The class $\mathrm{cl}(Z)$ of an algebraic cycle is a Hodge class: rational and of type $(p,p)$

Read-back: isHodgeClass_of_isCycleClass

Statement

Let $N, n, p$ be natural numbers with $p \le n$. Let $X$ and $Z$ be two "projective varieties in $\mathbb{P}^N$" in the sense defined below. Assume:

1. $X$ is nonsingular of dimension $n$: for every nonzero $v$ in the affine cone of $X$, the Zariski tangent space $T_vX \subseteq \mathbb{C}^{N+1}$ (defined below) has complex dimension exactly $n+1$;
2. the affine cone of $Z$ (minus the origin) is contained in the affine cone of $X$ (minus the origin): $C(Z) \subseteq C(X)$;
3. $Z$ is irreducible (defined below; this includes $C(Z) \neq \varnothing$);
4. $\dim Z = n - p$, where $\dim$ is the specific quantity defined below (natural-number subtraction, but exact here because $p \le n$);
5. $c$ is an element of $H^{2p}(X)$ (the specific cohomology group defined below) which is a cycle class of $Z$ relative to $(n,p)$ (defined below).

Then $c$ is a Hodge class of degree $p$ on $X$, i.e. $c$ is rational and of type $(p,p)$, both in the specific senses defined below.

Everything that follows unpacks the words in italics down to standard notions.

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Ambient objects

• $V := \mathbb{C}^{N+1}$, with coordinates indexed $0,\dots,N$.
• $\mathcal{A} := \mathbb{C}^{(N+1)\times(N+1)}$, the complex $(N+1)\times(N+1)$ matrices with the Hilbert–Schmidt (Frobenius) inner product $\langle a,b\rangle_{\mathbb{C}} = \sum_{i,j}\overline{a_{ij}}\,b_{ij}$; it is also regarded as a real Euclidean space of real dimension $2(N+1)^2$ with $\langle a,b\rangle_{\mathbb{R}} := \operatorname{Re}\langle a,b\rangle_{\mathbb{C}}$.
• $\pi : V \to \mathcal{A}$, $\pi(v)_{ij} := v_i\,\overline{v_j}\,/\,\sum_k |v_k|^2$. For $v \ne 0$ this is the orthogonal projector $vv^*/\|v\|^2$ onto the line $\mathbb{C}v$; for $v = 0$ the division-by-zero convention gives $\pi(0) = 0$ (irrelevant below, since $0$ is always excluded).
• For $P, A \in \mathcal{A}$: $J_P(A) := i\,(AP - PA)$.
• For $P \in \mathcal{A}$, $z \in \mathbb{C}$, $A \in \mathcal{A}$: $\mu_P(z, A) := (\operatorname{Re} z)\,A + (\operatorname{Im} z)\,J_P(A)$.

Varieties

A projective variety in $\mathbb{P}^N$ is a set $E$ of polynomials in $\mathbb{C}[x_0,\dots,x_N]$ such that each $Q \in E$ is homogeneous of some degree $d \in \mathbb{N}$ (the zero polynomial qualifies, being homogeneous of every degree; nonzero constants qualify with $d = 0$). No finiteness, radicality or nontriviality is required. For such an $X$ with defining set $E_X$:

• Cone. $C(X) := \{v \in V : v \ne 0,\ Q(v) = 0 \text{ for all } Q \in E_X\}$.
• Points. $\mathrm{pts}(X) := \pi(C(X)) \subseteq \mathcal{A}$.
• Vanishing ideal. $I(X) := \{Q \in \mathbb{C}[x_0,\dots,x_N] : Q(v) = 0 \text{ for all } v \in C(X)\}$ (all polynomials, not only homogeneous ones).
• Tangent cone. For any $v \in V$ (not only $v \in C(X)$),

$$T_vX := \bigcap_{Q \in I(X)} \ker\big(dQ_v\big), \qquad dQ_v(u) := \sum_{i=0}^{N} \frac{\partial Q}{\partial x_i}(v)\,u_i ,$$

a complex linear subspace of $V$.

