Generalized Thomas Sections and Density-Corrected Residue Selectors

Hongmin Li (Institute of Science Tokyo; The University of Tokyo). Developed with OpenAI Codex and GPT Pro under the author's direction. Submitted through the rStation account lihongmin.

1. Introduction

Thomas reformulated the rational Hodge conjecture in terms of the homology of sufficiently ample divisors with ordinary double points [1]. Saito made the relation-to-class direction precise: a type $(0,0)$, unipotent relation among local vanishing cycles maps to a primitive ambient Hodge class, and a primitive Hodge class restricts nontrivially exactly when it pairs with one of these relation classes [2].

This reformulation contains two logically different problems:

1. given a singular section, compute its relation space and its primitive image;
2. starting from a Hodge class, construct a singular section whose relation image detects it.

The first problem is geometric and can be completely explicit. The second is an inverse problem and is where period ideals, Hodge loci and support conjectures enter. The present paper gives three results about this distinction.

First, node data depend on normal-bundle information not determined by the cohomology class of the support. Second, an explicit plane-containing cubic fourfold shows that the complete Thomas--Saito chain can close in a controlled example. Third, a smooth cubic counterexample shows that the converse implication from a linear period annihilator to a singular support is false as stated. The final local calculation identifies a corrected selector for a generalized $D_4$ block. Appendix A records the related quadratic-polarization transfer criterion and a conditional smooth-support construction for known Weil classes. The latter is not a new case of the Hodge conjecture.

The paper is therefore a result about the scope and correct formulation of the Thomas--Saito mechanism. It is not a proof of the Hodge conjecture and does not assert that every primitive Hodge class is detected by the constructions below.

2. Thomas data and the relation-to-class map

Let $X$ be a smooth projective variety of dimension $2n$, let $L$ be ample, and let $Z^n\subset X$ be a smooth middle-dimensional subvariety. For a section

$$ s\in H^0(X,I_Z\otimes L^N), \qquad D=V(s), $$

the normal derivative is

$$ \sigma=ds|_Z\in H^0\left(Z,N_{Z/X}^{\vee}\otimes L^N|_Z\right). $$

If $\sigma$ is transverse to the zero section, the singular scheme of $D$ along $Z$ is the reduced zero scheme $\Sigma=Z_Z(\sigma)$ of ordinary double points.

The zero-point formula gives

$$ [\Sigma]=c_n\left(N_{Z/X}^{\vee}\otimes L^N|_Z\right)\cap[Z] $$

and hence

$$ \ell(\Sigma)=\int_Z\sum_{j=0}^{n}(-1)^j c_j(N_{Z/X})(N\ell|_Z)^{n-j}, \qquad \ell=c_1(L). \tag{2.1} $$

The normal bundle occurs explicitly. This already obstructs any claim that the node scheme is a function only of the ambient class $[Z]$.

Proposition 2.1 (node data are not cohomological support data)

Take $X=\mathbf P^4$, $L=\mathcal O(1)$, and general smooth complete-intersection surfaces

$$ Z_{1,6}=V(f_1,f_6), \qquad Z_{2,3}=V(f_2,f_3). $$

They have the same class $6H^2$, but for a general containing hypersurface of degree $N>6$, the corresponding node lengths are

$$ \ell(\Sigma_{1,6})=6(N-1)(N-6), \qquad \ell(\Sigma_{2,3})=6(N-2)(N-3). $$

Thus the node length is not determined by $(X,L,N,[Z])$.

Proof

The normal bundles are $\mathcal O(1)\oplus\mathcal O(6)$ and $\mathcal O(2)\oplus\mathcal O(3)$, respectively. Substitution into (2.1) gives the two formulas. Their difference has absolute value $12N$, hence is nonzero. $\square$

Even a fixed support does not determine the node positions. For

$$ X=\mathbf P^1\times\mathbf P^1, \qquad Z=\{t\}\times\mathbf P^1, \qquad L=\mathcal O(1,1), $$

write a general divisor containing $Z$ as $D=Z\cup C$, with $C\in|\mathcal O(N-1,N)|$. The node scheme is $Z\cap C$, and the restricted linear system is the complete system $|\mathcal O_{\mathbf P^1}(N)|$. Its support can therefore vary through any general reduced divisor of degree $N$ on $Z$.

