Less Perverse Filtration in Algebraic Geometry

• Less perverse filtration is a framework where opaque perverse Leray filtrations are reinterpreted as explicit, geometrically transparent weight filtrations.
• The Hilbert scheme example on T∨E and ℂ*×ℂ* demonstrates a concrete exchange between perverse filtrations and Hodge–theoretic weight filtrations.
• This concept extends to applications in flag theory, rigidity phenomena, and algebraic models including Chern, Fourier, and shuffle algebra settings.

“Less perverse filtration” denotes a recurrent research viewpoint in which a perverse Leray filtration, usually defined through perverse truncation in the constructible derived category, becomes explicitly calculable, geometrically transparent, or identifiable with a more classical filtration. The paradigmatic example is the exchange between the perverse Leray filtration on the cohomology of a Hilbert scheme of points on $X=T^\vee E\simeq E\times\mathbb{C}$ and the weight filtration on the corresponding Hilbert scheme on $Y=\mathbb{C}^*\times\mathbb{C}^*$, where “perversity becomes a genuine Hodge-theoretic weight” (Cataldo et al., 2010). Subsequent work broadens this viewpoint to flag descriptions, deformation-rigidity statements, Chern and Fourier-theoretic realizations, and algebraic models in CoHA and shuffle-algebra settings.

1. Prototype: Hilbert schemes on $T^\vee E$ and $\mathbb{C}^*\times\mathbb{C}^*$

The basic model fixes an integer $n\ge 0$ and considers two smooth complex surfaces. One is an elliptic curve $E$ together with its cotangent bundle

$X:=T^\vee E\simeq E\times \mathbb{C}.$

The other is

$Y:=\mathbb{C}^*\times \mathbb{C}^*,$

which is isomorphic to $\mathbb{C}^2$ minus the union of the two coordinate axes, and also minus the origin. Both are smooth, non-compact, holomorphic symplectic surfaces. They are noncanonically diffeomorphic, and a choice of diffeomorphism

$\phi:Y\xrightarrow{\sim}X$

induces an isomorphism of graded vector spaces on cohomology and, functorially, an isomorphism on Hilbert schemes of points

$Y=\mathbb{C}^*\times\mathbb{C}^*$0

Here $Y=\mathbb{C}^*\times\mathbb{C}^*$1 denotes the Hilbert scheme of $Y=\mathbb{C}^*\times\mathbb{C}^*$2 points on a smooth surface $Y=\mathbb{C}^*\times\mathbb{C}^*$3, a smooth variety of dimension $Y=\mathbb{C}^*\times\mathbb{C}^*$4, and

$Y=\mathbb{C}^*\times\mathbb{C}^*$5

is the Hilbert–Chow morphism (Cataldo et al., 2010).

The geometry on the $Y=\mathbb{C}^*\times\mathbb{C}^*$6-side is organized by the projection

$Y=\mathbb{C}^*\times\mathbb{C}^*$7

which induces a proper flat morphism

$Y=\mathbb{C}^*\times\mathbb{C}^*$8

This map has relative dimension $Y=\mathbb{C}^*\times\mathbb{C}^*$9, $T^\vee E$0, $T^\vee E$1, and general fiber isomorphic to a product of copies of $T^\vee E$2. It is therefore “Hitchin-like”: a holomorphic symplectic variety of dimension $T^\vee E$3 mapping to an affine space of dimension $T^\vee E$4 with Lagrangian fibers (Cataldo et al., 2010).

This concrete pair $T^\vee E$5 provides the canonical first meaning of a less perverse filtration. The two spaces are topologically identified, but their algebraic structures are very different: $T^\vee E$6 carries a nontrivial perverse Leray filtration from $T^\vee E$7, whereas $T^\vee E$8 carries a nontrivial mixed Hodge structure with weight filtration. The central phenomenon is that these two filtrations match under $T^\vee E$9 (Cataldo et al., 2010).