• Nonsingular of dimension $n$. $\forall v \in C(X)$: $\dim_{\mathbb{C}} T_vX = n+1$. (Vacuously true if $C(X) = \varnothing$.)
• Tangent space at $P \in \mathcal{A}$. $\tau_P X$ is the real linear span in $\mathcal{A}$ of the union, over all $v \in C(X)$ with $\pi(v) = P$, of the images $D\pi_v\big(T_vX\big)$, where $D\pi_v : V \to \mathcal{A}$ is the real Fréchet derivative of $\pi$ at $v$ and $T_vX$ is viewed as a real subspace of $V \cong \mathbb{R}^{2N+2}$. If $P \notin \mathrm{pts}(X)$ then $\tau_PX = \{0\}$.
• Irreducible. $C(X) \neq \varnothing$, and for all sets $S_1, S_2$ of polynomials (arbitrary, not necessarily homogeneous), if $C(X) \subseteq \mathcal{Z}(S_1)\cup\mathcal{Z}(S_2)$ then $C(X) \subseteq \mathcal{Z}(S_1)$ or $C(X) \subseteq \mathcal{Z}(S_2)$, where $\mathcal{Z}(S) := \{v \in V : Q(v) = 0\ \forall Q\in S\}$.
• Dimension.

$$\dim X := \Big(\min_{v \in C(X)} \dim_{\mathbb{C}} T_vX\Big) - 1,$$

with natural-number conventions: the minimum of the empty set is $0$, and subtraction truncates at $0$. So a variety with empty cone has $\dim = 0$. (When $C(X) \ne \varnothing$ the minimum is $\ge 1$, since $Q \in I(X)$ vanishes on the $\mathbb{C}^\times$-stable set $C(X)$, so each homogeneous component of $Q$ vanishes there and Euler's relation gives $dQ_v(v) = 0$, i.e. $v \in T_vX$; then the subtraction is exact.)

• Regular points. $\mathrm{reg}(X) := \pi\big(\{v \in C(X) : \dim_{\mathbb{C}} T_vX = \dim X + 1\}\big)$, the images of cone points where the tangent-cone dimension attains its minimum.

Forms and cohomology

• $\Omega^k$ := the set of all functions $\omega : \mathcal{A} \to \mathrm{Alt}^k_{\mathbb{R}}(\mathcal{A};\mathbb{C})$ (continuous, alternating, real-multilinear $k$-forms on $\mathcal{A}$ with complex values). No regularity is built into $\Omega^k$; smoothness is imposed pointwise below. $\omega_P(v_1,\dots,v_k)$ denotes evaluation.
• $\omega \in \Omega^k$ vanishes on $X$ iff $\forall P \in \mathrm{pts}(X)$, $\forall v_1,\dots,v_k \in \tau_PX$: $\omega_P(v_1,\dots,v_k) = 0$.
• $\omega$ is closed on $X$ iff $\omega$ is $C^\infty$ on all of $\mathcal{A}$ and $d\omega$ vanishes on $X$ ($d$ = exterior derivative of forms on the real normed space $\mathcal{A}$).
• $\omega \in \Omega^0$ is exact on $X$ iff it is $C^\infty$ and vanishes on $X$. $\omega \in \Omega^{k}$, $k \ge 1$, is exact on $X$ iff it is $C^\infty$ and there is a $C^\infty$ form $\alpha \in \Omega^{k-1}$ (on all of $\mathcal{A}$) such that $\omega - d\alpha$ vanishes on $X$.
• $\mathcal{Z}^k(X)$ := the $\mathbb{C}$-linear span of the forms closed on $X$; $\mathcal{B}^k(X)$ := the $\mathbb{C}$-linear span of the forms exact on $X$.
• $H^k(X)$ := the image of $\mathcal{Z}^k(X)$ in the quotient $\Omega^k/\mathcal{B}^k(X)$; for $\omega \in \mathcal{Z}^k(X)$, $[\omega]$ denotes its coset. Thus an element of $H^k(X)$ is a coset of $\mathcal{B}^k(X)$ containing at least one element of $\mathcal{Z}^k(X)$.
• $\omega \in \Omega^k$ has type $(p,q)$ on $X$ iff for all $P \in \mathrm{pts}(X)$, all $v_1,\dots,v_k \in \tau_PX$, and all $z \in \mathbb{C}$,

$$\omega_P\big(\mu_P(z,v_1),\dots,\mu_P(z,v_k)\big) = z^{p}\,\bar z^{\,q}\;\omega_P(v_1,\dots,v_k).$$

• A class $c \in H^k(X)$ has type $(p,q)$ iff there exists $\omega \in \mathcal{Z}^k(X)$ with $[\omega] = c$ and $\omega$ of type $(p,q)$ on $X$ (one representative suffices).

