Maximality of moduli spaces of vector bundles on curves

This paper proves that moduli spaces of semistable vector bundles with coprime rank and degree over a non-singular real projective curve are maximal real algebraic varieties if and only if the underlying curve is maximal, establishing this result through the stronger property of Hodge-expressivity.

The Gist

Imagine you are an architect trying to build the most efficient, spacious house possible within a fixed amount of land. In the world of mathematics, specifically Real Algebraic Geometry, there is a similar concept called a "Maximal Variety."

Think of a "variety" as a complex geometric shape (like a curve, a surface, or a higher-dimensional object) that exists in two worlds simultaneously:

1. The Complex World: A rich, colorful, multi-dimensional universe where the shape is fully realized.
2. The Real World: A shadow or a slice of that shape that exists in our familiar, everyday reality (using real numbers).

The Big Question: How "Full" is the Shadow?

Mathematicians have a rule called the Smith-Thom Inequality. It's like a budget limit. It says: "The amount of 'stuff' (topological complexity) you can see in the Real World shadow can never exceed the amount of 'stuff' in the Complex World."

Usually, the shadow is much smaller than the full object. But sometimes, the shadow is perfectly full. It captures every possible piece of information from the complex world. When a shape achieves this perfect maximum, we call it a Maximal Variety.

For a long time, mathematicians only knew a few examples of these "perfect" shapes: simple things like spheres, tori (doughnuts), and certain grid-like structures (Toric varieties). Finding new, complex examples was very hard.

The New Discovery: The "Vector Bundle" Factory

This paper, by Erwan Brugallé and Florent Schaffhauser, introduces a brand new family of these "perfect" shapes. They are Moduli Spaces of Vector Bundles.

Let's break that down with an analogy:

• The Base Curve: Imagine a rubber band (a curve) floating in space.
• Vector Bundles: Imagine wrapping different types of ribbons or strings around this rubber band.
• The Moduli Space: This is a giant "catalog" or "map" that lists every single unique way you can wrap those ribbons.

The authors prove a stunning fact: If your original rubber band (the curve) is a "Maximal" shape, then the entire catalog of ribbon-wrapping ways (the Moduli Space) is also a Maximal shape.

Even better, they show this works no matter how complex the catalog gets. You can have ribbons of any thickness or length, and the resulting "catalog" will still be a perfect, maximal shape. This gives mathematicians an infinite supply of new, high-dimensional "perfect houses."

The Secret Weapon: "Hodge-Expressivity"

How did they prove this? They didn't just count the rooms in the shadow and the complex world and see if they matched (which is incredibly difficult). Instead, they discovered a deeper, stronger property they call "Hodge-Expressivity."

Think of the complex shape as a symphony.

• The Hodge Numbers are the sheet music, describing the specific notes (frequencies) and how they harmonize.
• The Betti Numbers (of the real shadow) are the sound you actually hear when the symphony is played in a small room.

Usually, the sound in the small room is a muffled, incomplete version of the sheet music.
However, the authors found that for these specific "ribbon catalogs," the sound in the small room is exactly what you get if you take the sheet music, ignore the "left-right" balance of the notes, and just play the main melody.

They call this "Hodge-Expressivity." It's like saying: "The shadow isn't just as big as the object; it's a perfect, direct translation of the object's internal structure."

Because this property is so strong, it automatically guarantees that the shape is "Maximal." It's like finding a master key that opens every door in the building, proving the building is fully accessible.

Why Does This Matter?

1. New Territory: Before this, we only knew a handful of maximal shapes. Now, we have a whole new factory producing them in arbitrarily large dimensions.
2. A New Strategy: The authors flipped the script. Usually, you prove a shape is maximal first, and then check if it has this special "Hodge" property. Here, they proved the special property first, which forced the shape to be maximal. This gives mathematicians a new tool to find other perfect shapes.
3. Connecting Worlds: It beautifully connects the abstract, high-dimensional world of complex geometry with the tangible world of real shapes, showing that under the right conditions, the shadow is just as rich as the object casting it.

In a nutshell: The authors found a magical recipe. If you start with a "perfect" curve and build a catalog of all the ways to wrap ribbons around it, the resulting catalog is guaranteed to be a "perfect" shape too, revealing a hidden harmony between the complex and real worlds.

Technical Summary

1. Problem Statement

The paper addresses a central question in real algebraic geometry: the classification and construction of maximal real algebraic varieties.

• Maximality: A real algebraic variety $X$ (equipped with an anti-holomorphic involution $\tau$) is called maximal if the sum of its mod 2 Betti numbers equals the sum of the Betti numbers of its complexification. This is the case of equality in the Smith–Thom inequality:
  $$\sum b_i(RX) = \sum b_i(X)$$
  where $RX$ is the real locus (fixed points of $\tau$).
• The Gap: While maximal varieties are known to exist in specific low-dimensional cases (e.g., real projective spaces, Grassmannians, toric varieties, and certain curves), constructing families of maximal varieties in arbitrarily large dimensions has been difficult.
• Specific Object: The authors investigate the moduli spaces of semistable vector bundles ($M_C(r, d)$) over a non-singular real projective curve $C$, where the rank $r$ and degree $d$ are coprime. The goal is to determine under what conditions these moduli spaces are maximal.

