Regularity of the volume function

This paper establishes the optimal $C^{1,1}$ regularity of the volume function on the big cone of a projective manifold and examines its regularity when restricted to segments moving in ample directions.

The Gist

Imagine you are an architect trying to understand the "size" or "potential" of different shapes you can build on a complex, multi-dimensional landscape. In the world of advanced mathematics (specifically algebraic geometry), this landscape is called a projective manifold, and the shapes you build are defined by objects called divisors.

The authors of this paper, Junyu Cao and Valentino Tosatti, are investigating a specific tool they use to measure these shapes: the Volume Function.

Here is a breakdown of their discovery using simple analogies.

1. The Volume Function: Measuring "Potential"

Think of the Volume Function as a special ruler. If you point this ruler at a shape (a divisor), it tells you how much "stuff" (mathematical data) fits inside it as you scale it up infinitely.

• The Big Cone: Imagine a giant, open room filled with shapes that have a positive volume. Mathematicians call this the "Big Cone." Inside this room, the shapes are "big" and full of potential.
• The Edge: The walls of this room are the boundary. If you step outside, the volume drops to zero.

2. The Big Question: How Smooth is the Ruler?

Mathematicians love to know how "smooth" their functions are.

• Smooth (C∞): Like a polished marble statue. You can run your finger over it, and it feels perfectly even. You can take derivatives (slopes) forever.
• Rough (C1): Like a piece of sandpaper. It has a slope, but if you try to measure how the slope changes, it feels jagged.
• The Problem: For a long time, mathematicians knew the Volume Function was "smooth enough" to have a slope (it was differentiable). But they didn't know if the slope itself changed smoothly, or if it had tiny, invisible bumps.

The Analogy: Imagine driving a car on a road defined by this Volume Function.

• We knew the road wasn't a cliff; you could drive on it.
• We knew the steering wheel turned smoothly (the first derivative exists).
• The Mystery: Was the road perfectly paved (smooth), or was it a bumpy dirt track where the steering wheel jerked slightly every time you turned?

3. The Main Discovery: The "Goldilocks" Smoothness

The authors prove that the Volume Function is optimal. It is not perfectly smooth (you can't drive forever without hitting a bump), but it is as smooth as it possibly can be without being perfect.

They call this $C^{1,1}$ regularity.

• What does this mean? Imagine a road where the surface is paved, but the curvature of the road changes abruptly at certain points. It's not a jagged cliff, but it's not a perfect curve either.
• The Result: They proved that inside the "Big Cone" (the room full of shapes), the Volume Function is exactly this level of smoothness. It has a continuous slope, and that slope doesn't change too wildly—it's "Lipschitz continuous."
• Why it matters: Before this, people weren't sure if the function was even this smooth. They proved it is the best possible smoothness you can get. You can't make it smoother; the universe of these shapes simply doesn't allow it.

4. The Edge of the Room

They also looked at the walls of the Big Cone (where the volume is zero).

• The Finding: Here, the function is less smooth. It's like walking from a paved road onto a gravel path. The function is continuous (you don't fall off), but the slope can change instantly. It's only "Lipschitz" here, meaning it's rougher than inside the room.

5. The "One Step Forward" vs. "One Step Back"

The paper also looks at what happens when you walk in a straight line through this landscape.

• Walking Forward (Adding a "Big" shape): If you take a shape and add a little bit of "good stuff" (an ample class) to it, the volume function behaves nicely. It's smooth enough to have a second derivative, but not a third. It's like walking on a slightly bumpy path.
• Walking Backward (Subtracting a "Big" shape): This is the tricky part. If you start with a big shape and subtract good stuff, the behavior is mysterious.
  • The Conjecture: Another famous mathematician (Rob Lazarsfeld) guesses that if you subtract, the path might actually be perfectly smooth (like a glass slide) for a short distance.
  • The Reality: The authors couldn't prove this yet, but they showed that in the "Forward" direction, the path definitely gets bumpy after a while.

