Wall And Chamber Structure For A Special Biserial Algebra Coming From Perverse Sheaves on $\mathbb{P}^n$

Technical Summary: Wall and Chamber Structure for a Special Biserial Algebra from Perverse Sheaves on $\mathbb{P}^n$

Problem Statement
The paper addresses the structure of the space of Bridgeland stability conditions for the bounded derived category of constructible sheaves, $D^b_c(\mathbb{P}^n)$, on the complex projective space $\mathbb{P}^n$ with respect to the Schubert stratification. While stability spaces for derived categories of coherent sheaves have been extensively studied, the case for constructible complexes has received less attention. The authors focus on the abelian category of middle perverse sheaves on $\mathbb{P}^n$, which serves as a faithful heart for $D^b_c(\mathbb{P}^n)$. This category is known to be equivalent to the module category of a specific finite-dimensional algebra, $B(n)$, which is a special biserial algebra. The primary goal is to explicitly describe the wall and chamber structure of $B(n)$, which, by Bridgeland's results, corresponds to the chamber structure in an open subset of the stability space for $D^b_c(\mathbb{P}^n)$.

Methodology
The authors employ a combination of representation theory of special biserial algebras and combinatorial string theory.

1. Algebraic Setup: They define the quiver with relations $(Q(n), I(n))$ whose path algebra quotient $B(n) = kQ(n)/I(n)$ is special biserial. The quiver has vertices $[0, n]$ and arrows $\alpha_i: i+1 \to i$ and $\beta_i: i \to i+1$, subject to specific monomial and binomial relations.
2. Combinatorial Classification: Utilizing the classification of indecomposable modules for special biserial algebras (string modules, band modules, and non-uniserial injective-projective modules), the authors analyze the strings and bands on $(Q(n), I(n))$. They prove that no bands exist for this specific algebra, thereby establishing that $B(n)$ is of finite representation type.
3. Thin vs. Non-Thin Modules: The authors distinguish between "thin" modules (where the dimension vector has entries $\le 1$) and non-thin modules. They demonstrate that the stability spaces (walls) associated with non-thin indecomposable modules are contained within the stability spaces of thin modules. Consequently, the wall and chamber structure is entirely determined by the thin indecomposable modules.
4. Explicit Description: They provide an explicit combinatorial parameterization of the thin indecomposable modules (corresponding to trivial paths and alternating walks) and derive the defining inequalities for their stability spaces.
5. Geometric Interpretation: Finally, they translate these algebraic results back into the language of perverse sheaves, identifying the specific perverse sheaves (constructed via iterated extensions of simple, standard, and costandard objects) that correspond to the walls of the structure.

Key Contributions and Results

• Finite Representation Type: The paper proves that the algebra $B(n)$ associated with middle perverse sheaves on $\mathbb{P}^n$ is of finite representation type. This is achieved by showing the absence of band modules (Remark 3.8) and classifying the indecomposable modules as either string modules or specific projective-injective modules arising from binomial relations.
• Reduction to Thin Modules: A central result (Proposition 5.8) establishes that the wall and chamber structure of $B(n)$ is uniquely determined by the thin indecomposable modules. Non-thin modules do not generate new walls; their stability spaces are subspaces of those generated by thin modules.
• Explicit Wall Description: The authors provide a complete, explicit description of the walls in Theorem 5.13.
  • Walls corresponding to simple modules $S_i$ are hyperplanes defined by $v_i = 0$.
  • Walls corresponding to thin string modules $M(a, b, \eta)$ are defined by a system of linear equalities and inequalities involving sums of coordinates $v_k$ over specific intervals, dependent on the parity of the indices.
• Counting Chambers: As a consequence of the wall description, the number of chambers (and thus the number of isomorphism classes of 2-silting complexes for $B(n)$) can be determined combinatorially. For instance, in the case $n=2$, there are exactly 20 chambers.
• Geometric Realization: Proposition 6.7 explicitly constructs the perverse sheaves corresponding to these walls as iterated extensions of simple perverse sheaves ($IC_r$) and standard/costandard objects ($\Delta, \nabla$).

Significance and Claims
The paper claims to provide a self-contained, explicit description of the wall and chamber structure for the algebra governing perverse sheaves on $\mathbb{P}^n$. By leveraging the combinatorial nature of special biserial algebras, the authors avoid relying on external general theorems where possible, offering direct proofs for the irrelevance of non-thin modules and the facial description of the walls.

The significance of this work lies in its contribution to understanding the Bridgeland stability space for constructible sheaves, a domain with fewer results than the coherent sheaf case. The authors note that their main theorem allows for the control of wall-crossing phenomena in the stability space of $D^b_c(\mathbb{P}^n)$. Furthermore, they highlight that the chambers in this structure are in bijection with 2-silting complexes for $B(n)$, turning the problem of counting these complexes into a combinatorial counting problem of maximal cones in a polyhedral fan. The authors explicitly state that they do not investigate the consequences for silting theory beyond this bijection, leaving the detailed properties of these complexes for future work.

The paper does not claim to solve the stability problem for the entire derived category of constructible sheaves on arbitrary varieties, but rather provides a concrete, solved case for $\mathbb{P}^n$ with Schubert stratification, serving as a model for similar stratified spaces.