Zhou Weights: Complex Singularities & Valuations

• Zhou weights are plurisubharmonic weights with tightly controlled singularities that provide canonical envelopes for integration-theoretic thresholds and capture valuation structures.
• They link local and global behaviors in complex domains through tropical multiplicativity and additivity of relative types, bridging multiplier ideals, jumping numbers, and singularity exponents.
• Zhou weights exhibit robust approximation, convergence, and continuity properties that unify methods in pluripotential theory, valuation theory, and algebraic geometry.

Zhou weights constitute a class of plurisubharmonic (psh) weights with tightly controlled singularities, designed to refine notions of complex analytic singularities, provide canonical envelopes for integration-theoretic thresholds, and encode valuative invariants with optimal tropical behavior. There are both local and global (“multipoled”) variants, whose foundational properties clarify and unify several aspects of the interplay between plurisubharmonic functions, multiplier ideals, jumping numbers, and singularity exponents. The relative types to Zhou weights—termed Zhou numbers—realize these weights as a natural setting linking singularities in several complex variables, valuation theory, and degeneration/approximation in analysis and algebraic geometry (Bao et al., 2023, &&&1&&&).

1. Local Zhou Weights: Definition and Foundational Properties

Given a holomorphic mapping $f = (f_1,\ldots, f_m)$ near $o\in\mathbb{C}^n$ and a psh weight $p_0$ such that $\|f\|^2 e^{-2p_0}$ is locally integrable, the local Zhou weight associated to this data is the unique psh function $\varphi_{o,\max}$ (Definition 1.1) characterized by:

1. For large $N_0$, the weighted expression $$\|f\|^2 e^{-2p_0} |z|^{2N_0} e^{-2\varphi_{o,\max}}$$ is integrable near $o$.
2. $\|f\|^2 e^{-2p_0} e^{-2\varphi_{o,\max}}$ is not integrable near $o$.
3. If $\phi$ is psh near $o$, $\phi\geq\varphi_{o,\max}+O(1)$, and $\|f\|^2 e^{-2p_0} e^{-2\phi}$ is not integrable, then $\phi=\varphi_{o,\max}+O(1)$ near $o$.

Such $\varphi_{o,\max}$ is maximal away from the singularity and defines, for each psh $u$, the relative type

$$\tau(u, \varphi_{o,\max}) = \sup\{ c : u \leq c\varphi_{o,\max} + O(1) \text{ near } o \},$$

which is the prototype of a “Zhou number”. The relative types are both tropically multiplicative and additive (Corollary 2.5)—meaning

$$\tau(u_1 + u_2, \varphi_{o,\max}) = \tau(u_1, \varphi_{o,\max}) + \tau(u_2, \varphi_{o,\max})$$

and

$$\tau(\max(u_1, u_2), \varphi_{o,\max}) = \min\{ \tau(u_1, \varphi_{o,\max}), \tau(u_2, \varphi_{o,\max}) \}.$$

This makes $v(f) = \tau(\log|f|, \varphi_{o,\max})$ a valuation on the local ring of germs of holomorphic functions at $o$ (Bao et al., 2023, Bao et al., 2023).

2. Global and Multipoled Zhou Weights

Let $D \subset \mathbb{C}^n$ be a hyperconvex domain and $Z = \{z_1,\ldots,z_p\}$ a set of poles, each equipped with holomorphic data $f_{0,i}$ and psh weights $u_{0,i}$ ($\|f_{0,i}\|^2 e^{-2u_{0,i}}$ integrable). The multipoled global Zhou weight $\Phi_{P,\max}$ is the unique negative psh function on $D$ for which, at each $z_i$:

1. $$\|f_{0,i}\|^2 e^{-2u_{0,i}} |z-z_i|^{2N_i} e^{-2\Phi_{P,\max}}$$ is integrable for large $N_i$.
2. $\|f_{0,i}\|^2 e^{-2u_{0,i}} e^{-2\Phi_{P,\max}}$ is not integrable.
3. Any negative psh $\Psi>\Phi_{P,\max}$ with all such integrals still non-integrable must coincide with $\Phi_{P,\max}$.

Equivalently,

$$\Phi_{P,\max}(z) = \sup\left\{ \psi(z) : \psi\in PSH^-(D), (f_{0,i},z_i)\in\mathcal{I}(u_{0,i}+\psi)_{z_i}, \psi\ge\Phi_{P,\max}+O(1) \text{ at all } z_i \right\}.$$

For trivial data $$(z_i, f_{0,i} = z-z_i, u_{0,i} = (2n-1)\log|z-z_i|)$$, $\Phi_{P,\max}$ agrees with the classical pluricomplex Green function with logarithmic poles $Z$.

3. Approximation, Convergence, and Continuity

Zhou weights possess canonical approximation and limit properties. For strictly hyperconvex $D$, consider sequences

$$\Phi_m(z) = \sup \{ 1 - \log|f(z)| : f\in\mathcal{O}(D), \|f\|_\infty \leq 1, (f, z_i)\in \mathcal{I}(m\Phi_{P,\max})_{z_i} \}$$

and similarly $\Psi_m(z)$ defined using $L^2$-bounded holomorphic functions. Then

$$\lim_{m \to \infty} \Phi_m(z) = \lim_{m \to \infty} \Psi_m(z) = \Phi_{P,\max}(z)$$

pointwise and uniformly on compacta. Moreover, for each pole $z_i$, the relative type approximation satisfies

$$1 - \frac{C}{m} \leq \tau_{z_i}(\Phi_m, \Phi_{P,\max}) \leq 1,$$

for some $C>0$ independent of $m$ (Bao et al., 2023).

