Letters of a~Bi-rationalist V: Mld's and Termination of Log Flips

V. V. Shokurov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebraic geometry: Methods, relations, and applications, Tome 246 (2004), pp. 328-351.

Résumé

Termination of log flips and, more generally, of log quasiflips under the descending chain condition (dcc) of boundary multiplicities follows from two expected properties of the minimal log discrepancy (mld) function on algebraic log varieties: (1) the semicontinuity of mld's on any fixed log variety and (2) the ascending chain condition (acc) of mld's on the log varieties of given dimension with boundary multiplicities under the dcc. This reduces the global statement on termination to two local ones. All known cases of termination follow from this reduction. In particular, this gives the log termination in dimension 3, as well as the special and canonical termination up to dimension 4. To prove the log termination in dimension 4, one only needs the acc in dimension 4 for the mld values in the interval $[0,1]$.

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