Brauer group of moduli of parabolic symplectic bundles

Technical Summary: Brauer Group of Moduli of Parabolic Symplectic Bundles

Problem Statement
The paper addresses the computation of the cohomological Brauer group, denoted as $Br(Y) \cong H^2_{\text{\'et}}(Y, \mathbb{G}_m)_{\text{tor}}$, for the smooth locus of the moduli space of parabolic symplectic stable bundles. Let $X$ be a smooth connected complex projective curve of genus $g \geq 3$. The authors fix an even rank $r \geq 2$, a finite set of parabolic points $D \subset X$, and a line bundle $L$ on $X$. The objects of study are parabolic symplectic bundles $(E_*, \phi_*, L(D))$, where the symplectic form takes values in $L(D)$ (equipped with a special parabolic structure), and the parabolic data at $D$ satisfies specific symmetry conditions on multiplicities and weights. The goal is to determine the Brauer group of the smooth locus of the moduli space, denoted $(M^{m,\alpha}_{L(D)})^{\text{sm}}$, for both concentrated and arbitrary generic systems of weights.

Methodology and Setup
The authors employ a combination of geometric invariant theory (GIT), the theory of filtered sheaves, and wall-crossing techniques.

1. Parabolic Symplectic Structures: The paper utilizes the framework of parabolic vector bundles as filtered sheaves (following Maruyama and Yokogaya) to define parabolic symplectic bundles. A key setup involves a system of multiplicities $m$ and weights $\alpha$ that are "of symmetric type." This symmetry requires that for each parabolic point $p$, the multiplicities satisfy $m_{p,j} = m_{p, \ell(p)+1-j}$ and the weights satisfy $\alpha_{p,j} = 1 - \alpha_{p, \ell(p)+1-j}$. Under these conditions, a parabolic symplectic form $\phi_*$ induces a non-degenerate symplectic form $\phi$ on the underlying vector bundle $E$ taking values in $L$.

2. Reduction to Concentrated Weights: The proof strategy begins with the case where the system of weights $\alpha$ is "concentrated" (Definition 3.7), meaning the weights are sufficiently close to $1/2$ such that specific inequalities hold. In this regime, Lemma 3.8 establishes a correspondence between parabolic symplectic stability and standard symplectic stability. This allows the construction of a morphism $\pi_0$ from the parabolic moduli space $M^{m,\alpha}_{L(D)}$ to the moduli space $M_L$ of semistable symplectic bundles (twisted $Sp(r, \mathbb{C})$-bundles).

3. Fibration and Spectral Sequences: For concentrated weights, the map $\pi$ restricted to the stable locus is shown to be a fiber bundle with fibers isomorphic to a product of flag varieties $\prod Sp(r, \mathbb{C})/P_i$. The authors utilize the Leray spectral sequence for the sheaf $\mathbb{G}_m$ associated with this fibration. This yields a 5-term exact sequence relating the Picard groups of the fibers, the Brauer group of the base (the moduli of symplectic bundles), and the Brauer group of the total space.

4. Wall-Crossing for Generic Weights: To extend the result from concentrated weights to arbitrary generic weights, the authors invoke the variation of GIT (Geometric Invariant Theory) principles, specifically referencing Thaddeus's work on wall-crossing. They demonstrate that crossing a single wall in the ample cone of weights induces an isomorphism between the smooth loci of the moduli spaces up to subsets of codimension at least 2. Since the Brauer group is invariant under the removal of subsets of codimension $\geq 2$, the Brauer group remains invariant across chambers of generic weights.

Key Contributions and Results
The primary contribution is the explicit computation of the Brauer group for the smooth locus of the moduli space of parabolic symplectic bundles under symmetric weight conditions. The main result, Theorem 1.1 (detailed in Theorem 4.3 and Corollary 4.6), states:

Let $M^{m,\alpha}_{L(D)}$ be the moduli space with symmetric multiplicities $m$ and weights $\alpha$. The Brauer group of its smooth locus is:

1. If $\deg(L)$ is even:
   $$Br((M^{m,\alpha}_{L(D)})^{\text{sm}}) \simeq \mathbb{Z} / \gcd(2, m_{p_1,1}, \dots, m_{p_n, \ell(p_n)})\mathbb{Z}$$

2. If $\deg(L)$ is odd:
   $$Br((M^{m,\alpha}_{L(D)})^{\text{sm}}) \simeq \begin{cases} 0 & \text{if } r/2 \geq 3 \text{ is odd}, \\ \mathbb{Z} / \gcd(2, m_{p_1,1}, \dots, m_{p_n, \ell(p_n)})\mathbb{Z} & \text{if } r/2 \geq 3 \text{ is even}. \end{cases}$$

The authors note that the generator of the Brauer group in the non-trivial cases is related to the class of a projective Poincaré bundle restricted to a point on the curve, analogous to results for the moduli of symplectic bundles without parabolic structure (citing [BHl]).

Significance and Scope
The paper claims to generalize previous results on the Brauer group of moduli spaces of symplectic bundles (specifically the work in [BHl]) to the parabolic setting. The significance lies in:

• Establishing the relationship between parabolic symplectic stability and standard symplectic stability under symmetric weight conditions.
• Providing a complete calculation of the Brauer group for these moduli spaces, showing that it depends on the parity of the degree of the line bundle $L$, the rank $r$, and the greatest common divisor of 2 and the parabolic multiplicities.
• Demonstrating the invariance of the Brauer group under variation of weights within the ample cone, provided the weights remain generic.

The authors explicitly restrict their results to the case where the system of weights and multiplicities are of "symmetric type" and do not contain 0. They do not claim results for non-symmetric weights or for the singular locus of the moduli space. The work relies heavily on the simply connected nature of the symplectic group $Sp(r, \mathbb{C})$ and the known structure of the Brauer group for the underlying symplectic moduli space.