Zhou weights and their envelopes are continuous off their poles and are exhaustive: $\exp(\Phi_{P,\max})$ is continuous, tends to zero at $\partial D$, and the sublevel sets $\{z:\Phi_{P,\max}(z)<c\}$ exhaust $D$ as $c\to-\infty$.

4. Zhou Numbers and Valuation Structure

Given a (multi)pole Zhou weight $\Phi_{P,\max}$, the Zhou number at a pole $z_i$ for any $v \in PSH(D)$ is

$$\tau_{z_i}(v, \Phi_{P,\max}) = \sup\{ c \geq 0 : v \leq c\Phi_{P,\max} + O(1) \text{ near } z_i \}.$$

These quantities extend the classical relative types, and under addition and max operations they induce tropical multiplicativity and additivity, corresponding to group laws for valuations: $$\tau_{z_i}(u_1 + u_2, \Phi_{P,\max}) = \tau_{z_i}(u_1, \Phi_{P,\max}) + \tau_{z_i}(u_2, \Phi_{P,\max}),$$

$$\tau_{z_i}(\max\{u_1,u_2\}, \Phi_{P,\max}) = \min\{\tau_{z_i}(u_1, \Phi_{P,\max}), \tau_{z_i}(u_2, \Phi_{P,\max})\}.$$

Thus, for holomorphic germs $f$, the map $v(f) = \tau(\log|f|, \varphi_{o,\max})$ is a non-Archimedean, divisorial-type valuation on the local ring.

Zhou numbers admit an integral representation in terms of Tian functions: $$\sigma_{\varphi}(u) = \int_0^{+\infty} e^{-t} \, d\tau_u(t; \varphi),$$ where the Tian function is defined as

$\tau_u(t; \varphi) := \sup\{ c \geq 0 : |\fo|^2 e^{-2t u - 2c \varphi} \text{ integrable near } o \}, \quad t \in \mathbb{R}.$

This formalism links Zhou numbers directly to jumping numbers, multiplier ideals, and singularity exponents (Bao et al., 2023).

5. Relations to Multiplier Ideals, Jumping Numbers, and Singularity Theory

Let $\varphi$ be a Zhou weight. For any holomorphic germ $G$, the jumping number with respect to $\varphi$ is

$$c_G(\varphi) := \sup\{ c : |G|^2 e^{-2c\varphi} \in L^1_\text{loc}(o) \},$$

recovering the classical singularity exponent. The connections established include:

• For $|f_0|^2 e^{-2\varphi} = 1$, $c_G(\varphi) = v(G,\varphi) + 1$.
• Multiplier ideal equalities $\mathcal{I}(tu) = \mathcal{I}(tv)$ for all $t > 0$ are equivalent to all relative types ($\sigma_\psi(u)=\sigma_\psi(v)$, all local Zhou weights $\psi$), and equality of corresponding Tian functions.
• The division of holomorphic functions is captured by Zhou valuations:

$$f \mid g \iff v(f,\psi) \geq v(g,\psi)\ \forall \text{ local Zhou weights } \psi.$$

Moreover, the usual complex singularity exponent is given as

$$c(p) = \sup\{ \sigma_\varphi(p) : \varphi \text{ runs over local Zhou weights from } |I|^2 \}.$$

Thus, Zhou weights and their numbers encode fine analytic and algebraic features of singularities, divisibility, and valuation extraction (Bao et al., 2023).

6. Semi-Continuity, Global Properties, and Continuity in the Data

For any local Zhou weight $\varphi_{o,\max}$ at $o\in\mathbb{C}^n$ and $v \in PSH(D)$, the upper level sets

$$E_c(v, \varphi_{o,\max}) = \{ z\in D : \tau_z(v, T_z \varphi_{o,\max}) \geq c \}$$

are analytic subsets of $D$ (where $T_z \varphi_{o,\max}(w) = \varphi_{o,\max}(w-z)$). This semi-continuity property establishes stability of Zhou numbers under perturbations of singularities (Bao et al., 2023). Furthermore, global Zhou weights depend continuously on the location of the poles as long as no collision occurs (Theorem 1.14), with convergence uniform away from the poles.

7. Representative Examples and Connections to Classical Notions

• In the basic case $f(z) = z,\, p_0(z)=(2n-1)\log|z|$, the local Zhou weight is $\varphi_{o,\max}(z) = \log|z|$, recovering the prototypical logarithmic singularity.
• The global Zhou weight coincides with the classical pluricomplex Green function with multiple logarithmic poles when the holomorphic data and psh weights are trivial:

$$G_{D,Z}(z) = \sup\{ \psi(z) : \psi \in PSH^-(D), \psi(w)\leq \log|w-z_i| + O(1) \text{ near } z_i \}$$

and is recovered as the Zhou weight for $f_{0,i}=z-z_i,\, u_{0,i}=(2n-1)\log|z-z_i|$ (Bao et al., 2023).

Zhou weights thus generalize and interconnect the fundamental objects of pluripotential theory, complex analysis, and algebraic geometry, unifying Green-type extremal functions, multiplier ideal theory, and valuation structures with strong openness and approximation properties (Bao et al., 2023, Bao et al., 2023).