Fake Special Cycle Classes

• Fake special cycle classes are non-geometric imitations in arithmetic and algebraic geometry that mimic the deformation and categorical signatures of genuine special cycles.
• They are identified in multiple settings—Hodge theory on Fermat hypersurfaces, the geometric Langlands program via categorical trace, and as Selmer substitutes through theta cycles.
• Their study reveals that while first-order invariants imitate true cycles, higher-order data such as non-reduced Hodge loci and quadratic forms are essential to distinguish genuine geometric components.

Fake special cycle classes form a non-uniform but technically coherent motif across several areas of arithmetic and algebraic geometry. In the Hodge-theoretic setting of even-dimensional Fermat hypersurfaces, they appear as Hodge classes whose Hodge loci have the same first-order behavior as loci coming from genuine linear subvarieties, yet which are not classes of linear cycles and whose Hodge loci are scheme-theoretically non-reduced (Franco et al., 2021). In the geometric Langlands setting, the term denotes classes produced by categorical trace from Hecke eigensheaves and period sheaves; these are called “fake” because they are not initially defined as geometric fundamental classes of subvarieties in Shtuka stacks, although they are later identified with $\sigma$-isotypic projections of genuine special cycles (Wang, 5 Sep 2025). In the Selmer-theoretic setting, theta cycles serve as canonical cohomological substitutes for geometric special cycle classes when geometry or modularity is not fully available, and are explicitly described as “fake special cycle classes” in that broader sense (Disegni, 2023).

1. Terminological range and conceptual unity

The phrase “fake special cycle classes” does not have a single universal definition. Its meaning depends on the ambient theory, but the common pattern is stable: a class behaves like a special cycle at the level of deformation theory, categorical trace, or arithmetic regulator formulas, while failing to arise directly as the geometric class one would naively expect.

Setting	Representative object	“Fake” aspect
Fermat hypersurfaces	Hodge cycle $\lambda$	Same minimal tangent codimension as a linear cycle, but not a linear subvariety
Shtukas	$z_{\mathfrak c,\sigma}:V_\sigma^I\langle -d_I\rangle\to \uk_{C^I}$	Produced by categorical formalism rather than geometric fundamental classes
Selmer groups	$\Theta_\rho:\Lambda_\rho\to H^1_f(E,\rho)$	Canonical cohomological substitute for special cycle classes

In "On fake linear cycles inside Fermat varieties" (Franco et al., 2021), the paper does not use “special cycles” explicitly. Instead, linear cycles are special in the broader sense that they generate minimal codimension components of Hodge loci, and fake linear cycles are “fake special cycle classes” because they attain the same first-order codimension bound without corresponding to actual linear subvarieties. In "Special Cycle on Shtukas and Categorical Trace" (Wang, 5 Sep 2025), the terminology is literal: fake special cycle classes are defined by categorical trace and only subsequently identified with genuine special cycles. In "Theta cycles and the Beilinson--Bloch--Kato conjectures" (Disegni, 2023), the term is again broader: theta cycles are canonical Selmer classes that play the role of special cycle classes in settings where the geometric cycles or their modularity remain conjectural.

2. Fake linear cycles in Fermat varieties

The basic geometric setting is the even-dimensional Fermat hypersurface

$$X_d^n=\{F=0\}\subset \mathbb{P}^{n+1},\qquad F=x_0^d+x_1^d+\cdots+x_{n+1}^d,$$

with $n$ even so that $H^{\frac n2,\frac n2}$ is non-trivial. For a family of smooth degree-$d$ hypersurfaces, a Hodge cycle is a class

$$\lambda\in H^{\frac n2,\frac n2}(X)\cap H^n(X,\mathbb{Z}),$$

and its local Hodge locus $V_\lambda$ is the analytic locus in the parameter space where the flat transport of $\lambda$0 remains of type $\lambda$1 (Franco et al., 2021).