Chains, cycles, periods

• A $k$-simplex is any function $\sigma : \mathbb{R}^{k+1} \to \mathcal{A}$ (no continuity or injectivity required at this level).
• A $k$-chain is a finite formal $\mathbb{Z}$-linear combination $\gamma = \sum_\sigma m_\sigma\,\sigma$ of $k$-simplices.
• Faces. For a $(k+1)$-simplex $\sigma$ and $i \in \{0,\dots,k+1\}$, $\partial_i\sigma : \mathbb{R}^{k+1}\to\mathcal{A}$ is $t \mapsto \sigma(t \text{ with a } 0 \text{ inserted at position } i)$.
• Boundary. For a $(k+1)$-chain, $\partial\gamma := \sum_\sigma m_\sigma \sum_{i=0}^{k+1} (-1)^i\,[\partial_i\sigma]$; here $\partial\gamma = 0$ means that after collecting terms whose face-maps are literally the same function, all coefficients vanish (so degenerate or coincident faces may cancel).
• Cycle. Every $0$-chain is a cycle (by convention, no condition). A $k$-chain with $k \ge 1$ is a cycle iff $\partial\gamma = 0$.
• $\Delta_k := \{t \in \mathbb{R}^k : t_i \ge 0,\ \sum_i t_i \le 1\}$ and $a : \mathbb{R}^k \to \mathbb{R}^{k+1}$, $a(t) := \big(1-\sum_i t_i,\ t_1,\dots,t_k\big)$.
• Integral over a simplex.

$$\int_\sigma \omega := \int_{\Delta_k} \omega_{\sigma(a(t))}\Big(\partial_1(\sigma\circ a)(t),\dots,\partial_k(\sigma\circ a)(t)\Big)\,dt ,$$

a Bochner integral with respect to Lebesgue measure on $\mathbb{R}^k$, where $\partial_i(\sigma\circ a)(t)$ is the real Fréchet derivative of $\sigma\circ a$ at $t$ applied to the $i$-th standard basis vector (taken to be $0$ wherever $\sigma\circ a$ is not differentiable). If the integrand is not integrable, the integral is $0$ by convention. For $k = 0$: $\Delta_0$ is a single point carrying mass $1$, so $\int_\sigma\omega = \omega_{\sigma(1)}()$, the value of the $0$-form at $\sigma(1) \in \mathcal{A}$.

• Period. $\langle\omega,\gamma\rangle := \sum_\sigma m_\sigma \int_\sigma\omega \in \mathbb{C}$.
• A $k$-simplex $\sigma$ lies in $X$ iff $\sigma$ is $C^\infty$ on all of $\mathbb{R}^{k+1}$ and $\sigma(t) \in \mathrm{pts}(X)$ for every $t$ in the standard simplex $\{t_i \ge 0,\ \sum_i t_i = 1\}$. A chain lies in $X$ iff every simplex with nonzero coefficient lies in $X$.
• $c \in H^k(X)$ is rational iff: for every $\omega \in \mathcal{Z}^k(X)$ with $[\omega] = c$ (every representative in the span), and every $k$-chain $\gamma$ that lies in $X$ and is a cycle, there exists $q \in \mathbb{Q}$ with $\langle\omega,\gamma\rangle = q$ (as complex numbers; in particular the period is real). No normalizing factor such as $2\pi i$ appears.
• $c \in H^{2p}(X)$ is a Hodge class of degree $p$ iff $c$ is rational and $c$ has type $(p,p)$. Note $z^p\bar z^{\,p} = |z|^{2p}$.

Frames, measures, pairing, integral over $Z$

• For a real subspace $T \subseteq \mathcal{A}$, $P \in \mathcal{A}$ and $m \in \mathbb{N}$, an $m$-tuple $(u_1,\dots,u_m)$ is a $J$-frame in $T$ at $P$ iff all $u_i \in T$, $\langle u_i,u_j\rangle_{\mathbb{R}} = \delta_{ij}$, and $\langle u_i, J_P u_j\rangle_{\mathbb{R}} = 0$ for all $i,j$. (For $m = 0$ this is vacuous.)
• Frame. $\mathrm{fr}(T,P,d) \in \mathcal{A}^d$: let $m := \lfloor d/2\rfloor$ and let $(u_1,\dots,u_m)$ be a $J$-frame in $T$ at $P$ chosen by the axiom of choice if one exists; if none exists, $(u_1,\dots,u_m)$ is an arbitrary, unspecified $m$-tuple in $\mathcal{A}$ (a fixed but unknowable "default"). Then

$$\mathrm{fr}(T,P,d) = \big(u_1,\ J_Pu_1,\ u_2,\ J_Pu_2,\ \dots,\ u_m,\ J_Pu_m\big),$$

followed by a single zero vector if $d$ is odd. The vectors $J_Pu_j$ are not required to lie in $T$ or to have unit length.

