (e.g. of non-negative Kodaira dimension) in dimension d. In fact, we prove that the same conclusion follows from a weaker assumption, namely, the log minimal model program with scaling in dimension d−1. This enables us to prove that effective lc pairs in dimension five have log minimal models. We also give new proofs of the existence of log minimal models for effective lc pairs in dimension four and of the Shokurov reduction theorem.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/91616b6d0b7292d22f3f8243c063e49bb669ceeb

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Y. Odaka
2. Ingrid Bauer
3. F. Catanese
4. J. Waldron
5. C. Birkar
6. Chenyang Xu
7. O. Fujino
8. Matteo Penegini

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.”
• Burniat surfaces I: fundamental groups and moduli of primary Burniat surfaces (2009, methodology): “Proof. (of Theorem 1.4) We closely follow the proof of [ 2, Theorem 1.3 ].”
• Existence of flips and minimal models for 3-folds in char p (2013, methodology): “Alternatively, one can apply the methods of [3] to construct log minimal models for lc pairs (X,B) such that KX +B ≡ M/Z for some M ≥ 0.”
• Existence of Mori fibre spaces for 3-folds in char $p$ (2014, background): “(4) assume KX + B is nef, ∆ ∈ L, and that Xi 99K Xi+1/Zi is a sequence of KX + ∆-flips which are KX + B-trivial and X = X1; then for any curve Γ on any Xi, we have (KXi + Bi) · Γ > α if (KXi + Bi) · Γ > 0 where Bi is the birational transform of B; (5) in addition to the assumptions of (4) suppose th”
• On existence of log minimal models II (2009, methodology): “We analyse the various places in the proof of [2], Proposition 3.”
• The GIT-stability of Polarised Varieties via discrepancy (2008, methodology): “og canonical model Y 99K Y′. We note that Conjecture 1.4 has already been established for dim(X) ≤ 3. Professor O. Fujino has kindly communicated to the author that it holds for dim(X) = 4 as well by [Bir08] and [Fuj08]. Therefore, Conjecture 1.1 holds for normal Xif dim(X) ≤ 4. Theorem 1.3 is an ap”