2. Explicit perversity: decomposition, partitions, and geometric description

For a proper map $\mathbb{C}^*\times\mathbb{C}^*$0, the perverse Leray filtration on $\mathbb{C}^*\times\mathbb{C}^*$1 is defined from the perverse truncations of $\mathbb{C}^*\times\mathbb{C}^*$2. In the Hilbert-scheme model, the relevant complex is

$\mathbb{C}^*\times\mathbb{C}^*$3

and the filtration is

$\mathbb{C}^*\times\mathbb{C}^*$4

The decisive simplification comes from the decomposition theorem for $\mathbb{C}^*\times\mathbb{C}^*$5. One obtains an explicit decomposition indexed by partitions $\mathbb{C}^*\times\mathbb{C}^*$6 of $\mathbb{C}^*\times\mathbb{C}^*$7: $\mathbb{C}^*\times\mathbb{C}^*$8 where $\mathbb{C}^*\times\mathbb{C}^*$9 is the length of $n\ge 0$0, and each $n\ge 0$1 is a perverse sheaf on $n\ge 0$2, in fact an intersection cohomology complex supported on the stratum closure $n\ge 0$3 (Cataldo et al., 2010).

Taking hypercohomology yields a canonical splitting of the perverse Leray filtration. In degree $n\ge 0$4,

$n\ge 0$5

and one can say that a class has perversity $n\ge 0$6 precisely when it lies in the summands with

$n\ge 0$7

Moreover, there are canonical identifications

$n\ge 0$8

with $n\ge 0$9 a product of symmetric powers of $E$0. In this model the perverse filtration is therefore completely explicit, canonically split, and indexed by combinatorics of partitions of $E$1 (Cataldo et al., 2010).

This explicitness is closely related to a more general geometric reinterpretation of perverse filtrations. For affine bases, the perverse filtration can be expressed as a flag filtration by kernels of restriction maps to general linear sections. In the affine case,

$E$2

and similarly for perverse Leray filtrations after pulling back a general flag from the base (Cataldo, 2010). In the Hilbert-scheme example, the paper gives the corresponding geometric formula

$E$3

so the filtration can also be read by restriction to preimages of general linear sections (Cataldo et al., 2010). This is a central sense in which the filtration is “less perverse”: the categorical definition is replaced by a concrete restriction-theoretic criterion.

3. Weight filtration, Hodge–Tate structure, and the exchange theorem

On the $E$4-side, the cohomology of $E$5 is mixed rather than pure. For every partition $E$6 of $E$7, the mixed Hodge structure on $E$8 is split Hodge–Tate; indeed,

$E$9

so

$X:=T^\vee E\simeq E\times \mathbb{C}.$0

Using the Hilbert–Chow decomposition, one obtains a decomposition

$X:=T^\vee E\simeq E\times \mathbb{C}.$1

compatible with mixed Hodge structures, and the cohomology of $X:=T^\vee E\simeq E\times \mathbb{C}.$2 is itself split Hodge–Tate (Cataldo et al., 2010).

Because all weights are even, the paper introduces the halved weight filtration

$X:=T^\vee E\simeq E\times \mathbb{C}.$3

It satisfies

$X:=T^\vee E\simeq E\times \mathbb{C}.$4

with the same indexing condition that appears in the perverse formula on the $X:=T^\vee E\simeq E\times \mathbb{C}.$5-side (Cataldo et al., 2010).

The main theorem is then an identification of filtered structures: $X:=T^\vee E\simeq E\times \mathbb{C}.$6 for all $X:=T^\vee E\simeq E\times \mathbb{C}.$7. Equivalently,

$X:=T^\vee E\simeq E\times \mathbb{C}.$8

is an isomorphism of pure Hodge structures (Cataldo et al., 2010). Informally, a class on $X:=T^\vee E\simeq E\times \mathbb{C}.$9 of perversity $Y:=\mathbb{C}^*\times \mathbb{C}^*,$0 is sent to a class on $Y:=\mathbb{C}^*\times \mathbb{C}^*,$1 of Hodge type $Y:=\mathbb{C}^*\times \mathbb{C}^*,$2.

This exchange theorem supplies the foundational example of the phrase. It does not define a new universal filtration; rather, it exhibits a situation in which a perverse Leray filtration can be re-read as a Hodge-theoretic weight filtration on a different space. This suggests that “less perverse” refers to a structural simplification: perverse degree is no longer an opaque categorical index, but a genuine weight index (Cataldo et al., 2010).

