[Paper Review] Faisceaux pervers sur les variétés toriques lisses

Delphine Dupont|arXiv (Cornell University)|Mar 16, 2010

Algebraic structures and combinatorial models5 references4 citations

TL;DR

This paper establishes an equivalence between the category of perverse sheaves on a smooth toric variety $X$ stratified by the torus action and a full subcategory of representations of a quiver constructed from the fan of $X$. The construction uses local equivalences from Galligo-Granger-Maisonobe for $\mathbb{C}^n$ and $\mathbb{C}^* \times \mathbb{C}^{n-1}$, then glues them via combinatorial recollement data encoded in the fan, with compatibility conditions ensuring global perverse sheaf structure.

ABSTRACT

Let X be a smooth toric variety stratified by the torus action. This paper is a presentation of a description of the category Perv_X of perverse sheaves on X relatively to the fixed stratification. We define a category of representations of a quiver, defined thanks to the fan of X, equivalent to Perv_X.

Motivation & Objective

• To provide an explicit, combinatorial description of the category of perverse sheaves on smooth toric varieties with respect to the torus stratification.
• To extend local quiver representations of perverse sheaves on $\mathbb{C}^n$ and $\mathbb{C}^* \times \mathbb{C}^{n-1}$ to a global description on smooth toric varieties.
• To formalize the gluing of local perverse sheaf categories using the fan structure, ensuring compatibility across strata.
• To define a quiver with relations whose representation category is equivalent to the category of perverse sheaves on the toric variety.
• To establish that the global category of perverse sheaves arises as the global sections of a constructible stack of quiver representations.

Proposed method

• Construct a quiver $c_\Delta$ from the fan $\Delta$ of a smooth toric variety $X$, with vertices corresponding to cones and edges encoding codimension-one face relations.
• Assign $n - \ell$ loops to each vertex $s_I$, where $\ell$ is the maximal size of a set $I$ such that $\sigma_I$ is a maximal cone.
• Define a full subcategory $\mathcal{C}_\Delta$ of quiver representations with relations derived from local equivalences in Galligo-Granger-Maisonobe for $\mathbb{C}^n$ and $\mathbb{C}^* \times \mathbb{C}^{n-1}$.
• Introduce four conditions on representations: (i) compatibility of maps along faces, (ii) duality of maps in opposite directions, (iii) invertibility of certain compositions, and (iv) global consistency via recollement data from the fan.
• Use the fact that smooth toric varieties are locally isomorphic to $\mathbb{C}^k \times (\mathbb{C}^*)^l$ to define a constructible stack of quiver representations, then show global sections recover $\mathcal{Perv}_X$.
• Verify that the recollement conditions (iv) encode the toric gluing data, ensuring that the global category matches the perverse sheaf category.

Experimental results

Research questions

• RQ1 Can the category of perverse sheaves on a smooth toric variety be described combinatorially via quiver representations derived from the fan?
• RQ2 How can local equivalences of perverse sheaves on $\mathbb{C}^n$ and $\mathbb{C}^* \times \mathbb{C}^{n-1}$ be glued together to form a global description on a smooth toric variety?
• RQ3 What relations must quiver representations satisfy to correspond precisely to perverse sheaves under the torus stratification?
• RQ4 How do the combinatorics of the fan encode the gluing data for perverse sheaves via quiver representations?
• RQ5 Is the global category of perverse sheaves on a smooth toric variety equivalent to the category of global sections of a constructible stack of quiver representations?

Key findings

• The category $\mathcal{C}_\Delta$ of quiver representations defined from the fan $\Delta$ is equivalent to the category $\mathcal{Perv}_X$ of perverse sheaves on the smooth toric variety $X$.
• The quiver $c_\Delta$ is constructed from the fan: vertices correspond to cones, and edges encode codimension-one face relations with bidirectional arrows.
• Each vertex $s_I$ is equipped with $n - \ell$ loops, where $\ell$ is the maximal size of a set $I$ such that $\sigma_I$ is a maximal cone.
• The relations in $\mathcal{C}_\Delta$ include compatibility of maps along faces, duality of inverse maps, and invertibility of compositions like $M_{IJ} = v_{IJ} \circ u_{IJ} + \text{id}$.
• Condition (iv) ensures global consistency: for example, $M_{\emptyset 3} = M_{\emptyset 1}^{-1} M_{\emptyset 2}^{-1}$, reflecting toric recollement data.
• The global category $\mathcal{Perv}_X$ is isomorphic to the category of global sections of a constructible stack of quiver representations, constructed by gluing local models via the fan's combinatorics.