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Proper moduli spaces of orthosymplectic complexes

This paper constructs proper good moduli spaces for Bridgeland semistable orthosymplectic complexes on complex smooth projective varieties as candidates for compactifying moduli spaces of principal bundles for orthogonal and symplectic groups, while also establishing results on good moduli spaces for fixed point and mapping stacks from finite groupoids.

Original authors: Chenjing Bu

Published 2026-06-18

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CC BY 4.0

Original authors: Chenjing Bu

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

Imagine you are a mathematician trying to organize a massive, chaotic library. This library contains every possible "shape" or "structure" you can build using the rules of a specific type of geometry (specifically, shapes on a smooth, curved surface like a sphere or a torus).

In the world of standard shapes (like simple bundles of strings), mathematicians have already built a perfect, complete catalog. If you take a shape and stretch it until it breaks or collapses, you can always find a "limit" version of it in the catalog. This is called a compactification. It ensures your library is complete; no shape is ever "missing" just because it got a little weird or broken.

However, when it comes to more complex structures called principal bundles (which are like bundles of strings that also have a specific internal symmetry, like being perfectly symmetrical under rotation or flipping), the library was incomplete. If these complex shapes broke or got singular (developed sharp points or tears), there was no clear place to put them in the catalog. The catalog had holes.

The Problem: The Missing Pieces

For simple curves (like a circle), this wasn't a problem. But for surfaces (like a sphere) and higher dimensions, the existing ways of filling the holes in the catalog were clunky. They relied on choosing specific "representations" (like forcing the shapes to look a certain way), which felt artificial.

The Solution: "Orthosymplectic Complexes"

The author, Chenjing Bu, proposes a new, more natural way to fill these holes. Instead of looking at the shapes directly, the author looks at them through a special lens called orthosymplectic complexes.

Think of these complexes as mirrors.

In this mathematical world, every object has a "dual" version (like a reflection in a mirror).
An orthosymplectic complex is a special object that is its own reflection. It is "self-dual."
If you have an orthogonal shape (like a sphere), its mirror image is itself. If you have a symplectic shape (like a specific type of fluid flow), its mirror image is also itself.

The paper argues that by organizing the library based on these "self-reflecting" objects, we can naturally include the broken or singular versions without forcing them into an unnatural mold.

The Main Achievement: Building the "Good" Catalog

The core of the paper is a construction project. The author proves that:

The Library is Complete: You can build a "moduli space" (a master catalog) for these self-reflecting shapes.
It's "Proper": This is a technical term meaning the catalog is closed and complete. If you have a sequence of shapes getting closer and closer to a limit (even a broken one), that limit is guaranteed to be inside the catalog. You never fall off the edge.
It's "Good": The catalog behaves well mathematically. It doesn't have weird, messy overlaps that make it impossible to count or study the shapes.

How Did They Do It? (The Magic Trick)

The author didn't build this catalog from scratch. They used a clever mathematical shortcut involving symmetry.

Imagine you have a huge, messy room of objects (the moduli stack of all shapes).
You have a tiny group of "tweezers" (a finite group, specifically a group of two elements, like a switch that flips things on and off) that can rearrange these objects.
The "fixed points" are the objects that don't move when the tweezers flip them. These are exactly the self-reflecting (orthosymplectic) objects.

The paper proves a general rule: If you have a complete, well-behaved catalog for the whole messy room, then the "fixed points" (the self-reflecting objects) automatically have their own complete, well-behaved catalog.

Since mathematicians already knew how to build the catalog for the whole room (thanks to previous work by Alper, Halpern-Leistner, and Heinloth), the author simply applied this "fixed point" rule to show that the catalog for the self-reflecting shapes exists and is perfect.

Why Does This Matter?

The author suggests this new catalog is a better candidate for studying Orthogonal and Symplectic groups (which are fundamental in physics and geometry) than previous methods.

Furthermore, the paper hints at a beautiful geometric property:

If the surface you are studying is a K3 surface (a very special, complex type of shape), this new catalog doesn't just have a structure; it has a symplectic structure.
Imagine the catalog isn't just a static list, but a dynamic landscape where you can measure "areas" and "flows" between the shapes. This opens the door to studying these shapes using the powerful tools of symplectic geometry, potentially revealing deep connections between different types of mathematical objects.

Summary

In short, Chenjing Bu has found a new, elegant way to organize a chaotic mathematical library. By focusing on objects that are their own mirrors, they proved that you can create a complete, gap-free catalog for these complex shapes. They did this by showing that if the "whole library" is well-organized, then the "special self-reflecting section" is automatically well-organized too. This provides a solid foundation for future studies of these shapes, particularly in contexts where symmetry and geometry play a crucial role.

