Recent zbMATH articles in MSC 14M10https://www.zbmath.org/atom/cc/14M102021-04-16T16:22:00+00:00WerkzeugBirational superrigidity and \(K\)-stability of Fano complete intersections of index \(1\).https://www.zbmath.org/1456.140502021-04-16T16:22:00+00:00"Zhuang, Ziquan"https://www.zbmath.org/authors/?q=ai:zhuang.ziquanA Fano variety \(X\) is said to be birationally superrigid if it has terminal singularities,
it is \({\mathbb Q}\)-factorial of Picard number 1, and every birational map \(X\)
to a Mori fiber space is an isomorphism. On the other hand, \(X\) is \(K\)-stable
with respect to its anticanonical bundle if, essentially, it admits a Kähler-Einstein
metric, and \(K\)-stability is encoded in the positivity of the invariants \(\beta(F)\)
for \(F\) any dreamy prime divisor \(F\) over \(X\) (see 2.2 for details). In the paper under
review the author shows (see Thm. 1.2 and 1.3) that for a \(n\)-dimensional smooth Fano complete
intersection \(X \subset {\mathbb P}^{n+r}\) of index one, if \(n \geq 10r\) then \(X\) is birationally
superrigid and \(K\)-stable. Moreover, the smooth complete intersection of a quadric and a cubic
in \({\mathbb P}^5\) is also \(K\)-stable. For a Fano manifold (see Def. A.1 in the appendix of
the paper under review) \(X\) is said to be conditionally birationally superrigid if
every birational map from \(X\) to a Mori fiber space whose undefined locus has
codimension at least \(1\) plus the index of \(X\) is an isomorphism.
In the Appendix, the authors show that Fano complete intersections of higher index in large dimension (see Cor. A.3
for details) are conditionally birationally superrigid.
Reviewer: Roberto Muñoz (Madrid)