Perverse Coherent Sheaves: Functorial Insights

• Perverse coherent sheaves are objects in the derived category defined by a t-structure governed by a perversity function, linking cohomological constraints with algebraic structure.
• They are characterized globally by requiring that the derived local cohomology with respect to p‐measuring subvarieties is concentrated in degree 0, offering a functorial and testable criterion.
• This framework bridges coherent sheaf theory with microlocal analysis and symplectic geometry, enhancing connections to Cohen–Macaulay conditions and geometric applications.

Perverse coherent sheaves are a class of objects in the derived category of coherent sheaves on a scheme (or more generally, an algebraic variety or stack) that generalize the notion of perverse (constructible) sheaves to the algebro-geometric, coherent context. Fundamentally, they are defined via t-structures determined by a perversity function, and the heart of such a t-structure is the abelian category of perverse coherent sheaves. The central result of (Koppensteiner, 2013) is a new, functorial characterization of perverse coherent sheaves via local cohomology with respect to special subvarieties—"measuring subvarieties"—analogs of Lagrangian submanifolds in symplectic geometry. This provides a direct and global (rather than pointwise or microlocal) criterion for perversity, tightly linking the geometry of coherent sheaves to methods from symplectic and microlocal analysis.

1. Classical and Functorial Definitions of Perverse Coherent Sheaves

Let $X$ be a Noetherian scheme, possibly with an action by an algebraic group $G$ with finitely many orbits. The standard approach defines perverse coherent sheaves as the heart of a t-structure constructed by means of a perversity function $p: \{0,\ldots,\dim X\} \to \mathbb{Z}$, extended to points $x \in X$ by $p(x) = p(\dim \overline{\{x\}})$.

Explicitly, on the $G$-equivariant bounded derived category $D^b_c(X)^G$, the t-structure is specified as: $\begin{aligned} \perv[p] D^{\leq 0}(X)^G &= \{ F \mid \mathbf{i}_x^* F \in D^{\leq p(x)}(\mathcal{O}_x) \ \forall x \} \ \perv[p] D^{\geq 0}(X)^G &= \{ F \mid \mathbf{i}_x^! F \in D^{\geq p(x)}(\mathcal{O}_x) \ \forall x \} \end{aligned}$ where $\mathbf{i}_x^*$, $\mathbf{i}_x^!$ are the natural (derived) stalk and costalk functors at $x$. The heart is the abelian category of perverse coherent sheaves.

The alternative characterization in (Koppensteiner, 2013) replaces these local conditions by a functorial, global criterion:

Main theorem:

Suppose $X$ has enough measuring subvarieties (see below), and $p$ is strictly monotone and comonotone. Then: $\perv[p] D^{\leq 0}(X)^G = \{ F \mid R\Gamma_Z(F) \in D^{\leq 0}(X) \text{ for all } p\text{-measuring } Z \}$ and similarly for the $D^{\geq 0}$ part. Thus,

A coherent sheaf $F$ is perverse if and only if $R\Gamma_Z(F)$ is concentrated in degree 0 for every $p$-measuring subvariety $Z$.

Here, $R\Gamma_Z(F)$ denotes the derived local cohomology with support in $Z$, viewed as a functorial test.

2. Measuring Subvarieties: Geometric Analogs of Lagrangians

A measuring subvariety (for a given perversity $p$) is a closed subvariety $Z \subset X$ such that, for every point $x$ whose orbit closure intersects $Z$,

• $\dim(\overline{x} \cap Z) = p(x) + \dim x$,
• $\overline{x} \cap Z$ is a set-theoretic local complete intersection in $\overline{x}$, locally defined by exactly $-p(x)$ functions.

A collection of such subvarieties is a measuring family if every orbit closure is intersected. The existence of enough measuring subvarieties requires mild geometric input, and is always achievable under suitable conditions on $p$.

The key insight is that, in the symplectic setting, measuring subvarieties generalize the notion of Lagrangian submanifolds: they serve as test objects probing the microlocal perversity of a sheaf, analogous to how Lagrangians capture microlocal features of constructible sheaves.

3. Comparison with Classical Constructible Perverse Sheaves

Classically, perverse constructible sheaves are defined by cohomological inequalities on stalks and costalks at strata, and can also be characterized "microlocally" by vanishing cycles at Morse critical points along Lagrangian submanifolds. The work in (Koppensteiner, 2013) shows a parallel holds for the constructible setting: for perverse constructible sheaves, requiring that the costalk $i_Z^!F$ be concentrated in degree 0 for all Lagrangian measuring submanifolds $Z$ also suffices to characterize perversity.

The main advancement for the coherent theory is that:

• The test via stalks/costalks is replaced by a global, functorial test using local cohomology with respect to measuring subvarieties,
• This provides a "globalization" and "microlocalization" of the perverse criterion—rather than checking at all points, one checks for all test subvarieties.

This reinterpretation enables a direct analogy with microlocal characterizations in the topological theory.

4. Relations to Symplectic Geometry and Cohen-Macaulay Sheaves

The analogy with symplectic geometry is more than formal: in the presence of a symplectic or Poisson structure, the measuring subvarieties can be interpreted as analogs of Lagrangians, and perverse coherent sheaves as the objects whose local cohomology relative to these geometric test loci satisfies a purity property.

For the standard dual perversity $p(n) = -n$, measuring subvarieties correspond exactly to those used to characterize Cohen-Macaulay sheaves, and the local cohomology test recovers the depth conditions for Cohen-Macaulayness—confirming the naturality and generality of the framework.

5. Summary Table: Classical and Functorial Criteria

Setting	Classical Definition	Alternative (Functorial) Definition
Perverse constructible sheaf	Stalk/costalk bounds at strata; microlocal: vanishing cycles at Lagrangians	$i_Z^!F$ in degree 0 for all measuring Lagrangians $Z$
Perverse coherent sheaf	Stalk/costalk at generic points (orbit closures)	$R\Gamma_Z(F)$ in degree 0 for all measuring subvarieties $Z$

This table encapsulates the new paradigm: perverse conditions are encoded via local (co)homological constraints with respect to geometrically meaningful subvarieties.

6. Significance and Applications

The functorial criterion

$$F \text{ perverse } \iff R\Gamma_Z(F) \text{ in degree 0 for all measuring } Z$$

enables global and geometric methods for analyzing perverse coherent sheaves, enhances compatibility with stratifications arising in symplectic and representation-theoretic contexts, and establishes a direct link to the microlocal viewpoint in perverse sheaf theory. The approach is both more general and more flexible in situations where local pointwise analysis is unwieldy or insufficient, and builds a bridge between coherent and constructible, as well as between commutative and symplectic algebraic geometry.

The recognition that measuring subvarieties can play the role of Lagrangians opens further directions for microlocal analysis in the algebro-geometric category, as well as for comparisons between different flavors of t-structures and their abelian hearts.

7. References to Key Formulas

Explicitly, the main alternative criterion is: $$F \text{ is perverse } \iff \forall Z\ \text{(measuring)},\ R\Gamma_Z(F) \text{ concentrated in degree } 0.$$ And in the constructible setting (Appendix to (Koppensteiner, 2013)): $$F \text{ perverse } \iff i_Z^!F \text{ in degree } 0\ \text{for measuring Lagrangians } Z.$$

These formulas succinctly capture the central test for perversity in both contexts, abstracting the t-structure in terms of functorial local (co)homology.