Split Weil Type in Abelian Varieties

• Split Weil type is a refinement of Weil type for polarized abelian varieties with CM-field action, characterized by a split hermitian form admitting a maximal isotropic subspace.
• The construction employs techniques like the Fourier–Mukai transform and secant sheaf methods to control and deform Hodge Weil classes in families.
• The split condition isolates new Hodge classes not generated by divisors, underpinning key results for abelian fourfolds and sixfolds and supporting low-dimensional cases of the Hodge conjecture.

Split Weil type is a refinement of Weil type for polarized abelian varieties carrying extra endomorphisms by a CM-field. In the formulation developed in "Secant sheaves and Weil classes on abelian varieties" (Markman, 27 Sep 2025), one starts with a polarized abelian variety $(A,\eta,h)$, where $\eta:K\hookrightarrow \operatorname{End}^0(A)$ for a CM-field $K$ with totally real subfield $F$, and the polarization is compatible with the involution $\iota$ of $K/F$. The structure is of split Weil type when the associated $K$-valued hermitian form is split, in the precise sense that it contains a maximal isotropic subspace of half rank. The notion is designed to isolate the setting in which the Hodge Weil classes $HW(A,\eta)$ can be constructed, controlled in families, and in important cases proved algebraic (Markman, 27 Sep 2025).

1. Definition and basic cohomological structure

Let $K$ be a CM-field, with totally real subfield $F$, and write

$\eta:K\hookrightarrow \operatorname{End}^0(A)$0

Given a complex abelian variety $\eta:K\hookrightarrow \operatorname{End}^0(A)$1 together with an embedding

$\eta:K\hookrightarrow \operatorname{End}^0(A)$2

the action of $\eta:K\hookrightarrow \operatorname{End}^0(A)$3 on $\eta:K\hookrightarrow \operatorname{End}^0(A)$4 yields a decomposition

$\eta:K\hookrightarrow \operatorname{End}^0(A)$5

where $\eta:K\hookrightarrow \operatorname{End}^0(A)$6 is the set of embeddings $\eta:K\hookrightarrow \operatorname{End}^0(A)$7 and $\eta:K\hookrightarrow \operatorname{End}^0(A)$8 is the $\eta:K\hookrightarrow \operatorname{End}^0(A)$9-eigenspace. The pair $K$0 is of Weil type if for every $K$1,

$K$2

This forces $K$3 and $K$4 to be even, and

$K$5

The associated Weil classes are the $K$6-linear exterior top classes

$K$7

They form an $K$8-dimensional $K$9-subspace,

$F$0

and lie in Hodge degree $F$1: $F$2 For split Weil type, the central problem is the algebraicity of these Hodge classes (Markman, 27 Sep 2025).

A polarized abelian variety of Weil type is a triple $F$3, where

$F$4

is the class of a polarization satisfying

$F$5

Equivalently, $F$6 defines a nondegenerate $F$7-valued hermitian form

$F$8

The triple is of split Weil type when $F$9 admits an isotropic subspace of half dimension, or equivalently when $\iota$0 has a maximal isotropic subspace of dimension

$\iota$1

2. Construction from $\iota$2 and pure spinors

The general construction in (Markman, 27 Sep 2025) starts with an abelian variety $\iota$3 endowed with real multiplication

$\iota$4

and then passes to

$\iota$5

On

$\iota$6

there is a natural symmetric bilinear pairing, and the spin group attached to $\iota$7 acts on

$\iota$8

viewed as the spin representation in the sense of Chevalley, Mukai, Polishchuk, and Orlov.

The bridge from real multiplication to CM-action uses two pieces of data: an $\iota$9-bilinear polarization class

$K/F$0

and an element $K/F$1 such that

$K/F$2

From this, the authors construct a maximal isotropic subspace

$K/F$3

as the pure spinor associated to $K/F$4. Concretely,

$K/F$5

defines an even pure spinor, and the corresponding isotropic subspace is computed as

$K/F$6

for a suitable $K/F$7 in the spin group.

The resulting $K/F$8-action on $K/F$9 is defined by letting $K$0 act as multiplication by $K$1 on $K$2 and by $K$3 on $K$4, where $K$5 is complex conjugation in $K$6. This yields an embedding

$K$7

or equivalently into the Hodge endomorphisms of $K$8. The $K$9-eigenspace decomposition of $HW(A,\eta)$0 is thereby matched with the isotropic decomposition coming from $HW(A,\eta)$1, making $HW(A,\eta)$2 an abelian variety of Weil type (Markman, 27 Sep 2025).

3. The split condition and the position of Weil classes in the Hodge ring

Within the polarized setting, split Weil type is the condition that the $HW(A,\eta)$3-hermitian form attached to the polarization is split. In the formulation of (Markman, 27 Sep 2025), this is not merely a numerical condition on signature; it is the existence of a maximal isotropic $HW(A,\eta)$4-subspace of half rank. This distinguishes split Weil type from the more general Weil-type condition, which only requires balanced Hodge multiplicities for the $HW(A,\eta)$5-action.

