Flops Connect Minimal Models

By Y. Kawamata. 2007

169 citations recorded by the source corpus.

A result by Birkar-Cascini-Hacon-McKernan together with the boundedness of length of extremal rays implies that different minimal models can be connected by a sequence of flops. A flop of a pair (X, B) is a flip of a pair (X, B � ) which is crepant for KX +B

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9dc41e0ae30382f4927511e51a6f4ebc39495681

How researchers see this paper

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

In Computational Theory and Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Existence of minimal models for varieties of log general type (2006, unknown): “U. Let Γi= φi∗∆. Then the birational map Y 1 99K Y 2 is the composition of a sequence of (KY1 +Γ 1)-ﬂops over U. Note that (1.1.3) has been generalised recently to the case when ∆ is not assumed big, [14]. The next natural problem is to understand how many diﬀerent models there are. Even if log term”
Remarks on degenerations of hyper-K\"ahler manifolds (2017, background): “The following results are contained in [35], but not explicitly stated there.”
Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups (2011, background): “e is no codimension one face of Amp(X)other than the Fj (1 ≤ j ≤n+1), it follows that there is no ﬂop other than ιj (1 ≤ j ≤ n+1) up to Aut(X). On the other hand, by a fundamental result of Kawamata ([Ka3], Theorem 1), any birational map between minimal models is decomposed into ﬁnitely many ﬂops up”
Derived categories of toric varieties III (2006, methodology): “This is a generalization, in the toric case, of a theorem in [17], which states that any birational morphism between minimal models is decomposed into a sequence of flops.”
Varieties fibered by good minimal models (2009, unknown): “Proof By[15,Theorem3.52], (Xi ,� i )areisomorphicincodimensionone,andhence the argument in [ 14 ] applies.”
The Sarkisov program (2009, background): “4) and this implies that (4) and (5) hold.”
On existence of log minimal models (2007, methodology): “tly smaller. Note that we could also prove (2) in a similar way. ✷ In section 4, we will apply the proposition in a way similar to [9]. Proposition 3.2 easily implies the following result of Kawamata [6] on ﬂops connecting log minimal models. Corollary 3.3. Let (Y 1/Z,B 1) and (Y 2/Z,B 2) be two klt”
Infinite time singularities of the Kähler-Ricci flow (2014, background): “In [13], Kawamata showed that different minimal models can be connected by a sequence of flops.”
On existence of log minimal models II (2009, background): “2 easily implies the following result of Kawamata [8] on flops connecting log minimal models.”