Hessenberg Varieties: Geometry & Cohomology

• Hessenberg varieties are subvarieties of flag varieties defined by linear incidence conditions governed by a Hessenberg function and operator.
• They connect algebraic geometry, topology, representation theory, and combinatorics through properties like affine pavings and explicit cohomology ring presentations.
• Special instances such as Springer fibers, Peterson varieties, and permutohedral varieties illustrate diverse geometric behaviors and combinatorial structures.

Hessenberg varieties are subvarieties of flag varieties defined by linear incidence conditions controlled by an operator and a Hessenberg datum. In type $A_{n-1}$, a Hessenberg function is a map $h:[n]\to[n]$ satisfying $h(1)\le \cdots \le h(n)$ and $h(j)\ge j$, and the associated variety is

$$\operatorname{Hess}(X,h)=\{V_\bullet\in \operatorname{Fl}(\mathbb C^n)\mid XV_j\subseteq V_{h(j)}\text{ for all }j\in[n]\}.$$

Equivalently, in Lie-theoretic form, if $G$ is a complex reductive group, $B\subseteq G$ a Borel subgroup, and $H\subseteq\mathfrak g$ a Hessenberg subspace, then

$$\operatorname{Hess}(x,H)=\{gB\in G/B\mid \operatorname{Ad}(g^{-1})x\in H\}.$$

Introduced by De Mari–Procesi–Shayman, Hessenberg varieties unify and interpolate among the full flag variety, Springer fibers, Peterson varieties, and permutohedral varieties, while connecting algebraic geometry, topology, representation theory, hyperplane arrangements, and algebraic combinatorics (Abe et al., 2019).

1. Definitions and basic examples

A Hessenberg space with respect to $\mathfrak b$ is a subspace $h:[n]\to[n]$0 such that $h:[n]\to[n]$1 and $h:[n]\to[n]$2. In root-space form one may write

$h:[n]\to[n]$3

with $h:[n]\to[n]$4. In type $h:[n]\to[n]$5, a Hessenberg function $h:[n]\to[n]$6 determines the matrix subspace

$h:[n]\to[n]$7

and under $h:[n]\to[n]$8 this is exactly the flag condition $h:[n]\to[n]$9 (Abe et al., 2015).

The construction is monotone in the Hessenberg datum: if $h(1)\le \cdots \le h(n)$0, meaning $h(1)\le \cdots \le h(n)$1 for all $h(1)\le \cdots \le h(n)$2, then

$h(1)\le \cdots \le h(n)$3

It is also invariant under conjugation of the operator: $h(1)\le \cdots \le h(n)$4 Accordingly, one typically places $h(1)\le \cdots \le h(n)$5 in Jordan canonical form or, in Lie-theoretic settings, in a standard position adapted to a chosen Borel (Abe et al., 2019).

Several classical spaces occur as extremal or special cases. If $h(1)\le \cdots \le h(n)$6 or $h(1)\le \cdots \le h(n)$7, then $h(1)\le \cdots \le h(n)$8. If $h(1)\le \cdots \le h(n)$9 is nilpotent and $h(j)\ge j$0, then $h(j)\ge j$1 is a Springer fiber. If $h(j)\ge j$2 is regular nilpotent and $h(j)\ge j$3, then $h(j)\ge j$4 is the Peterson variety. If $h(j)\ge j$5 is regular semisimple and the same Hessenberg function is used, then $h(j)\ge j$6 is the permutohedral variety, a smooth toric variety associated with the fan of Weyl chambers of type $h(j)\ge j$7 (Abe et al., 2019).

2. Global geometric properties

A foundational theorem due to Tymoczko states that every Hessenberg variety admits an affine paving. As recorded in the survey literature, this implies that integral cohomology is torsion-free and odd cohomology vanishes. In particular, the Poincaré polynomial may be written as

$h(j)\ge j$8

where $h(j)\ge j$9 (Abe et al., 2019).