A genuine linear cycle is a linear subspace $\lambda$2 contained in $\lambda$3. For the Fermat hypersurface, a standard example is

$\lambda$4

The locus $\lambda$5 of hypersurfaces containing a $\lambda$6-linear subvariety has codimension

$\lambda$7

Fake linear cycles are Hodge cycles whose Hodge loci have the same first-order behavior as those of true linear cycles, namely the same minimal tangent codimension at the Fermat point, but which do not come from actual linear subvarieties and whose associated Hodge loci are non-reduced. The paper proves a precise characterization: if

$\lambda$8

then, up to permuting coordinates,

$\lambda$9

with

$z_{\mathfrak c,\sigma}:V_\sigma^I\langle -d_I\rangle\to \uk_{C^I}$0

where

$z_{\mathfrak c,\sigma}:V_\sigma^I\langle -d_I\rangle\to \uk_{C^I}$1

If all $z_{\mathfrak c,\sigma}:V_\sigma^I\langle -d_I\rangle\to \uk_{C^I}$2, then $z_{\mathfrak c,\sigma}:V_\sigma^I\langle -d_I\rangle\to \uk_{C^I}$3 is the class of a true linear cycle; fake linear cycles occur when the $z_{\mathfrak c,\sigma}:V_\sigma^I\langle -d_I\rangle\to \uk_{C^I}$4 are not all $z_{\mathfrak c,\sigma}:V_\sigma^I\langle -d_I\rangle\to \uk_{C^I}$5-th roots of $z_{\mathfrak c,\sigma}:V_\sigma^I\langle -d_I\rangle\to \uk_{C^I}$6 simultaneously.

The existence theorem is sharp: fake linear cycles exist only for degrees $z_{\mathfrak c,\sigma}:V_\sigma^I\langle -d_I\rangle\to \uk_{C^I}$7, and in each of these degrees there are infinitely many scheme-theoretically different Hodge loci $z_{\mathfrak c,\sigma}:V_\sigma^I\langle -d_I\rangle\to \uk_{C^I}$8 through the Fermat point that achieve the minimal tangent codimension (Franco et al., 2021). The Galois-invariance needed to make $z_{\mathfrak c,\sigma}:V_\sigma^I\langle -d_I\rangle\to \uk_{C^I}$9 a Hodge class is ensured through Hilbert’s Theorem 90.

3. Infinitesimal structure, Galois symmetry, and pathologies

The infinitesimal variation of Hodge structure controls the Zariski tangent space to the Hodge locus through the Griffiths Jacobian ring. For a hypersurface $\Theta_\rho:\Lambda_\rho\to H^1_f(E,\rho)$0, the Jacobian ring is

$\Theta_\rho:\Lambda_\rho\to H^1_f(E,\rho)$1

If $\Theta_\rho:\Lambda_\rho\to H^1_f(E,\rho)$2, then the associated Artinian Gorenstein ideal is

$\Theta_\rho:\Lambda_\rho\to H^1_f(E,\rho)$3

and Carlson–Green–Griffiths–Harris identify the tangent space by

$\Theta_\rho:\Lambda_\rho\to H^1_f(E,\rho)$4

(Franco et al., 2021).

For fake linear cycles, the ideal has the explicit form

$\Theta_\rho:\Lambda_\rho\to H^1_f(E,\rho)$5

This ideal mimics the period-theoretic footprint of a genuine linear $\Theta_\rho:\Lambda_\rho\to H^1_f(E,\rho)$6 cut out by

$\Theta_\rho:\Lambda_\rho\to H^1_f(E,\rho)$7

but that linear space is not contained in $\Theta_\rho:\Lambda_\rho\to H^1_f(E,\rho)$8 unless all $\Theta_\rho:\Lambda_\rho\to H^1_f(E,\rho)$9. This is the mechanism behind the statement that such cycles are not generated by their periods in the sense of Movasati–Sertöz.

The pathology is not merely first-order. Maclean’s quadratic fundamental form gives a second-order IVHS invariant in degree

$$X_d^n=\{F=0\}\subset \mathbb{P}^{n+1},\qquad F=x_0^d+x_1^d+\cdots+x_{n+1}^d,$$0

namely

$$X_d^n=\{F=0\}\subset \mathbb{P}^{n+1},\qquad F=x_0^d+x_1^d+\cdots+x_{n+1}^d,$$1

For linear forms $$X_d^n=\{F=0\}\subset \mathbb{P}^{n+1},\qquad F=x_0^d+x_1^d+\cdots+x_{n+1}^d,$$2, one computes

$$X_d^n=\{F=0\}\subset \mathbb{P}^{n+1},\qquad F=x_0^d+x_1^d+\cdots+x_{n+1}^d,$$3

Hence $$X_d^n=\{F=0\}\subset \mathbb{P}^{n+1},\qquad F=x_0^d+x_1^d+\cdots+x_{n+1}^d,$$4 for fake linear cycles, proving that the Hodge locus is non-reduced at the Fermat point; for genuine linear cycles the factor $$X_d^n=\{F=0\}\subset \mathbb{P}^{n+1},\qquad F=x_0^d+x_1^d+\cdots+x_{n+1}^d,$$5 vanishes.