For the generic triple in the relevant moduli space, the Hodge ring in middle degree decomposes as

$HW(A,\eta)$6

Thus the Weil classes form the complementary “new” Hodge classes not generated by divisors. A plausible implication is that the split condition isolates the part of the Hodge ring where divisor-generated algebraicity is insufficient and genuinely new cycle constructions are required.

The paper "Cycles on abelian $HW(A,\eta)$7-folds of Weil type from secant sheaves on abelian $HW(A,\eta)$8-folds" (Markman, 5 Feb 2025) treats the imaginary quadratic case in a closely related but not identical language. It develops Weil type via a rational $HW(A,\eta)$9-secant plane in the even spin representation and proves that the induced hermitian form has signature $K$0, with discriminant

$K$1

That paper does not develop a separate formal definition labeled split Weil type, and explicitly identifies explicit signature and discriminant control as the closest analogue of a split condition (Markman, 5 Feb 2025). This marks an important terminological distinction: in (Markman, 27 Sep 2025), split Weil type is a formal condition on the hermitian form, whereas in (Markman, 5 Feb 2025) the nearest parallel is the special oriented secant-plane situation.

4. Secant sheaves, semiregularity, and deformation

The principal strategy for proving algebraicity in (Markman, 27 Sep 2025) is deformation-theoretic and categorical. One constructs a coherent sheaf $K$2 on $K$3 with two properties:

1. $K$4 is semi-regular.
2. Its normalized Chern character

$K$5

remains of Hodge type under every deformation of the polarized Weil-type variety.

The formal input is the Semi-regularity Theorem of Buchweitz–Flenner: if $K$6 is semi-regular and $K$7 remains of Hodge type in a deformation family, then $K$8 deforms locally, hence the relevant Chern classes remain algebraic. The problem is therefore reduced to producing sheaves whose normalized Chern character has the required deformation-invariance.

This is achieved through secant constructions. One chooses coherent sheaves $K$9 on $F$0 whose Chern characters lie in a secant subspace

$F$1

and then forms

$F$2

where $F$3 is Orlov’s Fourier–Mukai equivalence

$F$4

The normalized class $F$5 is shown to be $F$6-invariant, where $F$7 is the special Mumford–Tate group attached to the secant datum. Its degree-$F$8 part decomposes as

$F$9

with

$\eta:K\hookrightarrow \operatorname{End}^0(A)$00

and $\eta:K\hookrightarrow \operatorname{End}^0(A)$01 lying in the algebra generated by divisor classes. If the linear-independence condition holds, then $\eta:K\hookrightarrow \operatorname{End}^0(A)$02, giving a nonzero Weil class. Because $\eta:K\hookrightarrow \operatorname{End}^0(A)$03 is one-dimensional over $\eta:K\hookrightarrow \operatorname{End}^0(A)$04 in the relevant sense, one nonzero algebraic class suffices to conclude that all Weil classes are algebraic (Markman, 27 Sep 2025).

The sixfold case in (Markman, 5 Feb 2025) implements this mechanism concretely. There, for a genus $\eta:K\hookrightarrow \operatorname{End}^0(A)$05 Jacobian example, the paper constructs sheaves

$\eta:K\hookrightarrow \operatorname{End}^0(A)$06

with

$\eta:K\hookrightarrow \operatorname{End}^0(A)$07

for $\eta:K\hookrightarrow \operatorname{End}^0(A)$08 the real and imaginary parts of $\eta:K\hookrightarrow \operatorname{End}^0(A)$09, and shows that the corresponding

$\eta:K\hookrightarrow \operatorname{End}^0(A)$10

is $\eta:K\hookrightarrow \operatorname{End}^0(A)$11-invariant and remains of Hodge type under every deformation of $\eta:K\hookrightarrow \operatorname{End}^0(A)$12 as a polarized abelian variety of Weil type (Markman, 5 Feb 2025).

5. Imaginary quadratic specialization and low-dimensional consequences

When $\eta:K\hookrightarrow \operatorname{End}^0(A)$13, the CM-field becomes an imaginary quadratic field

$\eta:K\hookrightarrow \operatorname{End}^0(A)$14

and $\eta:K\hookrightarrow \operatorname{End}^0(A)$15. In this case, (Markman, 27 Sep 2025) specializes the general CM-field framework to the classical setting of imaginary quadratic multiplication and surveys how the secant-sheaf strategy was used to prove algebraicity results in low dimensions.

The main theorem quoted there is:

Theorem 1.5.1. The Weil classes for abelian fourfolds of Weil type and abelian sixfolds of split Weil type with complex multiplication by a quadratic imaginary number field $\eta:K\hookrightarrow \operatorname{End}^0(A)$16 are algebraic.