Connectedness depends sharply on the Jordan type of the operator. For a noncentral semisimple element $$\operatorname{Hess}(X,h)=\{V_\bullet\in \operatorname{Fl}(\mathbb C^n)\mid XV_j\subseteq V_{h(j)}\text{ for all }j\in[n]\}.$$0, Precup proved that

$$\operatorname{Hess}(X,h)=\{V_\bullet\in \operatorname{Fl}(\mathbb C^n)\mid XV_j\subseteq V_{h(j)}\text{ for all }j\in[n]\}.$$1

Equivalently, a noncentral semisimple Hessenberg variety is connected exactly when the Hessenberg space contains all negative simple root spaces. The same paper emphasizes that this is a criterion for connectedness, not irreducibility. By contrast, for nilpotent $$\operatorname{Hess}(X,h)=\{V_\bullet\in \operatorname{Fl}(\mathbb C^n)\mid XV_j\subseteq V_{h(j)}\text{ for all }j\in[n]\}.$$2, every Hessenberg variety $$\operatorname{Hess}(X,h)=\{V_\bullet\in \operatorname{Fl}(\mathbb C^n)\mid XV_j\subseteq V_{h(j)}\text{ for all }j\in[n]\}.$$3 is rationally connected, hence connected (Precup, 2013).

Affine paving extends beyond type $$\operatorname{Hess}(X,h)=\{V_\bullet\in \operatorname{Fl}(\mathbb C^n)\mid XV_j\subseteq V_{h(j)}\text{ for all }j\in[n]\}.$$4. For a complex semisimple or reductive group, if $$\operatorname{Hess}(X,h)=\{V_\bullet\in \operatorname{Fl}(\mathbb C^n)\mid XV_j\subseteq V_{h(j)}\text{ for all }j\in[n]\}.$$5 is nilpotent and regular in some Levi subalgebra, then $$\operatorname{Hess}(X,h)=\{V_\bullet\in \operatorname{Fl}(\mathbb C^n)\mid XV_j\subseteq V_{h(j)}\text{ for all }j\in[n]\}.$$6 is paved by affines. More generally, if $$\operatorname{Hess}(X,h)=\{V_\bullet\in \operatorname{Fl}(\mathbb C^n)\mid XV_j\subseteq V_{h(j)}\text{ for all }j\in[n]\}.$$7 is the Jordan decomposition and $$\operatorname{Hess}(X,h)=\{V_\bullet\in \operatorname{Fl}(\mathbb C^n)\mid XV_j\subseteq V_{h(j)}\text{ for all }j\in[n]\}.$$8 is regular in some Levi subalgebra of $$\operatorname{Hess}(X,h)=\{V_\bullet\in \operatorname{Fl}(\mathbb C^n)\mid XV_j\subseteq V_{h(j)}\text{ for all }j\in[n]\}.$$9, then $G$0 is paved by affines. In particular, every Hessenberg variety associated to a semisimple element, and every Hessenberg variety associated to a regular element, is paved by affines; in all these cases odd cohomology vanishes (Precup, 2012).

These statements delimit a recurring phenomenon in the subject. Semisimple Hessenberg varieties can be disconnected and often reducible, nilpotent Hessenberg varieties are much more strongly connected, and affine paving supplies a uniform topological control across many otherwise disparate cases.

3. Regular nilpotent and regular semisimple regimes

The two principal families in type $G$1 are the regular nilpotent and regular semisimple Hessenberg varieties. Let $G$2 be a regular nilpotent matrix and $G$3 a diagonal matrix with distinct eigenvalues. Then $G$4 and $G$5 exhibit complementary geometric behavior (Abe et al., 2019).

For regular nilpotent Hessenberg varieties, the basic structural theorem states that $G$6 is irreducible and singular in general, with

$G$7

Its Poincaré polynomial has two equivalent forms: $G$8 and

$G$9

where $B\subseteq G$0 and $B\subseteq G$1 are defined by explicit permutation conditions. The cohomology ring also admits an explicit presentation: $B\subseteq G$2 and this makes the cohomology a complete intersection. A notable feature is that $B\subseteq G$3 is a Poincaré duality algebra even though $B\subseteq G$4 is generally singular (Abe et al., 2019).