Summary

Think of the Volume Function as a terrain map for a mathematical universe.

• Old View: We knew the terrain wasn't a cliff, but we didn't know if it was a smooth hill or a jagged mountain.
• New View (This Paper): The terrain is a perfectly paved road with occasional sharp turns. It is as smooth as math allows it to be ($C^{1,1}$), but no smoother.
• Why we care: Knowing exactly how "smooth" this function is helps mathematicians build better theories about the shape of the universe, solve complex equations, and understand the fundamental rules of geometry.

In short, Cao and Tosatti have mapped the "roughness" of this mathematical landscape with the highest precision possible, proving that while the road isn't perfect, it's as good as it gets.

Technical Summary

Here is a detailed technical summary of the paper "Regularity of the Volume Function" by Junyu Cao and Valentino Tosatti.

1. Problem Statement

The paper investigates the optimal regularity of the volume function, $\text{Vol}(\alpha)$, defined on the space of $(1,1)$-cohomology classes of a compact Kähler manifold $X$.

• Context: The volume function generalizes the asymptotic growth of sections of line bundles. It is defined on the big cone $B_X \subset H^{1,1}(X, \mathbb{R})$ (classes with positive volume) and vanishes outside this cone.
• Known Results:
  • $\text{Vol}$ is continuous and locally Lipschitz on the entire space $H^{1,1}(X, \mathbb{R})$.
  • On the big cone $B_X$, $\text{Vol}$ is known to be $C^1$ differentiable (proven by Boucksom-Favre-Jonsson, Lazarsfeld-Mustață, and Witt Nyström).
  • The gradient is given by the formula: $\frac{d}{dt}\big|_{t=0} \text{Vol}(\alpha + t\beta) = n \beta \cdot \langle \alpha^{n-1} \rangle$, where $\langle \cdot \rangle$ denotes the non-pluripolar product.
• The Open Question: It was unknown whether the volume function is $C^{1,1}$ (continuously differentiable with a Lipschitz gradient) on the big cone. Previous examples (e.g., blowups of $\mathbb{P}^2$) showed that $\text{Vol}$ is generally not $C^2$ (its Hessian is discontinuous). The authors address whether $C^{1,1}$ is the optimal regularity class.

2. Methodology

The authors employ a combination of convex analysis, transcendental methods in complex geometry, and specific geometric constructions.

A. Convexity and Concavity Arguments

• Proposition 2.1: The authors establish that the function $\alpha \mapsto (\omega \cdot \langle \alpha^{n-1} \rangle)^{\frac{1}{n-1}}$ is concave on the big cone for any Kähler class $\omega$.
• Proof Technique: They utilize the Fujita approximation method combined with the Khovanskii-Teissier inequalities. By approximating big classes with nef classes on modifications of $X$, they reduce the problem to standard intersection theory where these inequalities hold.
• Implication: Since the $(n-1)$-th root of the gradient component is concave, and concave functions on open sets are locally Lipschitz, the gradient itself is locally Lipschitz.

B. Abstract Functional Analysis

• Proposition 2.3: The authors prove a general lemma for functions $f: B \to \mathbb{R}_{\geq 0}$ defined on a convex cone $B$. If $f$ is:
  1. Homogeneous of degree $d \geq 1$,
  2. Non-decreasing in the directions of the cone,
  3. Locally bounded,
     Then $f$ is locally Lipschitz.
• Application: They apply this to the function $f(\alpha) = \omega \cdot \langle \alpha^{n-1} \rangle$ (which represents the components of the gradient of $\text{Vol}$). This provides a second, more elementary proof of the main theorem that does not rely on the concavity of the root function.