The correct invariant is instead the image of the vanishing-cycle relation space. For a nodal section $Y_0$, let $R(Y_0)$ denote the kernel of the map from the direct sum of local vanishing homology groups to the homology of a nearby smooth fibre. Saito constructs

$$ \Gamma_{Y_0}:R(Y_0)_1^{(0,0)}\longrightarrow H_{2n}(X,\mathbf Q(n))_{\mathrm{prim}}, \qquad \beta\longmapsto\gamma_\beta. \tag{2.2} $$

For ordinary double points, local vanishing cycles are of type $(0,0)$ and have unipotent monodromy. Therefore, for a primitive Hodge class $\zeta$,

$$ \zeta|_{Y_0}\ne0 \quad\Longleftrightarrow\quad \langle\zeta,\gamma_\beta\rangle\ne0 \text{ for some }\beta\in R(Y_0)_1^{(0,0)}. \tag{2.3} $$

This is a one-way usable construction from a section to a class. The global Hodge conjecture would follow from the much stronger assertion that the spans of all such $\gamma_\beta$ generate every primitive Hodge space; (2.3) does not provide that generating theorem.

3. A positive cubic-fourfold control model

We next verify the complete chain in a model where the target class is already known to be algebraic. In $\mathbf P^5$, set

$$ P=V(x_3,x_4,x_5) $$

and define

$$ \begin{aligned} q_3&=x_0^2-x_1^2+x_0x_3+x_1x_4+x_2x_5+x_3^2+x_4x_5,\\ q_4&=x_1^2-x_2^2+x_0x_4+x_1x_5+x_2x_3+x_4^2+x_3x_5,\\ q_5&=x_0^2+x_1^2+x_2^2+x_0x_5+x_1x_3+x_2x_4+x_5^2+x_3x_4, \end{aligned} $$

with

$$ X=V(F),\qquad F=x_3q_3+x_4q_4+x_5q_5. $$

Then $P\subset X$. The hyperplane $x_5=0$ cuts a cubic threefold $Y=X\cap V(x_5)$.

Theorem 3.1 (closed Thomas--Saito chain for a plane class)

Let $\delta=[P]_{\mathrm{prim}}=[P]-h^2/3$. Then:

1. $X$ is smooth;
2. the linear part of the period-annihilator is$$ \widetilde I(\delta)_1=I(P)_1=\langle x_3,x_4,x_5\rangle; $$
3. $Y$ has exactly four ordinary double points$$ p_{\varepsilon,\eta}=[\varepsilon:1:\eta:0:0], \qquad \varepsilon,\eta\in\{\pm1\}; $$
4. the four nodes impose only three independent conditions on linear forms, so the defect is one-dimensional, with relation $\lambda=(1,-1,-1,1)$;
5. the corresponding nonzero Saito image is a rational multiple of $\delta$, and after the natural scale normalization $\gamma_\beta=\delta$, with $\langle\delta,\gamma_\beta\rangle=\delta^2=8/3$.

Proof

The affine homogeneous Jacobian scheme of $F$ is zero-dimensional, supported at the origin; its projectivization is empty, so $X$ is smooth. On $P$, the normal derivative of $Y$ is

$$ (q_3|_P,q_4|_P)=(x_0^2-x_1^2,x_1^2-x_2^2), $$

whose zero scheme consists of the four displayed points. The local Hessian determinant is nonzero at each point, so they are ordinary double points and exhaust the Jacobian scheme.

