From stability of varieties to GIT-stability of Hilbert points

Algebraic Geometry Seminar

Organizers：

Caucher Birkar，贾甲

Speaker：

Behrouz Taji (UNSW)

Time：

Monday, 15:00 - 16:00

April 20, 2026

Venue：

C548, Shuangqing Complex Building

Title:

From stability of varieties to GIT-stability of Hilbert points

Abstract:

Deligne--Mumford showed that stability for curves is intimately connected to GIT‑stability of the corresponding Hilbert‑scheme points (for a suitable choice of linearization). Mumford's construction of the coarse moduli space of stable curves as a GIT‑quotient follows directly from this relationship. In higher dimensions, KSB‑stable varieties play the role of stable curves; however, the "standard'' linearizations are known to fail in realizing their coarse moduli spaces as GIT‑quotients. Kollár therefore asks whether alternative linearizations might still produce such quotients. Using new positivity results for pushforward sheaves, we give an affirmative answer: with an appropriate choice of linearization, KSB‑stability implies GIT‑stability. This is based on joint work with S. Kovács.