For regular semisimple Hessenberg varieties, De Mari–Procesi–Shayman proved the parallel theorem: $B\subseteq G$5 In contrast with the nilpotent case, $B\subseteq G$6 is smooth, and it is connected if and only if $B\subseteq G$7 for all $B\subseteq G$8. The regular semisimple case is therefore topologically controlled by the same Hessenberg function but geometrically smoother and more symmetric (Abe et al., 2019).

These two regimes are linked by an invariant-theoretic theorem: $B\subseteq G$9 Thus the cohomology of the regular nilpotent Hessenberg variety is the $H\subseteq\mathfrak g$0-invariant subring of the cohomology of the corresponding regular semisimple Hessenberg variety, with the symmetric-group action given by Tymoczko’s dot action (Abe et al., 2019).

4. Cohomology, equivariant methods, and combinatorics

The cohomological study of Hessenberg varieties is closely tied to Schubert calculus, GKM theory, hyperplane arrangements, and graph invariants. On the ambient flag variety one has

$H\subseteq\mathfrak g$1

where $H\subseteq\mathfrak g$2 denotes the $H\subseteq\mathfrak g$3-th elementary symmetric polynomial. Regular nilpotent Hessenberg varieties inherit explicit quotient presentations, while regular semisimple varieties are more naturally described equivariantly (Abe et al., 2019).

For $H\subseteq\mathfrak g$4, Tymoczko’s GKM description gives

$H\subseteq\mathfrak g$5

whenever $H\subseteq\mathfrak g$6 for some $H\subseteq\mathfrak g$7 with $H\subseteq\mathfrak g$8. This is encoded by a GKM graph $H\subseteq\mathfrak g$9 with vertex set $$\operatorname{Hess}(x,H)=\{gB\in G/B\mid \operatorname{Ad}(g^{-1})x\in H\}.$$0 and edges prescribed by the Hessenberg function. The same equivariant framework supports Tymoczko’s dot action: $$\operatorname{Hess}(x,H)=\{gB\in G/B\mid \operatorname{Ad}(g^{-1})x\in H\}.$$1 which turns cohomology into a graded $$\operatorname{Hess}(x,H)=\{gB\in G/B\mid \operatorname{Ad}(g^{-1})x\in H\}.$$2-representation (Abe et al., 2019).

The regular nilpotent side admits a parallel arrangement-theoretic model. To a Hessenberg function $$\operatorname{Hess}(x,H)=\{gB\in G/B\mid \operatorname{Ad}(g^{-1})x\in H\}.$$3 one associates the ideal arrangement

$$\operatorname{Hess}(x,H)=\{gB\in G/B\mid \operatorname{Ad}(g^{-1})x\in H\}.$$4

in

$$\operatorname{Hess}(x,H)=\{gB\in G/B\mid \operatorname{Ad}(g^{-1})x\in H\}.$$5

If $$\operatorname{Hess}(x,H)=\{gB\in G/B\mid \operatorname{Ad}(g^{-1})x\in H\}.$$6 denotes the logarithmic derivation module and $$\operatorname{Hess}(x,H)=\{gB\in G/B\mid \operatorname{Ad}(g^{-1})x\in H\}.$$7, then

$$\operatorname{Hess}(x,H)=\{gB\in G/B\mid \operatorname{Ad}(g^{-1})x\in H\}.$$8

where $$\operatorname{Hess}(x,H)=\{gB\in G/B\mid \operatorname{Ad}(g^{-1})x\in H\}.$$9. This identifies regular nilpotent Hessenberg cohomology with a quotient defined by logarithmic derivations of the corresponding ideal arrangement (Abe et al., 2019).

