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On \(K\)-stability and the volume functions of \(\mathbb{Q}\)-Fano varieties. (English) Zbl 1375.14139

Let \(X\) be a \(\mathbb Q\)-Fano variety, i.e. a complex projective variety with ample \(\mathbb Q\)-Cartier anticanonical divisor \(-K_X\) and log terminal singularities. This article introduces the notion of divisorial (semi)stability, a necessary condition for \(K\)-(semi)stability of the pair \((X, -K_X)\). \(K\)-(semi)stability being notoriously hard to decide, weaker notions that can be tested such as slope (semi)stability [J. Ross and R. Thomas J. Algebr. Geom. 16, No. 2, 201–255 (2007; Zbl 1200.14095)] were introduced. Divisorial (semi)stability along divisors is a refinement of slope (semi)stability; it is a strictly sharper notion and remains easy to test.
The author shows that divisorial (semi)stability can be formulated in terms of volume functions. When \(X\) is toric, it follows that \((X, -K_X)\) is divisorially semistable (or equivalently in this case \(K\)-semistable) precisely when the barycentre of the associated polytope is the origin. In the general case, divisorial (semi)stability can be related to properties of the Okounkov body of \(-K_X\).
Using the characterisation in terms of volume functions, the author further shows that divisorial semistability of \((X, -K_X)\) along a divisor \(D\) can be tested by computing intersection numbers on the varieties and divisors that arise when running a suitable Minimal Model Program (MMP) with scaling. He then proceeds to determine divisorial (semi)stability for all Fano manifolds of dimension at most \(3\) and produces a list of non-toric, non \(K\)-semistable Fano manifolds of dimension \(3\).

MSC:

14J45 Fano varieties
14L24 Geometric invariant theory

Citations:

Zbl 1200.14095
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References:

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