Finiteness of intermediate-dimensional optimal destabilizers

Establish the finiteness of optimally destabilizing subvarieties of intermediate dimensions, namely dimensions 2 through n−2, for a compact Kähler manifold equipped with Kähler classes α and β, without imposing the additional positivity hypotheses used for the established conditional finiteness result.

Background

The paper proves that optimally destabilizing curves and divisors are always finite on compact Kähler manifolds of arbitrary dimension. It also obtains finiteness of all optimally destabilizing subvarieties under additional assumptions that the classes cα−pβ belong to suitable modified Kähler cones for every intermediate dimension 2≤p≤n−2; in particular, this yields unconditional finiteness on Kähler threefolds, where every proper subvariety is a curve or a divisor.

For higher-dimensional manifolds, the unresolved issue is whether infinitely many optimal destabilizers can occur in intermediate dimensions. The paper establishes a rigidity theorem showing that, under a quantitative nef-threshold condition, intermediate-dimensional optimal destabilizers cannot vary in nonconstant compact one-parameter families. Thus, any possible failure of finiteness in these dimensions would have to arise from infinitely many isolated rigid subvarieties rather than from moving families.