For a compact complex manifold \(X\) with an ample line bundle \(L\), the notion of a test configuration is central to the definition of \(K\)-stability, which in turn is conjecturally related to the existence of a constant scalar curvature Kähler (cscK) metric in the first Chern class \(c_1(L)\). The author extends the concepts of test configurations in the several versions of \(K\)-stability. He introduces a generalized notion of \(K\)-stability that is defined using filtrations instead of test configurations. These filtrations are obtained by asking for a kind of uniform \(K\)-stability through a certain class of sequences of test configurations. This is motivated by several reasons. First, every convex function on the moment polytope of a toric variety can be thought of as a filtration, but only the rational piecewise linear convex functions give rise to test configurations [S. K. Donaldson, J. Differ. Geom. 62, No. 2, 289–349 (2002; Zbl 1074.53059)]. Second, in [V. Apostolov et al., Invent. Math. 173, No. 3, 547–601 (2008; Zbl 1145.53055)], an example of a manifold was obtained that does not admit an extremal metric, but does not appear to be destabilized by a test configuration.
This and other examples are described in detail in Section 4. The author defines a notion of Futaki invariant for filtrations, extending the usual definition. In Section 6, the main result of the paper is obtained:
Theorem A. Suppose that \(X\) admits a cscK metric in \(c_1(L)\), and the automorphism group of \((X,L)\) is finite. If \(\chi\) is a filtration for \((X,L)\) such that \(\|\chi \|_2>0\), then the Futaki invariant of \(\chi\) satisfies \(\mathrm{Fut}(\chi)>0\). A key ingredient in the proof of this result is the Okounkov body, and the concave transform of a filtration which is described in Section 5 of the paper. In the appendix, written by S. Boucksom, the following result is proved:
Theorem B. Suppose that \(S\subset \bigoplus_{k\geq 0}H^0(X,L^k)\) is a graded subalgebra which contains an ample series. In addition suppose that
\[ \lim_{k\to \infty}k^{-n}\dim S_k<\lim_{k\to \infty}k^{-n}\dim H^0(X,L^k), \]
where \(n\) is the dimension of \(X\). Then there is a point \(p\in X\) and a number \(\varepsilon>0\), such that \(S_k\subset H^0(X, L^k\otimes_p^{[k\varepsilon]})\), for all \(k\), where \(I_p\) is the ideal sheaf of the point \(p\).