Green currents of holomorphic correspondences on compact Kähler manifolds

This paper constructs Green currents associated with the dominant eigenspaces of holomorphic correspondences on compact Kähler manifolds under specific dynamical degree conditions, establishes the log-Hölder continuity of their super-potentials, and proves the exponential equidistribution of positive closed currents toward the main Green current when the correspondence exhibits simple cohomological action and satisfies a local multiplicity assumption.

The Gist

Imagine you are standing in a vast, complex city (the Compact Kähler Manifold). This city has a special rulebook for how things move and change. Usually, we study maps where every point goes to exactly one other point (like a standard function). But in this paper, the authors are studying Holomorphic Correspondences.

Think of a correspondence not as a single path, but as a magic multi-tool. When you stand at one spot in the city, this tool doesn't just point you to one destination; it points you to several possible destinations at once, like a "Choose Your Own Adventure" book where the story branches out.

The paper asks: If we keep using this multi-tool over and over again, where do we end up?

Here is the breakdown of their discovery, translated into everyday language:

1. The "Green Currents" (The City's Gravity)

In any chaotic system, there are usually "attractors"—places where things naturally settle down. In math, these are called Green Currents.

• The Analogy: Imagine pouring a bucket of water (representing all the possible paths in the city) onto a landscape. No matter where you pour it, the water eventually flows into specific rivers or lakes. These rivers are the Green Currents.
• The Problem: For simple maps, we know where the rivers are. But for these "multi-tool" maps (correspondences), the terrain is bumpy, and the water can split and merge in confusing ways.
• The Discovery: The authors proved that even with these confusing multi-tools, there are stable rivers (Green Currents). They figured out exactly how to calculate the shape of these rivers, even when the map is messy and non-reversible (you can't always trace your steps back).

2. The "Super-Potential" (The Smoothness of the River)

Once they found the rivers, they wanted to know: How smooth are the banks?

• The Analogy: Is the riverbank a perfectly smooth glass slide, or is it jagged and rough? In math, "smoothness" tells us how predictable the system is.
• The Discovery: They found that the banks of these rivers are "log-Hölder continuous."
  • Translation: This is a fancy way of saying the banks are very smooth, but not perfectly smooth. They have a tiny bit of "fuzziness" or "friction."
  • Why it matters: If the banks were too rough, the water (the dynamics) would behave erratically. Because they are "log-Hölder," the system is stable enough to predict, even if it's not perfectly rigid.

3. The "Exponential Equidistribution" (The Rush to the River)

The second major part of the paper is about speed. How fast does the water flow into the river?

• The Analogy: Imagine dropping a leaf in the city. Does it wander around for a century before finding the river? Or does it zoom straight there?
• The Discovery: The authors proved that if the "multi-tool" isn't too crazy (a condition they call "small multiplicity," meaning it doesn't get stuck in weird loops too often), the leaf zooms to the river.
• The "Exponential" Part: This means it happens incredibly fast. It's not just "fast"; it's like a rocket. The further you go in time, the closer you get to the final pattern, doubling your progress every step. This is a huge deal because it allows mathematicians to predict the long-term behavior of the system with high precision.

4. The "Generic" Truth (It Happens Most of the Time)

Finally, they asked: "Is this true for every multi-tool, or just the lucky ones?"

• The Analogy: If you pick a random multi-tool from a giant bin, will it behave nicely?
• The Discovery: Yes! They showed that for "generic" (randomly chosen) correspondences, these nice properties hold true. Even if you build a complex polynomial map (a specific type of multi-tool), as long as you run it a few times, it will settle into this predictable, smooth, fast-flowing pattern.

Summary

In short, Luo and Vergamini took a very complex, multi-branching mathematical system and proved:

1. Stability: There are always stable "rivers" (Green Currents) where the system settles.
2. Smoothness: These rivers have predictable, smooth edges.
3. Speed: The system rushes to these rivers incredibly fast.
4. Universality: This happens for almost all systems of this type, not just special cases.

They essentially built a map for the chaos, showing us that even in a world of "many choices," there is a clear, fast, and smooth path that everything eventually follows.

Technical Summary

Here is a detailed technical summary of the paper "Green Currents of Holomorphic Correspondences on Compact Kähler Manifolds" by Muhan Luo and Marco Vergamini.

1. Problem Statement and Context

The paper addresses the dynamical properties of holomorphic correspondences on compact Kähler manifolds. Unlike holomorphic maps (endomorphisms) or automorphisms, correspondences are multivalued maps defined by analytic cycles. They represent a broader class of dynamical systems that encompass both endomorphisms of projective spaces and automorphisms of Kähler manifolds, yet introduce significant complexities due to non-invertibility and the existence of critical points and values.

The central problem is to construct Green currents (invariant positive closed currents) for these correspondences and analyze their regularity and dynamical behavior. Specifically, the authors aim to:

1. Construct Green currents associated with the dominant eigenspaces of the pullback operator on cohomology.
2. Establish the regularity of these currents, specifically regarding their super-potentials.
3. Prove exponential equidistribution results for generic currents towards these Green currents under specific multiplicity conditions.