On the semisimple side, a Hessenberg function also determines a graph $\mathfrak b$0 on vertex set $\mathfrak b$1, with edges $\mathfrak b$2. Shareshian and Wachs associated to $\mathfrak b$3 the chromatic quasisymmetric function $\mathfrak b$4, and the theorem of Brosnan–Chow and Guay-Paquet identifies it with the graded Frobenius characteristic of regular semisimple Hessenberg cohomology: $\mathfrak b$5 This places regular semisimple Hessenberg varieties at the center of the interaction among geometry, symmetric-group representations, and chromatic symmetric-function theory (Abe et al., 2019).

A separate but complementary result concerns classes in the cohomology and $\mathfrak b$6-theory of the flag variety for regular $\mathfrak b$7. If $\mathfrak b$8 is the permutation defined by

$\mathfrak b$9

then the class of the regular Hessenberg variety $h:[n]\to[n]$00 is represented by substituting a specific list of variables into the Schubert polynomial $h:[n]\to[n]$01 or Grothendieck polynomial $h:[n]\to[n]$02. The resulting formulas depend only on $h:[n]\to[n]$03, not on the choice of regular $h:[n]\to[n]$04, and give a uniform description of Hessenberg classes in both cohomology and $h:[n]\to[n]$05-theory (Insko et al., 2018).

5. Explicit and singular special families

Several special families of Hessenberg varieties are unusually explicit. Among the most tractable are the Hessenberg varieties attached to the minimal nilpotent orbit. If $h:[n]\to[n]$06 is connected, simply connected, and simple, and $h:[n]\to[n]$07 is a highest root vector, then $h:[n]\to[n]$08 depends only on the minimal nilpotent orbit. A key structural fact is that $h:[n]\to[n]$09 is $h:[n]\to[n]$10-invariant and hence a union of Schubert varieties. Its $h:[n]\to[n]$11-fixed points are

$h:[n]\to[n]$12

and the variety itself is

$h:[n]\to[n]$13

In type $h:[n]\to[n]$14, if $h:[n]\to[n]$15 is the corresponding Hessenberg function, then

$h:[n]\to[n]$16

and the Poincaré polynomial has the explicit factorization

$h:[n]\to[n]$17

The irreducible components are classified by corners of a modified Hessenberg stair shape, and the varieties are GKM: their GKM graph is the full subgraph of the GKM graph of $h:[n]\to[n]$18 on the allowed vertices $h:[n]\to[n]$19. The ordinary and $h:[n]\to[n]$20-equivariant cohomology rings are quotient rings of those of the flag variety (Abe et al., 2015).

Codimension-one Hessenberg varieties form another especially explicit family. In type $h:[n]\to[n]$21, the maximal proper Hessenberg function is

$h:[n]\to[n]$22

and for non-scalar $h:[n]\to[n]$23, $h:[n]\to[n]$24 has codimension one in the flag variety. If $h:[n]\to[n]$25 has pairwise distinct eigenvalues $h:[n]\to[n]$26 with $h:[n]\to[n]$27, then

$h:[n]\to[n]$28

Moreover,

$h:[n]\to[n]$29

For $h:[n]\to[n]$30, the scheme $h:[n]\to[n]$31 is reduced for every $h:[n]\to[n]$32. If $h:[n]\to[n]$33 is nilpotent, then the singular locus satisfies

$h:[n]\to[n]$34

whereas for general $h:[n]\to[n]$35 only the containment

$h:[n]\to[n]$36

holds (Escobar et al., 2022).

A different special family arises from the minimal indecomposable Hessenberg space

$h:[n]\to[n]$37

For regular elements $h:[n]\to[n]$38, the corresponding regular Hessenberg varieties $h:[n]\to[n]$39 form a flat family of irreducible subvarieties that includes the Peterson variety and the regular semisimple toric variety associated to Weyl chambers. Their affine cells have closures that are themselves regular Hessenberg varieties in smaller Levi flag varieties, and all regular members are singular outside of the toric case (Insko et al., 2024).

