Monodromic Fourier Transform

• Monodromic Fourier transform is a specialized Fourier-type functor that restricts attention to objects with controlled monodromy along scalar orbits.
• It employs kernel constructions, such as exponential and homogeneous kernels, and utilizes vanishing cycles to ensure uniform behavior across different characteristics.
• The framework underpins key applications in microlocal geometry, mixed Hodge theory, and representation theory by preserving characteristic cycles and enabling categorical equivalences.

The monodromic Fourier transform is a family of Fourier-type functors adapted to the scalar-action symmetry on a vector space or vector bundle. In this setting, one restricts attention to objects whose behavior along $\mathbb{G}_m$- or $\mathbb{C}^*$-orbits is controlled by monodromy: for $\ell$-adic sheaves this means tame local systems along scaling orbits, for holonomic $\mathcal D$-modules it is expressed through the Euler vector field, and for mixed Hodge modules it is imposed on the underlying constructible or $\mathcal D$-module data. Across these frameworks, the central phenomenon is that Fourier transform becomes especially well behaved on the monodromic subcategory: it preserves monodromicity, often becomes an equivalence, and admits descriptions via exponential kernels, homogeneous kernels on quotient stacks, or vanishing cycles (Wang, 2014, Virk, 22 Jul 2025).

1. Monodromicity and the basic categorical setting

Let $V$ be a vector space, or more generally a vector bundle $V \to S$, endowed with the homothety action of $\mathbb{G}_m$ or $\mathbb{C}^*$. In the $\ell$-adic setting, a complex $\mathbb{C}^*$0 is monodromic if its cohomology sheaves restrict to tame local systems on every scalar orbit. For finite coefficients, Verdier’s criterion gives an equivalent condition: there exists an integer $\mathbb{C}^*$1, prime to the characteristic $\mathbb{C}^*$2, such that

$\mathbb{C}^*$3

where $\mathbb{C}^*$4. The minimal such $\mathbb{C}^*$5 is called the twist. For perverse irreducible monodromic sheaves, trivial twist is equivalent to $\mathbb{C}^*$6-equivariance, but monodromic does not in general mean equivariant; arbitrary local systems can occur along the orbits (Zhou, 2024).

In the sheaf-theoretic Fourier–Sato framework on a $\mathbb{C}^*$7-variety, monodromic means that each $\mathbb{C}^*$8 restricts to a local system on every $\mathbb{C}^*$9-orbit. The same works for constructible sheaves on a vector bundle with fiberwise scaling. In this algebraic constructible context, monodromicity is equivalent to the conic condition for the $\ell$0-action, so the classical conic Fourier–Sato theory and the monodromic theory coincide on the relevant subcategories (Virk, 22 Jul 2025).

For holonomic $\ell$1-modules, there are two complementary formulations. On a vector space over $\ell$2, a holonomic $\ell$3-module is monodromic if the Euler vector field

$\ell$4

acts locally finitely. On a vector bundle $\ell$5, a left $\ell$6-module $\ell$7 is monodromic if every local section is annihilated by some polynomial in the fiberwise Euler field $\ell$8. In that case one has a decomposition

$\ell$9

which plays the same structural role as tame-orbit decomposition in the sheaf setting (Saito, 2022).

2. Kernel constructions and the geometric transform $\mathcal D$0

Classical geometric Fourier transforms use an exponential object. For $\mathcal D$1-adic sheaves in characteristic $\mathcal D$2, the Fourier–Deligne transform is

$\mathcal D$3

For $\mathcal D$4-modules in characteristic $\mathcal D$5, the usual Fourier transform is

$\mathcal D$6

Both depend on an exponential kernel on $\mathcal D$7. Laumon’s homogeneous transform avoids this by passing to quotient stacks $\mathcal D$8 and $\mathcal D$9, thereby producing a transform that is uniform in any characteristic but only defined for homogeneous objects (Wang, 2014).