4. Lefschetz phenomena, rigidity, and the $Y:=\mathbb{C}^*\times \mathbb{C}^*,$3 paradigm

The same model also exchanges Lefschetz-type structures. On $Y:=\mathbb{C}^*\times \mathbb{C}^*,$4, the class

$Y:=\mathbb{C}^*\times \mathbb{C}^*,$5

induces a class $Y:=\mathbb{C}^*\times \mathbb{C}^*,$6 on $Y:=\mathbb{C}^*\times \mathbb{C}^*,$7, and cup product with powers of $Y:=\mathbb{C}^*\times \mathbb{C}^*,$8 satisfies a “curious Hard Lefschetz” property with respect to the halved weight filtration: $Y:=\mathbb{C}^*\times \mathbb{C}^*,$9 Transporting $\mathbb{C}^2$0 to $\mathbb{C}^2$1 gives $\mathbb{C}^2$2, and relative Hard Lefschetz for

$\mathbb{C}^2$3

takes the form

$\mathbb{C}^2$4

The theorem identifies these two statements: under $\mathbb{C}^2$5, curious Hard Lefschetz on $\mathbb{C}^2$6 is equivalent to relative Hard Lefschetz on $\mathbb{C}^2$7 (Cataldo et al., 2010).

The paper explicitly presents this example as an analogue of the $\mathbb{C}^2$8 theorem for character varieties. In the rank $\mathbb{C}^2$9, odd degree Higgs-bundle case, the same authors had shown that under the non-Abelian Hodge diffeomorphism between $\phi:Y\xrightarrow{\sim}X$0 and the twisted character variety $\phi:Y\xrightarrow{\sim}X$1, the halved weight filtration on $\phi:Y\xrightarrow{\sim}X$2 coincides with the perverse Leray filtration for the Hitchin map on $\phi:Y\xrightarrow{\sim}X$3 (Cataldo et al., 2010). The Hilbert-scheme example is therefore a simplified model of the same exchange.

Later work sharpened the “less perverse” motif by proving rigidity properties for Hitchin systems. For the Hitchin morphism in a smooth projective family, the perverse Leray filtration on intersection cohomology is locally constant in the base; equivalently, the filtration on $\phi:Y\xrightarrow{\sim}X$4 is independent of $\phi:Y\xrightarrow{\sim}X$5 (Cataldo et al., 2018). The paper formulates this as behavior that is “less perversely” variable than could have been feared: although the Hitchin fibration is highly singular, the perverse filtration does not jump in families (Cataldo et al., 2018). A further specialization framework shows that specialization morphisms can be made filtered for the perverse and perverse Leray filtrations, with applications to compactified Dolbeault moduli and Hitchin morphisms (Cataldo, 2021).

5. Later realizations: flags, Chern filtrations, Fourier transforms, and algebraic models

A second major meaning of “less perverse” is geometric replacement of perverse truncation by elementary restriction theory. For a quasi-projective base $\phi:Y\xrightarrow{\sim}X$6, after passing to a Jouanolou affine bundle and a general linear flag, the perverse filtration on $\phi:Y\xrightarrow{\sim}X$7 and the perverse Leray filtration on $\phi:Y\xrightarrow{\sim}X$8 can be identified with filtrations by kernels and images of restriction maps along the flag. In affine form,

$\phi:Y\xrightarrow{\sim}X$9

and similarly for perverse Leray filtrations on $Y=\mathbb{C}^*\times\mathbb{C}^*$00 (Cataldo, 2010). This makes the filtration “less perverse” in a literal sense: it is measured by when classes vanish upon restriction to general hyperplane sections.

A third meaning is Fourier-theoretic and tautological. For a dualizable abelian fibration $Y=\mathbb{C}^*\times\mathbb{C}^*$01, the interaction between Fourier–Mukai transforms and the perverse filtration yields multiplicativity of the perverse filtration and the “Perverse $Y=\mathbb{C}^*\times\mathbb{C}^*$02 Chern” phenomenon; this class includes families of compactified Jacobians of integral locally planar curves (Maulik et al., 2023). In that framework the perverse filtration is motivic, Fourier-compatible, and bounded above by a filtration generated by Chern or Fourier components, which is precisely the type of simplification suggested by the phrase.

A fourth realization arises in refined BPS theory for local $Y=\mathbb{C}^*\times\mathbb{C}^*$03. For the moduli $Y=\mathbb{C}^*\times\mathbb{C}^*$04 of $Y=\mathbb{C}^*\times\mathbb{C}^*$05-dimensional stable sheaves on $Y=\mathbb{C}^*\times\mathbb{C}^*$06, the paper formulates the $Y=\mathbb{C}^*\times\mathbb{C}^*$07 conjecture on the free part of cohomology: $Y=\mathbb{C}^*\times\mathbb{C}^*$08 where $Y=\mathbb{C}^*\times\mathbb{C}^*$09 is the Chern filtration generated by tautological classes (Kononov et al., 2022). The conjecture is proved for degrees $Y=\mathbb{C}^*\times\mathbb{C}^*$10, and later work shows asymptotically that the perverse filtration matches the Chern filtration in low degrees for ample classes on del Pezzo surfaces (Pi et al., 2024). In that asymptotic regime, the filtration is “as simple as a tautological filtration defined by Chern classes” (Pi et al., 2024).