2. Methodology

The authors employ a synthesis of complex dynamics, pluripotential theory, and linear algebra. Key methodological components include:

• Super-potentials: Utilizing the theory introduced by Dinh and Sibony, the authors define super-potentials for currents of arbitrary bidegree. These functionals generalize classical potentials and serve as the primary tool for measuring the regularity of currents.
• Spectral Analysis of Pullback Operators: The study relies on the spectral properties of the pullback operator $f^*$ acting on the cohomology groups $H^{q,q}(X, \mathbb{R})$. The authors analyze the dynamical degrees $d_q(f)$ (spectral radii) and the structure of the dominant eigenspaces.
• Lojasiewicz-type Inequalities: To handle the non-invertibility of the correspondence, the authors derive specific estimates using Lojasiewicz-type inequalities. These relate the distance between points in the graph of the correspondence to the distance between their images, controlling the local multiplicity (ramification).
• Skoda-type Estimates and Interpolation: The proof of regularity involves Skoda-type inequalities and interpolation between Banach spaces to establish Hölder and log-Hölder continuity of super-potentials.
• Family Arguments: To prove genericity results, the authors consider families of holomorphic correspondences and use slicing theory to show that properties like "small inverse multiplicity" hold for a Zariski open dense set of parameters.

3. Key Contributions and Results

A. Construction and Regularity of Green Currents

The paper generalizes previous results on automorphisms and endomorphisms to the setting of holomorphic correspondences.

• Theorem 1.1 (Existence and Regularity): For a holomorphic correspondence $f$ on a compact Kähler manifold $X$ of dimension $k$, if the dynamical degrees satisfy $d_{q-1} < d_q$, the authors construct Green currents $T_c$ for every class $c$ in the real dominant eigenspace $F$ of $f^*$ on $H^{q,q}(X, \mathbb{R})$.
  • These currents satisfy the invariance relation $f^*(T_c) = T_{f^*(c)}$.
  • Crucial Regularity Result: The super-potential of $T_c$ is log-Hölder-continuous. This is a weaker condition than standard Hölder continuity but is sufficient for many dynamical applications. The proof involves explicit construction of the super-potential via an infinite series and estimating the decay of terms using the gap between dynamical degrees.

B. Exponential Equidistribution

The second major contribution concerns the speed of convergence of iterated currents towards the Green currents.

• Theorem 1.2 (Exponential Convergence): Under the assumption that $f$ has simple action on cohomology (meaning there is a unique maximal dynamical degree $d_q$ with a simple eigenvalue) and $f^{-1}$ is also holomorphic:
  1. If $f$ has $q$-small inverse multiplicity, the sequence $d_q^{-n} (f^n)^*(S)$ converges exponentially fast to the Green current $T^+$ for any positive closed current $S$ with appropriate normalization.
  2. If $f$ has small multiplicity (a condition on the graph of the pair $(f, f^{-1})$), the sequence of graphs $d_q^{-n} [\Gamma_{f^n}]$ converges exponentially fast to the product current $T^+ \otimes T^-$.
• Significance of Multiplicity: The "small multiplicity" condition is an analogue to the absence of highly critical periodic orbits in dimension 1. The authors prove that this condition is satisfied by generic holomorphic correspondences (specifically, for polynomial correspondences up to finite iterations).

C. Examples and Genericity

The authors demonstrate the richness of the class of correspondences satisfying these conditions:

• Symmetric Products: They construct examples using symmetric products of maps on $\mathbb{P}^1$.
• Polynomial Correspondences: They show that for families of polynomial correspondences defined by $G_0(Z) = G_1(W)$, the condition of having small inverse multiplicity holds for a generic element after a finite number of iterations.

4. Technical Definitions and Tools

• Dynamical Degrees ($d_q$): Defined as the spectral radius of $f^*$ on $H^{q,q}$. The condition $d_{q-1} < d_q$ ensures the existence of a dominant direction in cohomology.
• Inverse Multiplicity ($\delta(f)$): The maximal local multiplicity of the projection $\pi_2$ restricted to the graph $\Gamma$. The condition of "$q$-small inverse multiplicity" requires $\delta(f)$ to be sufficiently small relative to the ratio of dynamical degrees ($d_q/d_{q-1}$).
• Log-Hölder Continuity: A function $u$ is log-Hölder continuous if $|u(R)| \leq \frac{L}{(1 + |\log \|R\|_{-2}|)^r}$. This regularity is shown to be the natural threshold for establishing exponential mixing and equidistribution in high-dimensional complex dynamics.

5. Significance and Impact

This paper represents a significant advancement in the field of complex dynamical systems:

1. Unification: It successfully extends the theory of Green currents from automorphisms and endomorphisms to the more general and complex setting of holomorphic correspondences.
2. Regularity Breakthrough: By establishing log-Hölder continuity of super-potentials, the authors provide the necessary analytical tools to handle the singularities inherent in non-invertible maps. This regularity is weaker than Hölder but strong enough to imply exponential decay of correlations.
3. Statistical Properties: The exponential equidistribution results are the key to establishing strong statistical properties (such as mixing, Central Limit Theorems, and large deviation principles) for these dynamical systems.
4. Genericity: The proof that "small multiplicity" is a generic property ensures that these strong dynamical results apply to a vast majority of holomorphic correspondences, including many polynomial systems, rather than just isolated examples.

In summary, Luo and Vergamini provide a rigorous framework for understanding the asymptotic behavior of holomorphic correspondences, bridging the gap between algebraic geometry (cohomology) and complex analysis (currents and potentials) to solve long-standing problems regarding invariant measures and equidistribution.