The evaluation matrix of $1,x_0,x_2$ at the four nodes has rank three and left kernel generated by $(1,-1,-1,1)$. Hence the linear defect is one. Saito's construction maps the unique nonzero relation to the primitive class supported by $P$, up to the independent choice of local vanishing-cycle scales. With the chosen normalization this is $\delta$. Finally,

$$ \delta^2=([P]-h^2/3)^2=3-2/3+1/3=8/3, $$

so the pairing is nonzero. $\square$

The normalization point matters: the relation vector itself is not canonical before local vanishing-cycle generators are scaled. The invariant conclusion is nonzero proportionality, not an unqualified equality of two canonically normalized classes.

4. A smooth counterexample to linear-support inversion

Consider

$$ f(u,v)=u(u-v)(u+v)=u^3-uv^2, $$

and the rational smooth cubic fourfold

$$ X=V\bigl(f(x_0,x_1)+f(x_2,x_3)+f(x_4,x_5)\bigr). $$

Let $Z$ be the integral join of the three zero-dimensional cycles $Z_i=[(0:1)]+4[(1:1)]$ on the cubic $V(f)\subset\mathbf P^1$. It is an integer combination of eight planes in $X$.

The join residue formula [3], together with the complete-intersection period normalization [4], gives the primitive period polynomial, up to a nonzero rational scalar,

$$ P_\delta=(6x_0+x_1)(6x_2+x_3)(6x_4+x_5). $$

The Jacobian-ring residue pairing identifies the period-annihilator with the colon ideal [5], and exact Gröbner computation gives

$$ I(\delta)=\langle x_0-2x_1,x_2-2x_3,x_4-2x_5, x_1^2,x_3^2,x_5^2\rangle. \tag{4.1} $$

Thus

$$ I(\delta)_1=\langle x_0-2x_1,x_2-2x_3,x_4-2x_5\rangle. $$

However, taking $g=x_0-2x_1$ gives

$$ X\cap V(g)=V\bigl(6x_1^3+f(x_2,x_3)+f(x_4,x_5)\bigr)\subset\mathbf P^4, $$

which is a smooth cubic threefold: its five partial derivatives have no common projective zero. The common zero plane of all three linear generators is not contained in $X$, since

$$ F|_{x_0=2x_1,\,x_2=2x_3,\,x_4=2x_5} =6(x_1^3+x_3^3+x_5^3)\ne0. $$

Since a smooth cubic threefold has $H_4(Y,\mathbf Q)\simeq\mathbf Q$, and its image in the cubic fourfold contributes only the ambient class, the primitive part of the smooth section cannot contain $\delta$. Therefore a nonzero linear period annihilator does not imply that the corresponding hyperplane section is singular, nodal, or supports the class. This supplies an explicit counterexample to the literal linear-support inversion principle.

The example is not a numerical anomaly: the smooth-section condition is Zariski open in the projective plane $\mathbf P(I(\delta)_1)$. It also shows why the Hilbert function of the Artinian annihilator, by itself, cannot be treated as a support theorem.

5. Generalized $D_4$ sections and the density correction

Let $V=V(F)\subset\mathbf P^4$ be a cubic threefold with isolated ordinary double points and $D_4$ singularities. Dimca's local-to-global map [6] (Theorem 2.4 and Corollary 2.5) is

$$ T^2:S_1\longrightarrow\bigoplus_{p\in\operatorname{Sing}V}H^4_p(V), $$

and its cokernel is primitive $H^4(V)$. The polynomial degree is $(4-2)3-5=1$. Since these singularities are rational, $b_2(V)=1$, so the cokernel dimension equals the topological defect $b_4(V)-b_2(V)$; see also [7] (Section 3).

For the standard $D_4$ equation

$$ g=U^2+V^2+X^3-XY^2, \qquad \operatorname{wt}(U,V,X,Y)=(3,3,2,2), $$

the relevant local block is

$$ (M_g)_2=\langle X,Y\rangle, \qquad M_g=\mathbf C[X,Y]/(3X^2-Y^2,XY). $$

Indeed, $\deg g=6$ and $2\deg g-\sum w_i=2$. Only the weight-two component of a residue numerator contributes to this block [6] (equation (1.7)).