C. Construction of Counter-Examples

• To prove that the regularity cannot be improved to $C^2$ (or even $C^{2,\gamma}$), the authors construct a specific example using projective bundles.
• They utilize a result from Filip, Lesieutre, and Tosatti [12] regarding a divisor $D$ on a hypersurface $B \subset (\mathbb{P}^1)^4$ where the volume along a segment is not $C^{1,\gamma}$.
• By lifting this to a $\mathbb{P}^1$-bundle $X$ over $B$, they relate the volume on $X$ to an integral of the volume on $B$ (using Wolfe's formula). This integration smooths the function by one degree of differentiability, transforming the $C^{1,\gamma}$ failure on the base into a $C^{2,\gamma}$ failure on the total space.

3. Key Contributions and Results

Theorem 1.1: Optimal $C^{1,1}$ Regularity on the Big Cone

• Statement: Let $X$ be a projective manifold (or a compact Kähler manifold satisfying the BDPP Conjecture). Then $\text{Vol} \in C^{1,1}_{\text{loc}}(B_X)$.
• Significance: This confirms that the volume function is twice differentiable almost everywhere (by Alexandrov's theorem on convex functions) and that its gradient is Lipschitz continuous. This is the optimal regularity, as $C^2$ regularity fails in general.
• Corollary: The volume function is 3-times differentiable Lebesgue almost everywhere on $B_X$.

Theorem 1.2: Global Lipschitz Continuity

• Statement: $\text{Vol} \in \text{Lip}_{\text{loc}}(H^{1,1}(X, \mathbb{R}))$.
• Significance: This extends the known Lipschitz property from the Néron-Severi group (for projective varieties) to the full real cohomology group for Kähler manifolds (under the BDPP conjecture). It confirms that the function remains Lipschitz even at the boundary of the big cone ($\partial B_X$), where the volume vanishes.

Theorem 1.4: Regularity Along Segments

• Statement: For $\alpha \in B_X$ and a Kähler class $\omega$, the function $t \mapsto \text{Vol}(\alpha + t\omega)$ is $C^{1,1}$ on $[0,1]$, but there exist examples where it is not $C^{2,\gamma}$ for any $\gamma > 0$.
• Contrast: The paper highlights a sharp contrast between moving in the "ample direction" ($\alpha + t\omega$) versus the "negative direction" ($\alpha - t\omega$).
  • $\alpha + t\omega$: Regularity is limited to $C^{1,1}$.
  • $\alpha - t\omega$: The authors conjecture (based on Lazarsfeld's work) that this might be smooth or real analytic for small $t$, as the path moves deeper into the big cone where the geometry is "nicer."

4. Significance and Implications

1. Resolution of an Open Problem: The paper definitively answers Question 1.4.4 from [12], establishing $C^{1,1}$ as the optimal regularity class for the volume function on the big cone.
2. Geometric Insight: The results clarify the behavior of the volume function near the boundary of the big cone and along specific paths. The failure of $C^2$ regularity is linked to the "walls" in the Zariski decomposition or the accumulation of singularities in the non-pluripolar products.
3. Methodological Advancement: The use of the abstract Lemma 2.3 provides a robust tool for proving Lipschitz continuity for homogeneous, monotonic functions on cones, which may have applications beyond algebraic geometry.
4. Connection to BDPP Conjecture: The results are conditional on the BDPP Conjecture (Boucksom-Demailly-Păun-Peternell) for general Kähler manifolds, which posits that the pseudo-effective cone is the closure of the movable cone. This underscores the deep link between the analytic regularity of the volume and the structural geometry of the cone of effective divisors.
5. Future Directions: The authors note that while $C^{1,1}$ is optimal globally, it remains an open and interesting question whether the volume function is $C^\infty$ (or real analytic) on the "chambers" of a wall-chamber decomposition of the big cone, a property known to hold for surfaces and toric manifolds.

In summary, Cao and Tosatti provide a rigorous proof that the volume function is as smooth as possible ($C^{1,1}$) given the geometric constraints of birational geometry, while demonstrating that higher regularity is obstructed by specific geometric phenomena.