These constructions also disprove Movasati’s conjecture that the lower bound

$$X_d^n=\{F=0\}\subset \mathbb{P}^{n+1},\qquad F=x_0^d+x_1^d+\cdots+x_{n+1}^d,$$6

for the codimension of $$X_d^n=\{F=0\}\subset \mathbb{P}^{n+1},\qquad F=x_0^d+x_1^d+\cdots+x_{n+1}^d,$$7 is attained only by classes of linear $$X_d^n=\{F=0\}\subset \mathbb{P}^{n+1},\qquad F=x_0^d+x_1^d+\cdots+x_{n+1}^d,$$8. Fake linear cycles attain the bound in all even dimensions for $$X_d^n=\{F=0\}\subset \mathbb{P}^{n+1},\qquad F=x_0^d+x_1^d+\cdots+x_{n+1}^d,$$9, but are not linear cycles (Franco et al., 2021).

The Galois-theoretic analysis uses the action of

$n$0

on cohomology and Shioda’s decomposition

$n$1

together with explicit period formulas over vanishing cycles and the normalized eigenclasses

$n$2

This Galois symmetry is what makes the classification and Hilbert-90 construction of $n$3 effective.

4. Minimal codimension Hodge loci and the role of fake classes

Despite the existence of fake linear cycles, the global minimal-codimension picture through the Fermat point remains rigid. For even $n$4 and

$n$5

the unique component of minimal codimension of the Hodge locus $n$6 passing through the Fermat point is $n$7, the locus of hypersurfaces containing a $n$8-linear subvariety, with codimension

$n$9

(Franco et al., 2021).

This theorem extends several earlier results. Green–Voisin showed in the surface case that for $H^{\frac n2,\frac n2}$0 the minimal component is the “contains a line” locus, while for $H^{\frac n2,\frac n2}$1 all components have the same codimension. Otwinowska obtained an asymptotic result for $H^{\frac n2,\frac n2}$2. Movasati proved the lower bound at the Fermat point. Villaflor had earlier treated $H^{\frac n2,\frac n2}$3. The new feature is that the low-degree exceptional cases $H^{\frac n2,\frac n2}$4 are resolved by analyzing fake linear cycles and proving their non-reducedness via the quadratic form $H^{\frac n2,\frac n2}$5.

The significance of this result is structural. Fake linear cycles show that first-order infinitesimal behavior does not determine geometry, but they do not produce new minimal-codimension components through the Fermat point. A plausible implication is that second-order IVHS data, rather than tangent-space dimension alone, is the correct level at which one separates genuine geometric components from pathological ones.

5. Categorical fake special cycle classes on Shtukas

In the geometric Langlands and Shtuka setting, fake special cycle classes are defined by categorical trace rather than by cycle-theoretic geometry. Let $H^{\frac n2,\frac n2}$6 be a smooth projective geometrically connected curve over $H^{\frac n2,\frac n2}$7, let $H^{\frac n2,\frac n2}$8 be a split reductive group, and let $H^{\frac n2,\frac n2}$9. The stack of $d$0-Shtukas with $d$1 legs is

$d$2

with projection $d$3 (Wang, 5 Sep 2025).

The relevant categorical framework is $d$4, the full subcategory of sheaves with nilpotent singular support. For a Hecke eigensheaf

$d$5

and a smooth affine spherical $d$6-variety $d$7, the period sheaf is

$d$8

Given a Hecke-type cohomological correspondence

$d$9

one obtains a canonical map

$$\lambda\in H^{\frac n2,\frac n2}(X)\cap H^n(X,\mathbb{Z}),$$0

and then the fake special cycle class

$$\lambda\in H^{\frac n2,\frac n2}(X)\cap H^n(X,\mathbb{Z}),$$1

defined by categorical trace. Equivalently,

$$\lambda\in H^{\frac n2,\frac n2}(X)\cap H^n(X,\mathbb{Z}),$$2

These classes are called “fake” because they are produced by Hecke eigensheaves, period sheaves, and categorical trace rather than by taking geometric fundamental classes of subvarieties in the Shtuka stack. The central theorem identifies them with genuine special cycles. For minuscule homogeneous $$\lambda\in H^{\frac n2,\frac n2}(X)\cap H^n(X,\mathbb{Z}),$$3, one has a special cycle class

