Supermoduli spaces are non-projected in every genus $g\geq 3$

Technical Summary: Non-Projectedness of Supermoduli Spaces in Genus $g \ge 3$

Problem Statement
The paper addresses the global geometric structure of the moduli superstack $\mathcal{M}^{\varepsilon}_g$ of smooth, unpunctured super Riemann surfaces of genus $g$ with spin parity $\varepsilon \in \{+, -\}$. A central question in supergeometry is whether these moduli spaces are "projected" or "split." A supermanifold is projected if it admits a retraction onto its reduced moduli space (the moduli space of spin curves, $\mathcal{S}^{\varepsilon}_g$). It is split if its structure sheaf is isomorphic to the exterior algebra of the odd cotangent bundle of the reduction.

While local deformation theory for these spaces is unobstructed, the existence of a global projection is obstructed by a primary cohomology class $\omega_2(\mathcal{M}^{\varepsilon}_g) \in H^1(\mathcal{S}^{\varepsilon}_g, \text{Hom}(\wedge^2 T^- \mathcal{M}^{\varepsilon}_g, T^+ \mathcal{M}^{\varepsilon}_g))$. Previous work established that for $g=2$, both parity components are split. For $g \ge 5$, non-projectedness was known, and the even genus-4 case was resolved. The threshold case of genus $g=3$ (and the odd genus-4 case) remained open. The authors aim to prove that for every $g \ge 3$, both parity components are non-projected and non-split in both algebraic and holomorphic categories.

Methodology
The proof proceeds through a multi-stage argument combining degeneration techniques, formal deformation theory, and algebraic-to-analytic comparison theorems.

1. Boundary Degeneration and Test Families:
   The authors construct a specific morphism from a "central source" superscheme $Z^{\varepsilon}_{\eta}$ to the separating Neveu–Schwarz boundary divisor $D$ of the stable supermoduli space.

   • The Core: The source is built upon a genus $h = g-1$ curve $C$ equipped with an ineffective even theta characteristic $L$ (where $H^0(C, L) = H^1(C, L) = 0$).
   • The Tail: An elliptic component is attached at a node. The spin structure of this tail determines the parity $\varepsilon$ of the resulting genus-$g$ surface.
   • Parity Distinction: For the even parity ($\varepsilon=+$), the construction includes a moving NS marking on the elliptic tail to eliminate invariant obstructions. For the odd parity ($\varepsilon=-$), the tail is kept constant.
   • Central Class: Using the ineffectivity of $L$, the authors show that the primary obstruction class of the source maps injectively into the boundary tangent bundle, yielding a non-zero "central comparison class" $c^{\varepsilon}_{\eta}$.

2. Transverse Formal Smoothing:
   Using the theory of transverse extensions, the authors extend the map $Z^{\varepsilon}_{\eta} \to D$ formally in a direction transverse to the boundary (smoothing the node).

   • They construct a formal family $\widehat{Y}^{\varepsilon}_{\eta}$ over a formal disk, mapping to the formal completion of the ambient superstack along $D$.
   • Obstruction Vanishing: A key technical step involves proving that the obstruction groups for this extension vanish. By analyzing the action of a finite group $A \cong \mu_4$ (acting on the elliptic tail) and utilizing Serre duality on the core, they demonstrate that all invariant obstruction classes vanish for all orders of the formal expansion. This ensures the existence of a coherent formal smoothing.

3. Laurent Localisation and Pole Removal:
   To move from the boundary to the smooth locus, the boundary parameter $t$ is inverted.

   • The authors analyze the behavior of the comparison class under Laurent localisation. They prove that if the class were to vanish after inverting $t$, the resulting primitive would require a pole.
   • By averaging under the parity involution and analyzing the associated graded bundles, they show that no non-zero invariant global sections exist for negative powers of the boundary parameter. Consequently, any primitive must be regular, contradicting the non-vanishing of the central class. This establishes that the extension remains non-split on the smooth locus.

4. Algebraic Non-Vanishing:
   The authors deduce that if the primary obstruction vanished on the smooth algebraic moduli space $\mathcal{M}^{\varepsilon}_g$, a splitting would exist. This splitting would pull back to a Laurent splitting of the constructed formal extension, contradicting the previous step. Thus, $\omega^{\text{alg}}_2(\mathcal{M}^{\varepsilon}_g) \neq 0$.

5. Algebraic-to-Analytic Comparison:
   To extend the result to the holomorphic category, the authors prove a general comparison theorem (Theorem 1.2).

   • They construct normal projective compactifications of spin-level covers of the moduli space where the complement has codimension at least two (using the Satake compactification of the Jacobian locus and the Torelli morphism).
   • They show that for such varieties, the analytification map on $H^1$ is injective. This relies on extending algebraic sections as coherent reflexive sheaves across the codimension-two complement and applying GAGA.
   • This injectivity ensures that the non-vanishing of the algebraic class implies the non-vanishing of the holomorphic class.

Key Contributions and Results

• Main Theorem (Theorem 1.1): For every genus $g \ge 3$ and both spin parities $\varepsilon \in \{+, -\}$, the primary obstruction class is non-zero in both the algebraic and holomorphic categories:
  $$\omega^{\text{alg}}_2(\mathcal{M}^{\varepsilon}_g) \neq 0, \quad \omega^{\text{hol}}_2((\mathcal{M}^{\varepsilon}_g)^{\text{an}}) \neq 0.$$
  Consequently, the smooth unpunctured supermoduli spaces are neither projected nor split.
• Resolution of the Genus-3 Threshold: The paper provides the first proof of non-projectedness for genus $g=3$, settling the lowest genus where this phenomenon occurs.
• Uniform Argument: The proof utilizes a single degeneration argument (attaching an elliptic tail to a core) that applies uniformly to all $g \ge 3$ and both parities, differing only in the specific handling of the elliptic tail's automorphisms.
• Comparison Theorem: The paper establishes a sufficient geometric condition (existence of a normal projective compactification with codimension-two complement) under which algebraic extension classes remain non-trivial after analytification, even for non-proper varieties.

Significance
The paper establishes a fundamental feature of the global geometry of supermoduli spaces: despite having unobstructed local deformation theory, they possess non-trivial global extension data over the classical spin moduli stack. This obstruction prevents the existence of a global holomorphic projection onto the moduli of spin curves for $g \ge 3$.

In the context of superstring perturbation theory, where amplitudes are defined by integration over supermoduli, this result implies that the construction used for genus 2 (which relies on a global projection via the super period matrix) cannot be generalized to higher genera. While the result does not immediately invalidate specific modular-form candidates for chiral measures proposed by other authors, it rules out the existence of a global holomorphic projection as a basis for analogous constructions. The work clarifies the geometric nature of the "Green obstruction" and confirms that the split model fails uniformly throughout the stable range above genus two.