Singularity invariants of plurisubharmonic functions and complex spaces
Pham Hoang Hiep
TL;DR
The paper develops a unified framework tying singularity invariants of plurisubharmonic functions to holomorphic invariants, yielding a sharp lower bound for the weighted log canonical thresholds and a suite of interrelated identities involving Monge-Ampère masses and Hilbert-Samuel multiplicities. Central results include a dimension-progression inequality $c_k(\varphi) \ge c_{k-1}(\varphi) + \frac{1}{\prod_{i=1}^{k-1}(c_i(\varphi)-c_{i-1}(\varphi))\, e_k(\varphi)}$, explicit toric formulas for $c_{||z||^{2t}dV}(\varphi)$, and a link between $MA_0$ and mixed HS multiplicities. The work also connects these invariants to the analytic and algebraic geometry of holomorphic functions via Demailly’s approximation and extends the theory to complex spaces, providing a robust toolset for studying singularities beyond smooth settings.
Abstract
In this paper, we combine tools from pluripotential theory and commutative algebra to study singularity invariants of plurisubharmonic functions. We establish several relationships between the singularity invariants of plurisubharmonic functions and those of holomorphic functions. These results yield a sharp lower bound for the log canonical threshold of a plurisubharmonic function. Our bound simultaneously improves upon the main result of Demailly and Pham (Acta Math. 212: 1--9, 2014), the classical result of Skoda (Bull. Soc. Math. France 100: 353--408, 1972), and the lower estimate of T. de Fernex, L. Ein and M. Mustaţǎ (Math. Res. Lett. 10: 219--236, 2003), which has played a crucial role in recent developments in birational geometry. Finally, we explore how singularity invariants associated with plurisubharmonic functions can be extended to complex spaces.