Theorem 5.1 (density-corrected local selector). Write the affine equation near $p$ as $f$, with volume density $b$, and choose a contact normalization

$$ f(\psi(z))=a(z)g(z),\qquad a(0)\ne0. $$

Put $B(z)=b(\psi(z))\det D\psi(z)a(z)^{-2}$. In the indicated local basis,

$$ T^2_p(h)=\left[B(0)\,dh_p(D\psi_0z)|_{\langle X,Y\rangle} +h(p)B_{[2]}(z)\right]\in(M_g)_2. $$

Here $B_{[2]}$ denotes the weight-two component. In particular a nonzero zero-order correction can occur even when the Hessian-null derivative of $h$ vanishes.

Proof. The pole-two residue form pulls back to

$$ \frac{h(\psi(z))B(z)}{g(z)^2}\,dU\wedge dV\wedge dX\wedge dY. $$

Take its numerator component of weight two. The variables $U,V$ have weight three; quadratic terms in $X,Y$ have weight four. The only contributions are therefore the displayed linear derivative and the constant $h(p)$ times $B_{[2]}$. Passing to $(M_g)_2$ gives the formula. A change of local trivialization changes its two displayed summands together, leaving the residue class unchanged. $\square$

The correction is nonzero in the explicit cubic

$$ F=ZU^2+XU^2+ZV^2+X^3-XY^2. $$

Its point $p=[1:0:0:0:0]$ is $D_4$. In the chart $Z=1$, set $U'=\sqrt{1+X}U$. The equation becomes $g$, while the density becomes $(1+X)^{-1/2}=1-X/2+\cdots$. For $h=Z$,

$$ dh_p|_{\ker\operatorname{Hess}f}=0, \qquad T^2_p(Z)=[-X/2]\ne0. $$

The four other singularities are $[0:1:\varepsilon i:0:\eta]$, with $\varepsilon,\eta\in\{\pm1\}$; their Hessian determinants in the $U=1$ chart are $-16$. Thus the cubic has only the isolated singularities allowed above. The supplement checks the Jacobian equations, local Tjurina dimension four, and the nonzero Milnor-algebra class.

For a complementary geometric example, return to the smooth cubic of Section 4 and put $\ell_j=x_{2j}-2x_{2j+1}$. The section $Y=X\cap V(\ell_0+\ell_1)$, after eliminating $x_0$, has equation

$$ G=(x_1+x_3)Q+x_4(x_4-x_5)(x_4+x_5), $$

where

$$ Q=6x_1^2-11x_1x_2+16x_1x_3+6x_2^2-13x_2x_3+8x_3^2. $$

Its two singular points are

$$ [-r:1+2r:r:0:0],\qquad r=\frac{-11\pm\sqrt{13}}{18}. $$

At each, the binary part in $(x_1,x_2,x_3)$ has a nondegenerate Hessian in local coordinates; splitting off those two variables leaves $x_4^3-x_4x_5^2$. Each point is $D_4$. Their Tjurina lengths sum to the global projective Jacobian length eight, so there are no other singularities. The Morse normalization depends only on the two Morse variables, whose weights are three. Consequently its density has no weight-two part, and there is no zero-order correction at these two points. In the basis $(x_1,x_2,x_3,x_4,x_5)$, compatible local bases give

$$ T^2=\begin{pmatrix} 0&0&0&1&0\\0&0&0&0&1\\ 0&0&0&1&0\\0&0&0&0&1 \end{pmatrix}. $$

It has rank two and cokernel dimension two. Thus $b_4(Y)=3$ and the defect is two.