• Measure. $\mathrm{vol}_d := \dfrac{\pi^{\lfloor d/2\rfloor}}{\lfloor d/2\rfloor!\;2^{d}}\;\mathcal{H}^d$, where $\mathcal{H}^d$ is the $d$-dimensional Hausdorff measure on $\mathcal{A}$ for its Hilbert–Schmidt metric, in Mathlib's unnormalized form ($\lim_{r\to0}\inf\sum_i \operatorname{diam}(U_i)^d$ over countable covers by sets of diameter $\le r$, with no volume-of-ball factor). For even $d = 2m$ the prefactor equals $\omega_{2m}/2^{2m}$ with $\omega_{2m} = \pi^m/m!$ the volume of the unit ball in $\mathbb{R}^{2m}$ (i.e. the classical normalization constant); for odd $d$ the floor makes it $\pi^{(d-1)/2}/(((d-1)/2)!\,2^d)$. $\mathcal{H}^0$ is counting measure, so $\mathrm{vol}_0$ is counting measure.
• Wedge value. For $\omega \in \Omega^k$, $\eta \in \Omega^l$, $P \in \mathcal{A}$, $f \in \mathcal{A}^{k+l}$:

$$(\omega\wedge\eta)_P(f) := \frac{1}{k!\,l!}\sum_{\sigma \in S_{k+l}} \operatorname{sgn}(\sigma)\;\omega_P\big(f_{\sigma(1)},\dots,f_{\sigma(k)}\big)\;\eta_P\big(f_{\sigma(k+1)},\dots,f_{\sigma(k+l)}\big).$$

• Pairing on $X$.

$$\langle\omega,\eta\rangle_X := \int_{\mathrm{pts}(X)} (\omega\wedge\eta)_P\big(\mathrm{fr}(\tau_PX,\,P,\,k+l)\big)\;d\mathrm{vol}_{k+l}(P),$$

a Bochner integral over the set $\mathrm{pts}(X)$ (measure restricted to that set); it is $0$ by convention if the integrand is not integrable there.

• Integral over $Z$. For $\eta \in \Omega^l$,

$$\int_Z \eta := \int_{\mathrm{reg}(Z)} \eta_P\big(\mathrm{fr}(\tau_PZ,\,P,\,l)\big)\;d\mathrm{vol}_l(P),$$

same conventions; note it uses $Z$'s regular points and $Z$'s tangent spaces.

• Cycle class. $c \in H^{2p}(X)$ is a cycle class of $Z$ relative to $(n,p)$ iff for every $\omega \in \mathcal{Z}^{2p}(X)$ with $[\omega] = c$ and every $\eta \in \Omega^{2n-2p}$ that is closed on $X$ (individual closed forms, not the span, and not modulo exact forms),

$$\langle\omega,\eta\rangle_X = \int_Z\eta .$$

The exponent $2n - 2p$ is natural-number subtraction; under $p \le n$ it equals $2(n-p)$, so the frames involved have lengths $2n$ (pairing, measure $\mathrm{vol}_{2n} = \frac{\pi^n}{n!\,4^n}\mathcal{H}^{2n}$) and $2(n-p)$ (integral over $Z$, measure $\frac{\pi^{n-p}}{(n-p)!\,4^{n-p}}\mathcal{H}^{2n-2p}$).

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The theorem, restated with everything unfolded

For all $N,n,p \in \mathbb{N}$ with $p \le n$, all sets $E_X, E_Z$ of homogeneous polynomials in $x_0,\dots,x_N$, if

• $\dim_{\mathbb{C}}T_vX = n+1$ for every $v \in C(X)$,
• $C(Z) \subseteq C(X)$,
• $C(Z) \ne \varnothing$ and $C(Z)$ satisfies the two-set irreducibility condition above,
• $\min_{v\in C(Z)}\dim_{\mathbb{C}}T_vZ - 1 = n-p$,
• $c \in H^{2p}(X)$ and, for all representatives $\omega \in \mathcal{Z}^{2p}(X)$ of $c$ and all $\eta\in\Omega^{2n-2p}$ closed on $X$, $\langle\omega,\eta\rangle_X = \int_Z\eta$,

then

• for all representatives $\omega\in\mathcal{Z}^{2p}(X)$ of $c$ and all $2p$-chains $\gamma$ lying in $X$ with $\partial\gamma = 0$ (no condition if $p=0$), $\langle\omega,\gamma\rangle\in\mathbb{Q}$; and
• some representative $\omega\in\mathcal{Z}^{2p}(X)$ of $c$ satisfies $\omega_P(\mu_P(z,v_1),\dots,\mu_P(z,v_{2p})) = |z|^{2p}\,\omega_P(v_1,\dots,v_{2p})$ for all $P\in\mathrm{pts}(X)$, $v_i\in\tau_PX$, $z\in\mathbb{C}$.

