57 citations · 268 across the 15 of their papers we have counts for
16 papers
Valuative independence for Calabi--Yau varieties
Harold Blum, Yuchen Liu
We construct valuatively independent bases for the space of sections of an ample line bundle on a log Calabi--Yau pair over a discretely valued field and the space of regular funct…
Relative stability theory and properness of K-moduli spaces
Harold Blum, Yuchen Liu, Chenyang Xu +1
We define the relative stability threshold of a family of Fano varieties over a DVR and show that it is computed by a divisorial valuation. In the case when the special fiber is K-…
Good moduli spaces for boundary polarized Calabi-Yau surface pairs
Harold Blum, Yuchen Liu
We construct projective asymptotically good moduli spaces parametrizing boundary polarized CY surface pairs, which are projective slc Calabi-Yau pairs such that is ampl…
Moduli of boundary polarized Calabi-Yau pairs
Kenneth Ascher, Dori Bejleri, Harold Blum +4
We develop the moduli theory of boundary polarized CY pairs, which are slc Calabi-Yau pairs such that is ample. The motivation for studying this moduli problem is to co…
Convexity of multiplicities of filtrations on local rings
Harold Blum, Yuchen Liu, Lu Qi
We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a conse…
The existence of the Kähler-Ricci soliton degeneration
Harold Blum, Yuchen Liu, Chenyang Xu +1
We prove an algebraic version of the Hamilton-Tian Conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a r…
On properness of K-moduli spaces and optimal degenerations of Fano varieties
Harold Blum, Daniel Halpern-Leistner, Yuchen Liu +1
We establish an algebraic approach to prove the properness of moduli spaces of K-polystable Fano varieties and reduce the problem to a conjecture on destabilizations of K-unstable…
Optimal destabilization of K-unstable Fano varieties via stability thresholds
Harold Blum, Yuchen Liu, Chuyu Zhou
We show that for a K-unstable Fano variety, any divisorial valuation computing its stability threshold induces a non-trivial special test configuration preserving the stability thr…
Openness of K-semistability for Fano varieties
Harold Blum, Yuchen Liu, Chenyang Xu
In this paper, we prove the openness of K-semistability in families of log Fano pairs by showing that the stability threshold is a constructible function on the fibers. We also pro…
Reductivity of the automorphism group of K-polystable Fano varieties
Jarod Alper, Harold Blum, Daniel Halpern-Leistner +1
We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness a…
Uniqueness of K-polystable degenerations of Fano varieties
Harold Blum, Chenyang Xu
We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with [Jia17,…
Openness of uniform K-stability in families of -Fano varieties
Harold Blum, Yuchen Liu
We show that uniform K-stability is a Zariski open condition in Q-Gorenstein families of Q-Fano varieties. To prove this result, we consider the behavior of the stability threshold…
The normalized volume of a singularity is lower semicontinuous
Harold Blum, Yuchen Liu
We show that in any -Gorenstein flat family of klt singularities, normalized volumes are lower semicontinuous with respect to the Zariski topology. A quick consequence…
Thresholds, valuations, and K-stability
Harold Blum, Mattias Jonsson
Let X be a normal complex projective variety with at worst klt singularities, and L a big line bundle on X. We use valuations to study the log canonical threshold of L, as well as…
Existence of valuations with smallest normalized volume
Harold Blum
Li introduced the normalized volume of a valuation due to its relation to K-semistability. He conjectured that over a klt singularity there exists a valuation with smallest normali…
On divisors computing MLD's and LCT's
Harold Blum
We show that if a divisor centered over a point on a smooth surface computes a minimal log discrepancy, then the divisor also computes a log canonical threshold. To prove the resul…