The planes

$$ P_r=V(x_0+x_2,x_1+x_3,x_4-rx_5),\qquad r=-1,0,1, $$

lie in $Y$ and sum to its hyperplane section. Their primitive Gram matrix in $X$ is

$$ \frac13\begin{pmatrix}8&-4&-4\\-4&8&-4\\-4&-4&8\end{pmatrix}. $$

Its rank is two: the plane self-intersection is three and distinct planes meet in a line with intersection number $-1$. They therefore span the extra rational $H_4(Y)$ modulo the ambient class. For the integral cycle $Z$ of Section 4, $\deg Z=125$ and direct plane intersections give

$$ \langle[Z]_{\mathrm{prim}},[P_r]_{\mathrm{prim}}\rangle =\frac13(130,52,-182). $$

The supplement independently gives proportional residue pairings $(-5,-2,7)$. The nonzero pairing with this Tate extra part gives a nonzero Saito relation image detecting the target class [2] (Theorem 1). This is a generalized Thomas section with two $D_4$ points, rather than an ordinary nodal section.

6. Consequences and limitations

The results establish the following hierarchy:

$$ \begin{array}{c} \text{singular section + vanishing-cycle relation}\\ \Downarrow\\ \text{primitive Hodge class via Saito}$$4pt] \not\Uparrow\$$-2pt] \text{linear period annihilator alone}. \end{array} $$

The positive cubic model proves that the full chain can close when the target class is known. The fake linear-cycle model proves that the linear annihilator cannot, without additional singularity or support hypotheses, construct the required section. The generalized $D_4$ calculation shows how the local selector, with its density term included, computes the defect when the singularity is actually present.

What remains open is a constructive theorem that starts from a general primitive Hodge class and produces a singular section and a relation with nonzero Saito pairing. Neither the counterexample nor the corrected selector supplies that theorem.

Appendix A. Quadratic transfer and conditional smooth Weil support

The quadratic-polarization and Weil-support material is included here because it concerns the forward construction of a supporting cycle and does not give a separate new case of the Hodge conjecture. In particular, van Geemen--Rapagnetta already describe the normal bundle and its positive-twist cycle construction [8] (Introduction). The distinct purpose here is to state the exact algebraicity interface and to make the additional local-freeness hypothesis for the stronger smooth representative explicit.

A.1. The type correction for the three Weil constructions

The comparison uses [9], [10], [11], [8].

Let $X$ be a $K3^{[3]}$-type hyperkähler sixfold and let $B$ be a Weil-type abelian fourfold. Suppose a Kummer variety $K(B)$ is embedded in $X$. The three constructions mentioned above should be separated as follows.

A.1.1 Secant sheaves

For $n=3$, Markman's explicit semiregular secant sheaf is placed on

$$ A=X_0\times\widehat X_0, $$

where $X_0$ is an abelian threefold with the relevant real multiplication. The normalized Chern character produces a class

$$ \kappa_3(E)\in H^6(A,\mathbf Q), $$

and its field translates together with the cubic polarization span the relevant Weil subspace. This is a six-dimensional construction. The resulting statement for Weil fourfolds is not an identification of $\kappa_3(E)$ with a specified class on an independently chosen fourfold $B$; it uses additional degeneration and classification input.

The secant-sheaf construction is consequently a mechanism for producing Hodge classes and, under its stated semiregularity/algebraicity hypotheses, algebraic classes. It is not automatically a coordinate description of the Kummer class below.

A.1.2 The OG6/Kuga--Satake correspondence

Let $K$ be the singular OG6-type variety in the Floccari--Fu construction and let $Z_K$ be the associated $K3^{[3]}$-type double cover. The Kummer locus in $\operatorname{Sing}(K)$ gives an algebraic correspondence whose action is

$$ \Psi=p_*q^*j^*: H^2_{\mathrm{tr}}(Z_K,\mathbf Q)\hookrightarrow H^2(B_K,\mathbf Q). $$

This is a map of Hodge structures induced by a correspondence. It does not by itself preserve cup products and therefore does not, by itself, turn the inverse polarization on $H^2_{\mathrm{tr}}(Z_K)$ into an algebraic class in $H^4(B_K)$.