The proof proceeds in two steps. First, the secant sheaves and semi-regularity argument establish algebraicity for all polarized abelian sixfolds of split Weil type. Second, multiplicativity of the discriminant under products is used to pass from arbitrary abelian fourfolds of Weil type to sixfolds with split discriminant by multiplying with a suitable polarized abelian surface; algebraicity in dimension $\eta:K\hookrightarrow \operatorname{End}^0(A)$17 then forces algebraicity in dimension $\eta:K\hookrightarrow \operatorname{End}^0(A)$18 (Markman, 27 Sep 2025).

The paper further states that the algebraicity of the Weil classes on all abelian fourfold of Weil type follows, and concludes: $\eta:K\hookrightarrow \operatorname{End}^0(A)$19 The closely related paper (Markman, 5 Feb 2025) presents the sixfold theorem in the discriminant $\eta:K\hookrightarrow \operatorname{End}^0(A)$20 case for all imaginary quadratic number fields, deduces algebraicity for all abelian fourfolds of Weil type by a degeneration argument of C. Schoen, and concludes that the Hodge conjecture for abelian fourfolds follows (Markman, 5 Feb 2025). Together, these results place split Weil type at the center of the current low-dimensional evidence for the Hodge conjecture in the abelian setting.

6. Related meanings, adjacent constructions, and terminological scope

The expression split Weil type is not uniform across the literature. In the recent abelian-variety papers, it is tied either to a split hermitian form or to trivial discriminant; in several neighboring literatures, the words split and Weil occur together in unrelated representation-theoretic or automorphic senses.

Setting	Meaning of “split”	Source
Polarized abelian varieties with CM-field action	Hermitian form contains a maximal isotropic subspace of half rank	(Markman, 27 Sep 2025)
Imaginary quadratic Weil type via secant planes	No separate formal definition; closest analogue is signature $\eta:K\hookrightarrow \operatorname{End}^0(A)$21 and discriminant $\eta:K\hookrightarrow \operatorname{End}^0(A)$22	(Markman, 5 Feb 2025)
Fourfolds with trivial discriminant	“Split” reflected by trivial discriminant or standard split hermitian class	(Geemen, 2022)

Several papers elaborate neighboring aspects of this picture. "Fourfolds of Weil type and the spinor map" (Geemen, 2022) studies abelian fourfolds of Weil type with trivial discriminant and explains that the “split” nature is reflected by the fact that the resulting family is a complete unitary family with Mumford–Tate group $\eta:K\hookrightarrow \operatorname{End}^0(A)$23, and that the hermitian form can be placed in a standard split class. The same paper connects these fourfolds to a rank-six weight-two Hodge structure and a Kuga–Satake variety, with

$\eta:K\hookrightarrow \operatorname{End}^0(A)$24

"Weil Classes and Decomposable Abelian Fourfolds" (Geemen, 2021) does not formalize split Weil type, but indicates that the decomposable geometry of $\eta:K\hookrightarrow \operatorname{End}^0(A)$25 may reasonably be regarded as a split or decomposable Weil-type structure in modern language. "Abelian Fourfolds of Weil type and certain K3 Double Planes" (Lombardo et al., 2012) similarly does not use the phrase as a standard named notion, but identifies a split phenomenon in the isogeny sense: when the quaternion algebra $\eta:K\hookrightarrow \operatorname{End}^0(A)$26 is split, the associated abelian fourfold is isogenous to a square $\eta:K\hookrightarrow \operatorname{End}^0(A)$27.

Outside Hodge theory, the phrase has distinct meanings. In "A Generalized Weil Representation for the finite split orthogonal group $\eta:K\hookrightarrow \operatorname{End}^0(A)$28" (Gajardo, 2013), the relevant object is a generalized Weil representation for a finite split orthogonal group, not a Weil-type abelian variety. In "A minimal even type of the 2-adic Weil representation" (Wood, 2012), the terminology concerns a minimal even type whose Hecke algebra is that of the split adjoint orthogonal group $\eta:K\hookrightarrow \operatorname{End}^0(A)$29. In automorphic theory, "A new regularized Siegel-Weil type formula, part I" (Ginzburg et al., 2022) and "Poles, Residues and Siegel-Weil Identities of Degenerate Eisenstein Series on Split Exceptional Groups of Type $\eta:K\hookrightarrow \operatorname{End}^0(A)$30" (Halawi et al., 2023) use Weil type in the sense of Siegel–Weil analogies on split groups. These usages are terminologically adjacent but mathematically separate from split Weil type for polarized abelian varieties.

In the current arithmetic-geometric literature, the most precise and systematic definition is therefore the one in (Markman, 27 Sep 2025): a polarized abelian variety of Weil type with CM action is of split Weil type when its associated $\eta:K\hookrightarrow \operatorname{End}^0(A)$31-hermitian form is split, and this condition is the structural input for the secant-sheaf approach to algebraicity of Hodge Weil classes.