Harold Blum

University of Utah - Department of Mathematics

About Me

I am an Assistant Professor in the Department of Mathematics at the University of Utah. From 2018 to 20201, I was an NSF postdoc at Stony Brook University and the University of Utah. In 2018, I completed my Ph.D. at the University of Michigan under the supervision of Mircea Mustaţă.

My research is partially supported by NSF grant DMS-2200690.

Research interests

Algebration geometry: birational geometry, singularities, Fano varieties, K-stability, and moduli.

Contact Info

Email: blum [at] math.utah.edu
Office: LCB 203
Mail: University of Utah
Department of Mathematics
Salt Lake City, UT 84112


Good moduli spaces for boundary polarized Calabi-Yau surface pairs (joint with Yuchen Liu), arXiv:2407.00850.

Moduli of boundary polarized Calabi-Yaur pairs (joint with Kenneth Ascher, Dori Bejler, Kristin DeVleming, Giovanni Inchiostro, Yuchen Liu, Xiaowei Wang), arXiv:2307.06522.

Convexity of multiplicities of filtrations on local rings (joint with Yuchen Liu and Lu Qi), Compos. Math. 160 (2024), 878-914.

The existence of the Kähler-Ricci soliton degeneration (joint with Yuchen Liu, Chenyang Xu, and Ziquan Zhuang), Forum of Math. Pi. To 11 (2023), e9.

On properness of K-moduli spaces and optimal degenerations of Fano varieties (joint with Daniel Halpern-Leistner, Yuchen Liu and Chenyang Xu), Selecta Math. 27 (2021).

Optimal destablization of K-unstable Fano varieties via stability thresholds (joint with Yuchen Liu and Chuyu Zhou), Geom. Topol.. 26 (2022), 2507-2564.

Openness of K-semistability for Fano varieties (joint with Yuchen Liu and Chenyang Xu), Duke Math. J.. 171 (2022), 2753-2797.

Reductivity of the automorphism group of K-polystable Fano varieties (joint with Jarod Alper, Daniel Halpern-Leistner, and Chenyang Xu), Invent. Math. 222 (2020), 995-1032.

Uniqueness of K-polystable degenerations of Fano varieties (joint with Chenyang Xu), Ann. of Math. 190 (2019), 609-656.

Openness of uniform K-stability in families of Q-Fano varieties (joint with Yuchen Liu), Ann. Sci. Éc. Norm. Supér. 55 (2022), 1-41

The normalized volume of a singularity is lower semicontinuous (joint with Yuchen Liu), J. Eur. Math. Soc. 23 (2021), 1225-1256.

Thresholds, valuations, and K-stability, (joint with Mattias Jonsson), Adv. Math. 365 (2020).

Existence of valuations with smallest normalized volume, Compos. Math. 154 (2018), 820-849.

On divisors computing mld's and lcts's, Bull. Korean Math. Soc. 58 (2021), 113-132.

Other Writing

Singularities and K-stability, Ph.D. Thesis, University of Michigan, link.

K-stability notes. These are notes from a course I taught in the fall of 2022. I plan to make significant edits and additions to the document.


This semester I am teaching Math 6180 - Complex Geometry. The course website is on Canvas.

A list of my past teaching can be found here.