A.1.3 The hyperkähler $c_2$-class

Van Geemen--Rapagnetta use an embedding $K(B)\hookrightarrow X$. On the transcendental part, the induced weight-two map may be written

$$ \psi:T_X\hookrightarrow H^2(B,\mathbf Q). $$

Let $q_X^\vee\in\operatorname{Sym}^2T_X$ be the inverse Beauville--Bogomolov tensor. Its image under cup product and $\psi$ is

$$ w_X=\operatorname{cup}_B\bigl(\operatorname{Sym}^2(\psi)(q_X^\vee)\bigr) \in H^4(B,\mathbf Q). $$

For the relevant $K3^{[3]}$-type sixfold, the image of $c_2(T_X)$ is a nonzero rational multiple of this class. The resulting $w_X$ is a nonzero Hodge--Weil class, and the representation-theoretic action of $K$ generates the Weil subspace.

The common content of the second and third constructions is therefore

$$ \boxed{ \text{polarized K3-type }T_X\hookrightarrow H^2(B,\mathbf Q) \Longrightarrow \text{a canonical quadratic Weil Hodge tensor in }H^4(B,\mathbf Q).} $$

The algebraicity interface must be stated separately.

A.2. A quadratic transfer criterion

We formulate the elementary criterion used repeatedly below.

Proposition A.2 (quadratic transfer)

Let $X$ and $Y$ be smooth projective varieties. Let

$$ T\subset H^2(X,\mathbf Q) $$

be a polarized sub-Hodge structure with inverse polarization tensor

$$ Q_T\in T\otimes T\subset H^4(X\times X,\mathbf Q). $$

Suppose an algebraic correspondence $\Gamma\in CH^*(X\times Y)_{\mathbf Q}$ induces a Hodge map

$$ \psi:T\longrightarrow H^2(Y,\mathbf Q). $$

Define

$$ w_{\Gamma,T}:=\Delta_Y^*(\psi\otimes\psi)(Q_T)\in H^4(Y,\mathbf Q). $$

Then $w_{\Gamma,T}$ is a Hodge class. It is algebraic if either of the following additional conditions holds:

1. $Q_T$ is algebraic as a Künneth class on $X\times X$; or
2. $\psi$ is induced by a geometric pullback that preserves the relevant cup product, and $\operatorname{cup}_X(Q_T)\in H^4(X,\mathbf Q)$ is algebraic.

Proof

The tensor $Q_T$ is of Hodge type $(2,2)$ on $X\times X$, and $\psi\otimes\psi$ is a morphism of Hodge structures. Pullback by the algebraic diagonal $\Delta_Y$ preserves Hodge type, so $w_{\Gamma,T}$ is a Hodge class.

Under condition 1, the algebraic correspondence $\Gamma\times\Gamma$ sends the algebraic Künneth class $Q_T$ to an algebraic class on $Y\times Y$, and $\Delta_Y^*$ preserves algebraic cycles. Under condition 2,

$$ w_{\Gamma,T}=\psi^{(4)}\bigl(\operatorname{cup}_X(Q_T)\bigr), $$

and geometric pullback sends algebraic classes to algebraic classes. $\square$

The converse is deliberately absent. An algebraic correspondence can induce a Hodge map on $H^2$ without being a ring homomorphism, and an inverse polarization tensor need not be algebraic on the source square. This is the precise reason why the Floccari--Fu weight-two correspondence and the van Geemen--Rapagnetta pullback should not be conflated.

A.3. The Kummer normal bundle and an algebraic lift

We now specialize to a Kummer--hyperkähler model satisfying the geometric and nonzero Weil-class conclusions of [8] (Theorems 0.2--0.3), after the isogeny specified there. These are additional geometric hypotheses; no assertion is made that every discriminant-one fourfold admits this model. Let $B$ be a Weil-type abelian fourfold of discriminant $1$, let

$$ U=B\setminus B[2], $$

and suppose there is a closed embedding

$$ i:K(B)\hookrightarrow X $$

into a projective $K3^{[3]}$-type hyperkähler sixfold. Write $f:U\to X$ for the composite of the quotient map $U\to K(B)$ with $i$. On the open Kummer locus the map is an immersion after the etale quotient map, so there is an exact sequence