The construction of “A new Fourier transform” introduces a kernel

$\mathcal D$0

and defines

$\mathcal D$1

This functor is defined on all objects of $\mathcal D$2 over any field and in any characteristic, because it uses $\mathcal D$3 rather than an exponential local system. Its basic square is

$\mathcal D$4

Since $\mathcal D$5 is not invertible in general, $\mathcal D$6 is not an autoequivalence on all constructible complexes. The monodromic subcategory is the decisive exception: after introducing universal monodromy objects $\mathcal D$7 on $\mathcal D$8, one proves

$\mathcal D$9

with quasi-inverse

$V$0

Thus $V$1 realizes the monodromic Fourier transform as an honest equivalence in a form that is uniform in characteristic. The same work proves that $V$2 extends Laumon’s homogeneous transform after pullback from the quotient stack, and that on monodromic objects it agrees with the Fourier–Deligne transform in characteristic $V$3 and with the usual Fourier transform for $V$4-modules in characteristic $V$5 (Wang, 2014).

3. Fourier–Sato transform and monodromic mixed Hodge modules

For a complex vector bundle $V$6 with dual $V$7, the Fourier–Sato transform of a constructible sheaf $V$8 is defined by the half-space kernel

$V$9

where $V \to S$0. In the monodromic setting this transform admits a vanishing-cycle description. Writing

$V \to S$1

and letting $V \to S$2 be projection, one has the canonical isomorphism

$V \to S$3

This reformulation is the bridge from constructible sheaves to mixed Hodge modules, because vanishing cycles, pullback, and pushforward are all available in Saito’s formalism (Virk, 22 Jul 2025).

A monodromic mixed Hodge module on a $V \to S$4-variety $V \to S$5 is an object $V \to S$6 whose underlying rational constructible complex $V \to S$7 is monodromic. The monodromic Fourier–Sato transform is then defined by the same formula,

$V \to S$8

and satisfies

$V \to S$9

The compatibility with duality is

$\mathbb{G}_m$0

for $\mathbb{G}_m$1. In rank one, the transform admits a quiver description: monodromic perverse sheaves on $\mathbb{G}_m$2 are encoded by diagrams

$\mathbb{G}_m$3

with the standard invertibility condition, and the shifted Fourier–Sato transform interchanges nearby and vanishing cycles. The mixed Hodge module version has the analogous form

$\mathbb{G}_m$4

which expresses the transform as an involution on the corresponding glued category (Virk, 22 Jul 2025).

4. Hodge filtrations, gluing data, and irregular Hodge theory

For monodromic $\mathbb{G}_m$5-modules on $\mathbb{G}_m$6, the decomposition by generalized eigenspaces of $\mathbb{G}_m$7,

$\mathbb{G}_m$8

is extremely rigid. Multiplication by $\mathbb{G}_m$9 and differentiation by $\mathbb{C}^*$0 shift the indices,

$\mathbb{C}^*$1

and are isomorphisms away from the exceptional indices $\mathbb{C}^*$2 and $\mathbb{C}^*$3. Nearby and vanishing cycles are identified by

$\mathbb{C}^*$4

For monodromic mixed Hodge modules, the Hodge filtration decomposes compatibly: $\mathbb{C}^*$5 This leads to an equivalence between monodromic mixed Hodge modules on $\mathbb{C}^*$6 and a category of gluing data built from nearby cycles, vanishing cycles, semisimple monodromy, nilpotent logarithm, and can/var morphisms (Saito, 2020).

That gluing formalism gives a direct construction of the Fourier–Laplace transform at the level of mixed Hodge modules. If $\mathbb{C}^*$7 denotes the usual algebraic Fourier–Laplace transform of the underlying $\mathbb{C}^*$8-module, then

$\mathbb{C}^*$9

and on the fundamental strip $\ell$0 the Hodge filtration is

$\ell$1

In higher rank, for a monodromic mixed Hodge module on a vector bundle $\ell$2 of rank $\ell$3, the Fourier transform again preserves monodromicity and reindexes the Euler decomposition by

$\ell$4

The same theory proves that the irregular Hodge filtration on $\ell$5 is explicit: $\ell$6 and, at integer indices,

$\ell$7

Thus, for Fourier transforms of monodromic mixed Hodge modules, the integer-indexed irregular Hodge filtration coincides with the genuine Hodge filtration (Saito, 2022).

A closely related description is obtained from the $\ell$8-filtration along the zero section of a vector bundle. For monodromic mixed Hodge modules on $\ell$9, the Hodge filtration splits along the monodromic decomposition, and the weight filtration is the relative monodromy filtration for the nilpotent operator induced by the Euler field. In this framework the Fourier–Laplace transform can be realized as

$\mathbb{C}^*$00

with explicit formulas for the Hodge and weight filtrations on each monodromic component, together with Fourier inversion up to the antipodal map and compatibility with duality (Chen et al., 2021).

