Math Conjectures

Termination of Flips Conjecture

OPENLandmarkConjectureProposed c. 1988 · Full conjecture

Canonical statement

Let (X0,Δ0)(X_0,\Delta_0) be a projective Q\mathbb Q-factorial Kawamata-log-terminal pair over C\mathbb C. Every sequence
(X0,Δ0)⇢(X1,Δ1)⇢(X2,Δ2)⇢⋯ (X_0,\Delta_0)\dashrightarrow(X_1,\Delta_1) \dashrightarrow(X_2,\Delta_2)\dashrightarrow\cdots
in which each arrow is a (KXi+Δi)(K_{X_i}+\Delta_i)-flip terminates after finitely many steps.
View source LaTeX
Let \((X_0,\Delta_0)\) be a projective \(\mathbb Q\)-factorial Kawamata-log-terminal pair over \(\mathbb C\). Every sequence
\[
  (X_0,\Delta_0)\dashrightarrow(X_1,\Delta_1)
    \dashrightarrow(X_2,\Delta_2)\dashrightarrow\cdots
\] in which each arrow is a \((K_{X_i}+\Delta_i)\)-flip terminates after finitely many steps.
AG-MMP-012:[KollarMori1998Birational][BCHM2010][Birkar2012LogFlips]
Algebraic geometryGeometry and topology

Notes

Flips are the delicate surgeries of the minimal model program: small birational modifications that replace a (KX+Δ)(K_{X}+\Delta)-negative contraction by a positive one. The termination conjecture asserts that, starting from a projective Q\mathbb Q-factorial Kawamata log terminal pair over C\mathbb C, every sequence of flips stops after finitely many steps. It took shape around 1988 as the program itself was being formulated, rather than at a single announced moment; the standard reference for the framework is [KollarMori1998Birational].

Termination is known in dimension at most three. In higher dimensions the existence of the required flips is now a theorem, for klt pairs by Birkar, Cascini, Hacon and McKernan [BCHM2010] and for log canonical flips by Birkar [Birkar2012LogFlips], and suitably directed runs of the program are known to terminate in broad settings, as are sequences satisfying special-termination hypotheses.

Arbitrary flip sequences in dimension four and higher remain uncontrolled, and proving that they always terminate is the chief missing step toward running the minimal model program in full generality.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.