6. Extensions, variants, and active directions

Regular semisimple Hessenberg varieties also support a refined birational and positivity theory. In type $h:[n]\to[n]$40, if

$h:[n]\to[n]$41

then

$h:[n]\to[n]$42

Writing

$h:[n]\to[n]$43

one has a complete classification: $h:[n]\to[n]$44 is weak Fano if and only if $h:[n]\to[n]$45 for all $h:[n]\to[n]$46, and it is Fano if and only if $h:[n]\to[n]$47 for some $h:[n]\to[n]$48 with

$h:[n]\to[n]$49

A distinctive feature of this setting is that nefness of the anticanonical bundle already implies bigness (Abe et al., 2020).

Semisimple Hessenberg varieties with exactly two eigenvalues admit another explicit description. For $h:[n]\to[n]$50 with multiplicities $h:[n]\to[n]$51, the corresponding varieties are $h:[n]\to[n]$52-stable for $h:[n]\to[n]$53. The irreducible ones are exactly the closures of certain $h:[n]\to[n]$54-orbits indexed by $h:[n]\to[n]$55-avoiding permutations in $h:[n]\to[n]$56, so their number is the Catalan number

$h:[n]\to[n]$57

For these irreducible varieties one has

$h:[n]\to[n]$58

and Brion’s theorem yields explicit multiplicity-free Schubert-polynomial representatives for their cohomology classes (Can et al., 2023).

Partial Hessenberg varieties extend the theory from $h:[n]\to[n]$59 to $h:[n]\to[n]$60. If $h:[n]\to[n]$61 is a $h:[n]\to[n]$62-Hessenberg space for a parabolic $h:[n]\to[n]$63, then

$h:[n]\to[n]$64

fits into a bundle

$h:[n]\to[n]$65

The cohomology of the partial variety is recovered from the full-flag variety by star invariants: $h:[n]\to[n]$66 In the regular setting this combines with dot-action invariants to give

$h:[n]\to[n]$67

This places partial Hessenberg cohomology in the same invariant-theoretic framework as the full-flag theory (Horiguchi et al., 31 Jul 2025).

Questions about torus actions and GKM structures remain delicate outside the classical regular semisimple and minimal nilpotent settings. In type $h:[n]\to[n]$68, some nilpotent Hessenberg varieties are $h:[n]\to[n]$69-stable and hence GKM, some are only stable under a proper subtorus $h:[n]\to[n]$70, and some have torus actions with isolated fixed points but still fail to be GKM because they have infinitely many one-dimensional torus orbits. In particular, not all Hessenberg varieties with torus actions and finitely many fixed points are GKM, and not every torus-stable Hessenberg variety is a union of Schubert varieties (Goldin et al., 2023).

The subject also admits Lie-theoretic reinterpretations beyond equivariant topology. For the distinguished Hessenberg space

$h:[n]\to[n]$71

the total space $h:[n]\to[n]$72 carries a natural Poisson structure, its unique open dense symplectic leaf is $h:[n]\to[n]$73, and there is a symplectomorphism

$h:[n]\to[n]$74

where $h:[n]\to[n]$75 is a regular Slodowy slice. The Mishchenko–Fomenko polynomials pull back to a completely integrable system on the Poisson variety $h:[n]\to[n]$76, extending the Toda lattice and recovering open subsets of regular Hessenberg fibres as

$h:[n]\to[n]$77

for regular $h:[n]\to[n]$78 (Abe et al., 2018).

Taken together, these developments show that Hessenberg varieties are not a single rigid class but a framework with several sharply distinct geometric regimes. Regular nilpotent varieties are generally singular but admit explicit complete-intersection cohomology rings; regular semisimple varieties are smooth and representation-theoretically rich; minimal-orbit, codimension-one, two-eigenvalue, partial, and Poisson-theoretic variants each reveal additional structure. The common thread is that the defining incidence condition $h:[n]\to[n]$79, or its Lie-theoretic analogue $h:[n]\to[n]$80, encodes a remarkably broad range of geometry in a form that remains computable across many of the subject’s most important examples.