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Genus 4 Supermoduli Space Is Not Projected

This paper proves that the even-spin component of the supermoduli space for genus 4 super Riemann surfaces is non-projected by constructing a specific compact curve and demonstrating that the obstruction class to projectedness does not vanish when restricted to it.

Original authors: Ron Donagi, Simone Noja

Published 2026-08-27

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CC BY 4.0

Original authors: Ron Donagi, Simone Noja

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Mathematics often deals with shapes that exist in dimensions beyond our everyday sight, where the rules of geometry become far more flexible and strange. In one branch of this field, researchers study "super" versions of familiar surfaces. Imagine a standard sheet of paper, but with an extra, invisible layer of information attached to every point, a layer that behaves differently from the paper itself. These are called super Riemann surfaces. For decades, mathematicians have tried to organize all possible shapes of these surfaces into a single, coherent map, known as a moduli space. A key question has been whether this map can be flattened out into a simpler, more predictable structure called a "projected" space. For surfaces with five or more holes, it was already known that this flattening is impossible. The question remained open for surfaces with exactly four holes, a case that sits right on the edge of what is possible.

In a recent study, Ron Donagi and Simone Noja settled this question for the four-hole case, proving that the map cannot be flattened. They focused on a specific type of these surfaces, those with a particular kind of symmetry known as "even spin." To do this, they did not try to examine every possible four-hole surface at once, which would be an impossible task. Instead, they constructed a specific, compact family of these surfaces. They started with a fixed, two-hole surface and created a new family of four-hole surfaces by taking double covers of it. You can think of this as taking a two-sided sheet and gluing it to itself along specific lines to create a new, more complex shape. By carefully choosing where to make these cuts and glues, they ensured the resulting family of surfaces stayed within the interior of the mathematical map, avoiding the messy edges where shapes break apart.

The researchers then tested whether this specific family of surfaces could be flattened. They looked for a mathematical obstruction, a kind of hidden tension that would prevent the surface from being projected. To find this tension, they examined how the surfaces in their family twisted and turned relative to one another. They focused on a specific part of the family that remained unchanged under a particular symmetry operation, effectively isolating a clean, manageable piece of the problem. By restricting their attention to this piece, they were able to translate the complex, high-dimensional problem into a calculation on the original, simpler two-hole surface.

The final step was a direct calculation on this simpler surface. The researchers computed a specific value that would be zero if the surfaces could be flattened, and non-zero if they could not. They found that this value was indeed non-zero. This result proved that the obstruction to flattening exists and is real. Because this obstruction was found within a specific family of four-hole surfaces, it proves that the entire map of four-hole surfaces is not projected. The authors showed that the full map, which includes both the even and odd types of these surfaces, cannot be flattened. This confirms that the geometry of these four-hole super surfaces is inherently complex and resists simplification, just as the five-hole versions do. The work closes a gap in our understanding, showing that the complexity of these mathematical objects begins at genus four, rather than waiting until genus five.

Technical Summary: Genus 4 Supermoduli Space is Not Projected

Problem Statement
The paper addresses the projectedness of the supermoduli stack $\mathcal{M}_g$ , which parametrizes smooth unpunctured super Riemann surfaces of genus $g$ . A supermanifold is projected if it admits a splitting of its structure sheaf, effectively allowing it to be viewed as a vector bundle over its reduced space. It was previously established (cf. [4]) that $\mathcal{M}_g$ is non-projected for $g \geq 5$ . The specific problem tackled in this work is to determine whether this non-projectedness extends to genus $g=4$ , specifically to the even-spin component $\mathcal{M}^+_4$ .

Methodology
The authors employ obstruction theory to demonstrate non-projectedness. The primary obstruction to projectedness is the class $\omega_2(S) \in H^1(M, T_M \otimes \Lambda^2 E^\vee)$ , where $S$ is the supermanifold and $E^\vee$ is the dual of the odd ideal. If this class is non-zero, the supermanifold is not projected.

To prove $\omega_2(\mathcal{M}^+_4) \neq 0$ , the authors restrict the obstruction class to a specific compact curve $T$ contained within the moduli space. The methodology proceeds through the following steps:

Construction of a Test Family: The authors construct a family of genus-4 curves $(X_T, \eta)$ over a base curve $T$ . These curves are defined as branched double covers of a fixed genus-2 curve $C: y^2 = x^6 - 1$ . The branch locus consists of pairs $(p, \sigma p)$ , where $\sigma$ is a fixed-point-free automorphism of order 6 on $C$ . This construction avoids the need to lift the family to a family of super Riemann surfaces explicitly, working instead within the spin moduli space $SM^+_4$ .
Reduction to a Mixed Atiyah Class: The obstruction class $\omega_2$ is represented by an extension of vector bundles (the "diagonal extension"). The authors isolate the $J$ -invariant part of this extension, where $J$ is a lift of the deck involution to the spin line bundle. By restricting the middle sheaf of this extension to the second infinitesimal neighborhood of the diagonal and taking the direct image, they reduce the problem to computing the categorical pushout of a "cross-block" extension.
Descent to the Fixed Curve: Through a finite étale map $h: T \to C$ , the problem is descended to the fixed genus-2 curve $C$ . The obstruction class is identified with the direct image of a mixed Atiyah class associated with a specific line bundle $L$ on $C \times C$ .
Explicit Čech Calculation: The non-vanishing of the obstruction is reduced to an explicit calculation on the genus-2 curve $C$ . This involves computing the action of the mixed Atiyah class on vector bundles constructed from the cohomology of powers of the canonical bundle, utilizing the specific automorphism $\sigma$ and its action on the basis of differential forms.

Key Contributions and Results

Theorem 1.1: The even component $\mathcal{M}^+_4$ of the smooth unpunctured genus-four supermoduli stack is not projected. Consequently, the full supermoduli stack $\mathcal{M}_4$ is not projected.
Technical Shortcut: Unlike previous work for $g \geq 5$ (specifically [4]), which required lifting a family of curves to a family of super Riemann surfaces and verifying constraints on ramification points (e.g., odd order), this paper bypasses the lifting step. It works directly with a compact curve in the spin moduli space $SM^+_4$ . Since $SM^+_4$ maps to $\mathcal{M}^+_4$ , the non-vanishing of the obstruction on the spin curve implies non-projectedness of the supermoduli space.
Explicit Non-Vanishing: The paper provides a concrete, non-zero calculation of the obstruction class. By analyzing the action of the automorphism $\sigma$ on the cohomology of the curve $C$ , the authors show that the relevant extension class does not split. Specifically, they demonstrate that the equation governing the splitting of the extension has no solution due to a contradiction in the lower-right block of the resulting matrix (specifically, a residual class of $\text{diag}(4, -4) \neq 0$ ).

Significance
The paper establishes that the phenomenon of non-projectedness, previously known only for genus $g \geq 5$ , also occurs in genus 4. This closes a gap in the understanding of the geometry of supermoduli spaces. The authors note that their approach offers a "shortcut" by avoiding the complex lifting procedures required in earlier proofs, relying instead on the geometry of the underlying spin moduli space and explicit cohomological calculations on a fixed curve. The work relies on and extends the framework of obstruction theory developed in [3, 5, 6] and utilizes the specific geometry of curves with automorphisms as described in [1]. The authors also mention that their methods are currently being applied to the case of genus 3 in ongoing joint work.

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