Technical Summary: Proper Moduli Spaces of Orthosymplectic Complexes

Problem Statement
The paper addresses the problem of compactifying moduli spaces of principal $G$ -bundles on a complex smooth projective variety $X$ when $\dim(X) \geq 2$ . While moduli spaces of semistable principal bundles are compact for curves (Ramanathan), higher-dimensional cases require a notion of "singular principal bundles" to achieve compactness. Existing approaches utilize principal $\rho$ -sheaves (depending on a representation $\rho$ of $G$ ). The author proposes an alternative compactification specifically for the orthogonal ( $O_n$ ) and symplectic ( $Sp_{2n}$ ) groups, utilizing orthosymplectic complexes. These are complexes of coherent sheaves isomorphic to their own derived duals. The central challenge is to establish the existence of proper good moduli spaces for the moduli stacks of Bridgeland semistable orthosymplectic complexes.

Methodology
The paper employs a two-pronged approach combining abstract stack theory with specific applications to derived categories:

General Theory of Fixed Loci and Mapping Stacks:
The author establishes a general result regarding the preservation of "good moduli spaces" under finite group actions and mapping stacks.
- Fixed Loci: For an algebraic stack $\mathcal{X}$ with a good moduli space $\mathcal{X} \to X$ , the paper investigates the fixed locus $\mathcal{X}^\Gamma$ under a finite group $\Gamma$ .
- Mapping Stacks: The paper analyzes the mapping stack $\text{Map}(B\Gamma, \mathcal{X})$ , showing it is equivalent to the fixed locus when the action is trivial.
- Valuative Criterion: The core of the proof relies on verifying the valuative criterion for universal closedness. The author adapts results from Alper, Halpern-Leistner, and Heinloth [2] to show that if $\mathcal{X}$ has a good moduli space, then $\text{Map}(B\Gamma, \mathcal{X})$ (and consequently $\mathcal{X}^\Gamma$ under separation assumptions) also admits a good moduli space that is finite over the original base. This involves lifting group homomorphisms $\rho: \Gamma \to \text{Aut}(x_K)$ from the generic fiber to an extension of discrete valuation rings.
Application to Orthosymplectic Complexes:
- Definition: An orthosymplectic complex is defined as a fixed point of a $\mathbb{Z}/2$ -action on the moduli stack of perfect complexes $\text{Perf}(X)$ . This action is induced by a self-dual structure $D = R\text{Hom}(I^*(-), L)[s]$ on the dg-category, where $I$ is an involution, $L$ a line bundle, and $s$ an integer.
- Stability: The author considers Bridgeland stability conditions $\tau = (Z, \mathcal{P})$ that are self-dual (i.e., $(Z, \mathcal{P}) = (Z^\vee, \mathcal{P}^\vee)$ ).
- Construction: The moduli stack of semistable orthosymplectic complexes is identified as the fixed locus of the moduli stack of semistable complexes under the $\mathbb{Z}/2$ -action.

Key Contributions and Results

Theorem 3.1 (General Result): Let $\mathcal{X}$ be an algebraic stack locally of finite type with affine diagonal over a noetherian algebraic space $S$ of characteristic zero, admitting a good moduli space $\mathcal{X} \to X$ .
- For any finite group $\Gamma$ , the mapping stack $\text{Map}(B\Gamma, \mathcal{X})$ admits a good moduli space finite over $X$ .
- If $X$ is separated, the fixed locus $\mathcal{X}^\Gamma$ also admits a good moduli space finite over $X$ .
- Note: The assumption that $X$ is separated is necessary for the fixed locus result; counterexamples exist if $X$ is not separated (e.g., the line with two origins).
Theorem 4.4 (Main Application): Let $X$ be a connected smooth projective $\mathbb{C}$ -variety with fixed self-dual data $(I, L, s, \epsilon)$ . Let $\tau$ be a self-dual Bridgeland stability condition with rational central charge satisfying generic flatness and boundedness.
- For any Chern character $\alpha$ such that $\alpha = D(\alpha)$ and $Z(\alpha) \in \mathbb{R}_{>0}$ , the moduli stack of $\tau$ -semistable self-dual complexes with Chern character $\alpha$ admits a proper good moduli space.
- Consequently, every connected component of the moduli stack of semistable orthosymplectic complexes has a proper good moduli space.

Significance and Claims
The paper claims that these proper good moduli spaces serve as a candidate for compactifying moduli spaces of principal bundles for $O_n$ and $Sp_{2n}$ . The author notes that in the context of motivic Donaldson–Thomas theory, orthosymplectic complexes appear more convenient than principal $\rho$ -sheaves.

The paper highlights specific geometric properties in the case where $X$ is a K3 or abelian surface:

The moduli stack admits a derived enhancement with a 0-shifted symplectic structure.
The resulting proper good moduli space admits a Poisson structure, which restricts to a symplectic structure on the stable locus.
The author suggests these spaces are locally modeled on symplectic reductions of moduli of representations of self-dual quivers, though the paper does not resolve whether they possess symplectic singularities or admit symplectic resolutions (questions left open for future analysis).

The work relies on the existence of proper good moduli spaces for the underlying moduli stack of semistable complexes (established in [2]) and extends this existence to the fixed loci via the general machinery developed in Section 3.