On properness of K-moduli spaces and optimal degenerations of Fano varieties (Q2047272)

The authors establish an algebraic approach for proving the properness of moduli spaces \(M^{\mathrm{Kps}}_{n,V}\) of K-polystable Fano varieties of dimension \(n\) and volume \(V\). They reduce the problem to a conjecture on the destabilizations of K-unstable Fano varieties. They show that if the stability threshold of every K-unstable Fano variety is computed by a divisorial valuation, then such K-moduli spaces are proper. The argument relies on studying certain optimal destabilizing test configurations and constructing a \(\Theta\)-stratification on themoduli stack of Fano varieties.