$$ 0\longrightarrow T_U\longrightarrow f^*T_X\longrightarrow N_U\longrightarrow0, $$

where $N_U$ is a rank-two vector bundle. Since $T_B$ is trivial and $c_1(T_X)=0$,

$$ c_1(N_U)=0,\qquad c_2(N_U)=f^*c_2(T_X). $$

Let $j:U\hookrightarrow B$. Because $B\setminus U=B[2]$ has codimension four, the pushforward

$$ \mathcal F:=j_*N_U $$

is a rank-two reflexive sheaf [12] on $B$, locally free away from the finite set $B[2]$. The determinant extends uniquely and $c_1(\mathcal F)=0$. The localization sequence for Chow groups [13] gives

$$ CH^2(B)_{\mathbf Q}\xrightarrow{\sim}CH^2(U)_{\mathbf Q}, $$

so $c_2(N_U)$ has a unique codimension-two extension

$$ \widetilde w:=c_2(\mathcal F)\in CH^2(B)_{\mathbf Q}. $$

Its cohomology class is the nonzero Weil Hodge class

$$ \operatorname{cl}(\widetilde w)=w=f^*c_2(T_X)\in H^4(B,\mathbf Q). $$

This construction supplies an actual algebraic lift because it starts from a vector bundle on the open locus and extends the bundle reflexively. It does not assert that every Hodge class admits such an extension.

A.4. Smooth surface representatives after a positive twist

Additional hypothesis for this subsection and A.5. Assume the reflexive extension $\mathcal F$ constructed in A.3 is locally free on all of $B$. This is not implied by its being reflexive, and is not established here for the Kummer model.

Fix an ample line bundle $L$ on $B$, and put $\ell=c_1(L)$. For $m$ sufficiently large, $\mathcal F\otimes L^m$ is generated by global sections on $B$. Choose a general section $s$ which is nonzero at every point of $B[2]$. Its zero locus lies in $U$, where the sheaf is a rank-two vector bundle. Bertini's theorem for vector-bundle sections gives a smooth codimension-two subvariety

$$ Z_m=Z(s)\subset U\subset B. $$

It is a projective smooth surface. Its class is the top Chern class:

$$ \begin{aligned} [Z_m] &=c_2(\mathcal F\otimes L^m)\\ &=c_2(\mathcal F)+m\ell\,c_1(\mathcal F)+m^2\ell^2\\ &=\widetilde w+m^2\ell^2 \end{aligned} $$

in $CH^2(B)_{\mathbf Q}$, and hence

$$ \operatorname{cl}(Z_m)=w+m^2\ell^2 $$

in $H^4(B,\mathbf Q)$. For every sufficiently large $m$, this gives a smooth effective representative of the algebraic lift of the Weil class modulo a divisor square. The statement is not that $w$ itself is represented by a smooth surface.

A.5. Nodal support in the sense of Thomas

Under the additional local-freeness hypothesis of A.4, apply Thomas's nodal-section theorem to the smooth middle-dimensional surface $Z_m^2\subset B^4$. For $N$ sufficiently large, a general hypersurface

$$ D_{m,N}\in |L^N|,\qquad Z_m\subset D_{m,N}, $$

is smooth away from finitely many ordinary double points, all of which lie on $Z_m$. Thus $D_{m,N}$ is a nodal threefold.

Let $H\in|L|$ be general. Inside $B$, the surface $D_{m,N}\cap H$ has class

$$ [D_{m,N}\cap H]=N\ell^2. $$

Since $Z_m\subset D_{m,N}$, both classes are represented by four-cycles on $D_{m,N}$. Consequently,

$$ w=[Z_m]-\frac{m^2}{N}[D_{m,N}\cap H] $$

lies in the image

$$ H_4(D_{m,N},\mathbf Q)\longrightarrow H_4(B,\mathbf Q). $$

Under the stated additional hypothesis, this is a positive realization of Thomas support for the Weil Hodge class. It should be separated from the stronger and generally false expectation that the class determines its node set. The choice of $s$ and the general containing hypersurface both vary, so the nodes are auxiliary data rather than an invariant of $w$.