A fifth realization is algebraic. In the preprojective CoHA, a “less perverse filtration” is introduced on the Borel–Moore homology of the stack of $Y=\mathbb{C}^*\times\mathbb{C}^*$11-representations, built from the decomposition theorem for the affinization morphism, and the zeroth piece of this filtration is isomorphic to the universal enveloping algebra of an associated BPS Lie algebra $Y=\mathbb{C}^*\times\mathbb{C}^*$12 (Davison, 2020). For quivers with zero potential, the perverse filtration on the CoHA is later identified explicitly with degree bounds on shuffle-algebra polynomials, so that the BPS Lie algebra becomes the degree-$Y=\mathbb{C}^*\times\mathbb{C}^*$13 piece of a concrete combinatorial filtration (Jindal et al., 31 Mar 2026). This replaces an abstract perverse truncation by explicit limit conditions on symmetric functions.

6. Scope, multiplicativity, and conjectural general picture

The Hilbert-scheme prototype already proposed a broader conjectural framework. The paper isolates a class $Y=\mathbb{C}^*\times\mathbb{C}^*$14 of smooth quasi-projective holomorphic symplectic varieties $Y=\mathbb{C}^*\times\mathbb{C}^*$15 of dimension $Y=\mathbb{C}^*\times\mathbb{C}^*$16 with a $Y=\mathbb{C}^*\times\mathbb{C}^*$17-action scaling the symplectic form, finitely generated coordinate ring, and proper affine reduction

$Y=\mathbb{C}^*\times\mathbb{C}^*$18

with fibers of dimension $Y=\mathbb{C}^*\times\mathbb{C}^*$19. It then speculates that there should exist a dual $Y=\mathbb{C}^*\times\mathbb{C}^*$20 with split Hodge–Tate cohomology and a natural isomorphism

$Y=\mathbb{C}^*\times\mathbb{C}^*$21

under which the perverse filtration for $Y=\mathbb{C}^*\times\mathbb{C}^*$22 corresponds to the halved weight filtration on $Y=\mathbb{C}^*\times\mathbb{C}^*$23 (Cataldo et al., 2010). The two examples explicitly cited are $Y=\mathbb{C}^*\times\mathbb{C}^*$24 and $Y=\mathbb{C}^*\times\mathbb{C}^*$25. The authors also emphasize that they have no general mechanism to produce such a $Y=\mathbb{C}^*\times\mathbb{C}^*$26 or to prove the principle beyond these specific cases (Cataldo et al., 2010).

Later results show that multiplicativity is an important but nonautomatic feature of being “less perverse.” For generalized Kummer varieties of fibered surfaces, the perverse filtration associated with

$Y=\mathbb{C}^*\times\mathbb{C}^*$27

is multiplicative and admits a natural strongly multiplicative splitting (Zhang, 2022). For compactified Jacobians $Y=\mathbb{C}^*\times\mathbb{C}^*$28 of integral planar curves, the perverse filtration is opposite to the Lefschetz filtration defined by a theta class, as conjectured by Maulik–Yun (Yuan, 7 Mar 2026). By contrast, for a proper holomorphic fibration $Y=\mathbb{C}^*\times\mathbb{C}^*$29 from a Kähler surface to a curve, the local perverse filtration on the fibers of

$Y=\mathbb{C}^*\times\mathbb{C}^*$30

is multiplicative on collision fibers if and only if $Y=\mathbb{C}^*\times\mathbb{C}^*$31 is an elliptic fibration (Zhang, 28 Apr 2025). This indicates that “less perverse” behavior is structural rather than automatic.

Taken together, these works suggest a precise informal meaning. A less perverse filtration is not a separate canonical object, but a regime in which a perverse filtration becomes rigid, multiplicative, explicitly split, or identifiable with a weight, Chern, Lefschetz, or shuffle-theoretic filtration. The model exchange on Hilbert schemes of $Y=\mathbb{C}^*\times\mathbb{C}^*$32 and $Y=\mathbb{C}^*\times\mathbb{C}^*$33 remains the clearest instance: under a topological identification, the complexity of perversity is re-encoded as pure Hodge-theoretic weight (Cataldo et al., 2010).