5. Microlocal geometry, characteristic cycles, and regularity

In the $\mathbb{C}^*$01-adic setting over an algebraically closed field of characteristic $\mathbb{C}^*$02, monodromic sheaves are stable under Fourier transform, and their twist is preserved. The principal microlocal statement is the $\mathbb{C}^*$03-adic analogue of the Brylinski–Malgrange theorem: for a monodromic $\mathbb{C}^*$04,

$\mathbb{C}^*$05

under the canonical symplectic identification $\mathbb{C}^*$06. More precisely, the paper proves that monodromic sheaves are $\mathbb{C}^*$07-good, meaning every irreducible perverse constituent $\mathbb{C}^*$08 satisfies $\mathbb{C}^*$09. This establishes that Fourier transform preserves the microlocal support and multiplicities of monodromic $\mathbb{C}^*$10-adic sheaves exactly as in the holonomic $\mathbb{C}^*$11-module case (Zhou, 2024).

The $\mathbb{C}^*$12-module side supplies a complementary topological regularity statement. For an algebraic regular holonomic $\mathbb{C}^*$13-module $\mathbb{C}^*$14 on $\mathbb{C}^*$15, the solution complex of its Fourier transform is always monodromic: $\mathbb{C}^*$16 The same work reproves Brylinski’s theorem and strengthens it to a converse: $\mathbb{C}^*$17 It also gives applications to direct images of exponential twists, especially in codimension one, where monodromicity of the Fourier transform forces non-characteristic behavior for affine hyperplanes not passing through the origin (Ito et al., 2018).

These microlocal results delimit the scope of the monodromic theory. In particular, the $\mathbb{C}^*$18-adic characteristic-cycle theorem does not imply that all tame phenomena are preserved under Fourier transform: the paper explicitly notes that the analogue of “monodromic tame is preserved under Fourier transform” is false in the $\mathbb{C}^*$19-adic setting (Zhou, 2024).

6. Monodromy at infinity, isomonodromy, and broader significance

For holonomic $\mathbb{C}^*$20-modules on the affine line with moderate irregularity at infinity, the monodromic content of Fourier transform can be read directly from monodromy at infinity. If $\mathbb{C}^*$21 denotes the multiplicity of $\mathbb{C}^*$22 as an eigenvalue of the monodromy at infinity of $\mathbb{C}^*$23, then for $\mathbb{C}^*$24 one has the reciprocity law

$\mathbb{C}^*$25

In higher dimensions, for a regular holonomic $\mathbb{C}^*$26-module $\mathbb{C}^*$27 on $\mathbb{C}^*$28, the monodromy at infinity of $\mathbb{C}^*$29 along a conic line is expressed through the monodromy zeta function of the pushforward $\mathbb{C}^*$30 by a linear form $\mathbb{C}^*$31. In a smooth hypersurface case, the zeta function can be written explicitly and yields closed formulas for the eigenvalues and their multiplicities (Kudomi et al., 2024).

A different but related manifestation appears in the theory of meromorphic connections and isomonodromic deformations. Given an AHHP representation

$\mathbb{C}^*$32

Yamakawa defines the Harnad dual

$\mathbb{C}^*$33

Under explicit normal-form and nondegeneracy assumptions, this dual realizes the Fourier–Laplace transform of the original connection. The same framework shows that admissible isomonodromic families with unramified irregular singularities are carried to dual isomonodromic families, and that additive middle convolution

$\mathbb{C}^*$34

preserves isomonodromy under the stated hypotheses (Yamakawa, 2013).

Taken together, these results position the monodromic Fourier transform as a common structure across geometric representation theory, microlocal geometry, and Hodge theory. The cited works identify applications to character sheaves, Whittaker sheaves, equivariant sheaves under tori, quotient stacks $\mathbb{C}^*$35, and character sheaves on reductive Lie algebras in positive characteristic. This suggests a unifying principle: once scalar-orbit monodromy is isolated as the relevant symmetry, Fourier transform ceases to be merely an integral transform with exponential kernel and becomes a categorical operation that preserves the essential orbitwise, microlocal, and Hodge-theoretic data of the objects on which it acts (Wang, 2014, Zhou, 2024).