A.6. Scope and limitations

The appendix separates the following statements:

1. the three recent Weil constructions have a common polarized weight-two core, but are not automatically three representatives of one $H^4$-class;
2. the inverse polarization tensor has a precise quadratic transfer criterion for algebraicity;
3. in the discriminant-one Kummer--hyperkähler model, reflexive extension of the normal bundle gives an algebraic codimension-two lift of the Weil class;
4. if the reflexive extension is locally free on $B$, positive twists give smooth surface representatives modulo divisor squares, and Thomas gives nodal hypersurface support.

No assertion is made of:

• a comparison in $CH^2$ between Markman's secant class and the van Geemen--Rapagnetta class;
• that every algebraic $H^2$-correspondence preserves the quadratic tensor;
• a smooth surface representing $w$ itself without the added $m^2\ell^2$ term;
• a converse construction from an unknown Hodge class;
• a new proof of the Hodge conjecture beyond the already known discriminant-one Weil-fourfold case.

The quartic-CM determinant and semiabelic/theta--Prym mechanisms investigated for $X_{33}^{4}$ are different open problems and are not used as premises here.

Reproducibility

The supplement contains five standalone Singular programs, the corresponding outputs rerun on October 22, 2026, and an English README with commands. They check the smooth cubic control, the colon ideal and smooth counterexample, the discriminant screen, the two-$D_4$ geometry and pairings, and the density-corrected local residue matrices. The computations use rational coefficients, with a quadratic extension only for the four displayed nodes of the density example. The residue derivation and the Saito and Thomas implications are the arguments in the text, rather than consequences of a program returning a rank.

The manuscript is submitted for mathematical review; independent human verification of its complete argument is not asserted. Copyright remains with the author, subject to rStation's publication and review terms.

References

1. R. P. Thomas, “Nodes and the Hodge conjecture,” Journal of Algebraic Geometry 13 (2004), 231–252. DOI: 10.1090/S1056-3911-04-00378-9; open version: arXiv:math/0212216.

2. M. Saito, “Generalized Thomas hyperplane sections and relations between vanishing cycles,” arXiv:0806.1461.

3. J. Duque Franco and R. Villaflor Loyola, “Periods of join algebraic cycles,” arXiv:2312.17222.

4. R. Villaflor Loyola, “Periods of complete intersection algebraic cycles,” manuscripta mathematica (2022). DOI: 10.1007/s00229-021-01290-x; open version: arXiv:1812.03964.

5. H. Movasati and E. C. Sertoz, “On reconstructing subvarieties from their periods,” Rendiconti del Circolo Matematico di Palermo Series 2 (2021). DOI: 10.1007/s12215-020-00562-x; open version: arXiv:1908.03221.

6. A. Dimca, Betti numbers of hypersurfaces and defects of linear systems, Duke Mathematical Journal 60 (1990). https://doi.org/10.1215/S0012-7094-90-06010-7.

7. L. Marquand and S. Viktorova, The defect of a cubic threefold and applications to intermediate Jacobian fibrations, International Mathematics Research Notices 2025 (16), rnaf244. https://doi.org/10.1093/imrn/rnaf244; https://arxiv.org/abs/2312.05118.

8. B. van Geemen and A. Rapagnetta, “Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class,” arXiv:2607.18341.

9. E. Markman, “Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds,” arXiv:2502.03415.

10. E. Markman, “Secant sheaves on abelian $n$-folds with real multiplication and Weil classes on abelian $2n$-folds with complex multiplication,” arXiv:2509.23079.

11. S. Floccari and L. Fu, “The Hodge conjecture for Weil fourfolds with discriminant 1 via singular OG6-varieties,” arXiv:2504.13607.

12. The Stacks Project, Lemma 31.13.12, Tag 0EBJ, reflexive extension across codimension at least two.

13. The Stacks Project, Lemma 82.15.2, Tag 0EQ5, localization for Chow groups.