Edge cases, conventions and junk values to be aware of

• Non-vacuity of the hypotheses. Irreducibility of $Z$ forces $C(Z)\neq\varnothing$, hence $C(X)\neq\varnothing$ via $C(Z)\subseteq C(X)$, so the nonsingularity hypothesis on $X$ is not vacuous, and $\mathrm{reg}(Z)\neq\varnothing$ (the minimum over a nonempty set of naturals is attained). Since $I(Z)\supseteq I(X)$, $T_vZ\subseteq T_vX$ for all $v$, so $\dim Z\le n$ is automatic; $\dim Z = n-p$ is a genuine (non-truncated) equation.
• Tangent-space dimension. For $v\in C(X)$, $v\in T_vX$ (Euler relation) and $\pi$ is constant on $\mathbb{C}^\times v$, so $D\pi_v$ annihilates $\mathbb{C}v$; hence under nonsingularity $\tau_PX$ has real dimension at most $2n$. The pairing asks for a $J$-frame of length $n$ inside $\tau_PX$; $\int_Z$ asks for one of length $n-p$ inside $\tau_PZ$.
• Choice with no witness. Whenever no $J$-frame of the required length exists in $\tau_PX$ (resp. $\tau_PZ$) at some $P$, the frame there consists of arbitrary unspecified vectors (not in the tangent space), and the integrand takes correspondingly unspecified values.
• Non-integrable integrands. Both $\langle\omega,\eta\rangle_X$ and $\int_Z\eta$ are $0$ whenever their integrand fails to be integrable (or a.e. strongly measurable) on the given set; the same holds for each $\int_\sigma\omega$. In particular, the cycle-class hypothesis is an equality of two numbers each of which may be a conventional $0$; if for a given $X,Z$ both sides are conventionally $0$ for all $\omega,\eta$, then every $c\in H^{2p}(X)$ is a "cycle class of $Z$".
• $p = 0$. $c\in H^0(X)$; every $0$-chain in $X$ is a cycle, and a $0$-simplex lies in $X$ iff it is a smooth map $\mathbb{R}\to\mathcal{A}$ with $\sigma(1)\in\mathrm{pts}(X)$. The period is $\sum_\sigma m_\sigma\,\omega(\sigma(1))$; taking single constant simplices, rationality forces $\omega(P)\in\mathbb{Q}$ for every $P\in\mathrm{pts}(X)$ and every representative $\omega$. The type-$(0,0)$ condition is automatically true (empty tuples). The cycle-class condition pairs $0$-forms with $2n$-forms.
• $p = n$. $\dim Z = 0$; $\eta$ is a $0$-form; $\int_Z\eta = \int_{\mathrm{reg}(Z)}\eta\,d(\text{counting measure})$, i.e. $\sum_{P\in\mathrm{reg}(Z)}\eta(P)$ when this converges absolutely and $0$ otherwise; the frame has length $0$.
• Boundary sign convention. $\partial\sigma = \sum_{i=0}^{k+1}(-1)^i\partial_i\sigma$ with $\partial_i$ inserting the coordinate $0$ at position $i$; the $k=0$ convention makes every $0$-chain a cycle.
• What "rational period" ranges over. All $\mathbb{Z}$-combinations of $C^\infty$ maps $\mathbb{R}^{2p+1}\to\mathcal{A}$ whose values on the standard simplex lie in $\mathrm{pts}(X)$ (values elsewhere unconstrained; degenerate simplices allowed; cancellation of literally equal faces allowed), all closed cycles among them, and all representatives of $c$ in the $\mathbb{C}$-span of closed forms.
• Measure normalization. $\mathrm{vol}_d$ is Mathlib's unnormalized Hausdorff measure on $\mathcal{A}$ (Hilbert–Schmidt metric) times $\pi^{\lfloor d/2\rfloor}/(\lfloor d/2\rfloor!\,2^d)$; only even $d$ ($2n$ and $2n-2p$) occur in this theorem.
• Natural subtraction. Appears in $\dim$ (exact when the cone is nonempty, $0$ for an empty cone), in $n-p$ and in $2n-2p$ (both exact under $p\le n$).
• Sets of polynomials. $E_X, E_Z$ may be infinite, may contain $0$, and may contain nonzero constants (making the cone empty—excluded here for $Z$ and hence for $X$). Irreducibility is tested against arbitrary (non-homogeneous) polynomial sets $S_1,S_2$.