$$\lambda\in H^{\frac n2,\frac n2}(X)\cap H^n(X,\mathbb{Z}),$$4

and the paper proves

$$\lambda\in H^{\frac n2,\frac n2}(X)\cap H^n(X,\mathbb{Z}),$$5

where $$\lambda\in H^{\frac n2,\frac n2}(X)\cap H^n(X,\mathbb{Z}),$$6 is the $$\lambda\in H^{\frac n2,\frac n2}(X)\cap H^n(X,\mathbb{Z}),$$7-isotypic part map arising from the Hecke eigenrelation and the Serre local term isomorphism $$\lambda\in H^{\frac n2,\frac n2}(X)\cap H^n(X,\mathbb{Z}),$$8 (Wang, 5 Sep 2025). Thus the fake class is exactly the $$\lambda\in H^{\frac n2,\frac n2}(X)\cap H^n(X,\mathbb{Z}),$$9-isotypic projection of a genuine special cycle on Shtukas.

In the Rankin–Selberg case $V_\lambda$0 and $V_\lambda$1, the resulting classes satisfy

$V_\lambda$2

The main formula of the paper relates the self-intersection number of the $V_\lambda$3-isotypic part of these special cycles to higher derivatives of Rankin–Selberg $V_\lambda$4-functions: $V_\lambda$5 This is proved under the hypotheses $V_\lambda$6 and geometric irreducibility of $V_\lambda$7.

The conceptual point is that “fake” does not mean spurious. In this setting, it means that the construction begins in categorical formalism and only later recovers geometry.

6. Theta cycles as cohomological substitutes for special cycles

Theta cycles arise in the arithmetic setting of conjugate-symplectic Galois representations over CM fields. They are canonical Selmer classes obtained by pairing Kudla’s special cycles on unitary Shimura varieties with automorphic forms through a theta kernel, and then projecting to the relevant isotypic Galois piece (Disegni, 2023). The paper states that these classes provide cohomological substitutes for geometric special cycle classes—“fake special cycle classes”—in contexts where the geometry or its modularity is not fully available.

The geometric input is the tower of unitary Shimura varieties

$V_\lambda$8

with codimension-$V_\lambda$9 special cycles

$\lambda$00

defined by Kudla–Rapoport–Yang. Their cohomology classes are assembled into Kudla’s generating series

$\lambda$01

Under the modularity hypothesis, there exists

$\lambda$02

whose $\lambda$03-expansion is Kudla’s generating series.

Passing to Selmer groups uses the arithmetic theta lift

$\lambda$04

and then a Galois projection $\lambda$05 produces

$\lambda$06

The canonicality is packaged by the one-dimensional theta line

$\lambda$07

well-defined up to isomorphism. The residual ambiguity comes from the choice of Whittaker datum, but the paper states that this ambiguity is innocuous for nonvanishing and rank-one consequences.

When $\lambda$08, theta cycles recover Heegner points: the unitary Shimura varieties are curves, codimension-$\lambda$09 special cycles are divisors supported on CM points, and the arithmetic theta lift produces the Abel–Jacobi images of Heegner divisors in Selmer groups. In higher rank, theta cycles are the proposed analogues of Heegner classes.

Their significance lies in the Beilinson–Bloch–Kato framework. Under the stated hypotheses, the paper proves height formulas

$\lambda$10

and, under ordinarity and $\lambda$11-splitting,

$\lambda$12

(Disegni, 2023). These formulas imply nonvanishing criteria: $\lambda$13 and, via the Euler-system argument of the paper, $\lambda$14 under large image hypotheses.

The limitations are explicit. The construction depends on Hypothesis 4.1.1 on Hecke/Galois decomposition of cohomology, Hypothesis 4.1.2 on modularity of generating series, Conjecture 5.3 on Abel–Jacobi injectivity for the complex height formula, and Assumption 5.2 for the $\lambda$15-adic height formula. Within those constraints, theta cycles illustrate a broad version of the fake-special-cycle phenomenon: arithmetic classes with the regulator and local-global behavior expected of genuine special cycles, but obtained through cohomological and automorphic machinery rather than directly from established cycle theory.

Taken together, these three settings exhibit a common pattern. Fake special cycle classes can be first-order imitations of genuine geometric loci, categorical traces that later recover geometric cycles, or canonical Selmer classes replacing unavailable cycle classes. The unifying issue is not falsity but indirectness: the class has the deformation-theoretic, categorical, or arithmetic signature of a special cycle while the underlying geometry is hidden, degenerate, or only recovered after additional structure is brought to bear.