2. Methodology

The authors employ a strategy that reverses the typical logical flow in this field. Instead of proving maximality directly by comparing Betti numbers (which is computationally intricate for these spaces), they prove a stronger property called Hodge-expressivity.

Key Definitions and Tools:

• Hodge-Expressivity: A non-singular projective real algebraic variety $X$ is Hodge-expressive if:

  1. Its integral cohomology $H^*(X; \mathbb{Z})$ is torsion-free.
  2. Its mod 2 Poincaré polynomial of the real locus equals the Hodge polynomial of the complex locus evaluated at $y=1$:
     $$P_t(RX) = H(t, 1)(X)$$
     Note: Hodge-expressivity implies maximality because $\sum b_i(RX) = P_1(RX) = H(1,1)(X) = \sum b_i(X)$.

• Recursive Formulas: The proof relies on comparing recursive formulas for:

  1. Complex Hodge Numbers: Derived from the work of Earl and Kirwan [EK00], who computed the Hodge polynomial $H(x,y)(M_C(r,d))$ using the Harder–Narasimhan stratification and gauge theory.
  2. Real Betti Numbers: Derived from the work of Liu and Schaffhauser [LS13], who computed the mod 2 Poincaré polynomial $P_t(RM_C(r,d))$ using a real analogue of the Atiyah–Bott approach.

• Harder–Narasimhan Stratification: Both the complex and real moduli spaces are stratified by the Harder–Narasimhan type (filtrations of unstable bundles). The authors show that the recursive relations governing the polynomials for the complex Hodge numbers and the real Betti numbers are structurally identical when the base curve is maximal.

3. Key Contributions

A. Introduction of Hodge-Expressivity as a Primary Tool

The authors formalize the concept of Hodge-expressivity and argue that it is a more fundamental property than maximality. They demonstrate that for many known maximal varieties, Hodge-expressivity holds, and it serves as a robust mechanism to prove maximality for new, high-dimensional families.

B. Main Theorem (Theorem 1.2)

The central result establishes a precise equivalence between the maximality of the base curve and the Hodge-expressivity of the moduli space:

Theorem: Let $C$ be a non-singular real projective curve of genus $g \ge 1$. Let $r, d$ be coprime integers. The moduli space $M_C(r, d)$ (and the fixed-determinant moduli space $M_C(r, \Lambda)$) is Hodge-expressive if and only if the base curve $C$ is maximal.

Consequently, if $C$ is maximal, $M_C(r, d)$ is a maximal real algebraic variety of dimension $r^2(g-1) + 1$.

C. Proof of the Converse (Proposition 1.3)

The authors prove that if the base curve $C$ is not maximal, then the moduli space $M_C(r, d)$ is not maximal. This establishes that the maximality of the moduli space is strictly dependent on the maximality of the underlying curve.

D. Unification of Recursive Structures

A significant technical contribution is the demonstration that the recursive formula for the real Betti numbers (dependent on the number of connected components of the real curve, $b_0(RC)$) matches the recursive formula for the complex Hodge numbers (dependent on the genus $g$) if and only if $b_0(RC) = g+1$ (the condition for $C$ being maximal).

4. Results

1. New Family of Maximal Varieties: The paper provides an infinite family of maximal real algebraic varieties with arbitrarily large dimensions. By choosing a maximal curve $C$ of genus $g$ and coprime integers $r, d$, one generates a maximal moduli space of dimension $r^2(g-1)+1$.
2. Fixed Determinant Case: The result extends to moduli spaces with fixed determinant $M_C(r, \Lambda)$. These are also Hodge-expressive if $C$ is maximal.
3. Limitations: The authors note that while the result holds for $M_C(r, d)$, the case for $M_C(r, \Lambda)$ when $g, r \ge 2$ and $C$ is non-maximal remains an open question regarding non-maximality, due to the complexity of the explicit formulas for higher ranks.
4. Torsion-Free Cohomology: The proof relies on the fact that the integral cohomology of these moduli spaces is torsion-free (a result by Atiyah and Bott [AB83]), which is a necessary condition for the Hodge-expressivity definition used.

5. Significance

• Expansion of Known Maximal Varieties: Prior to this work, the list of known maximal varieties in high dimensions was sparse (limited mostly to hypersurfaces constructed via Viro's patchworking or specific low-dimensional classifications). This paper significantly expands the landscape by linking vector bundle moduli spaces to the theory of maximal curves.
• Methodological Shift: The paper advocates for a shift in strategy: proving Hodge-expressivity first, which automatically implies maximality. This approach bypasses the difficult direct comparison of Betti sums and utilizes the richer structure of Hodge polynomials.
• Connection to Topology: The work deepens the understanding of the relationship between the topology of the real locus of a moduli space and the complex geometry of the underlying curve. It confirms that the "maximality" property is preserved under the operation of taking moduli spaces of vector bundles, provided the base is maximal.
• Broader Context: The authors place their results in a broader panorama (Section 3), showing that Hodge-expressivity is a common feature of "basic" maximal varieties (projective spaces, Grassmannians, toric varieties) and suggesting that this property might be a defining characteristic of "natural" maximal varieties in real algebraic geometry.

In summary, Brugallé and Schaffhauser successfully identify a vast new class of maximal real algebraic varieties by proving that the moduli spaces of vector bundles over maximal curves inherit a strong structural property (Hodge-expressivity) that guarantees their maximality.