The Hodge conjecture (Clay Millennium Prize Problem, Deligne's statement)

Read-back of HodgeConjecture.hodge_conjecture

Throughout, $N, n, p$ are natural numbers (so $0$ is allowed for each), and everything below is stated over the following fixed ambient objects.

Ambient objects.

• $\mathbb{C}^{N+1}$ = functions $\{0,\dots,N\}\to\mathbb{C}$ ("coordinates" $v=(v_0,\dots,v_N)$).
• $\mathcal{A}_N$ = the space of $(N+1)\times(N+1)$ complex matrices $A=(A_{ij})$ equipped with the Hilbert–Schmidt Hermitian inner product $\langle A,B\rangle_{\mathbb C}=\sum_{i,j}\overline{A_{ij}}B_{ij}$ (an $\ell^2$ Euclidean space over $\mathbb C$; as a real vector space it has dimension $2(N+1)^2$). Write $\langle A,B\rangle_{\mathbb R}=\operatorname{Re}\langle A,B\rangle_{\mathbb C}$.
• Polynomials: $\mathbb{C}[x_0,\dots,x_N]$.
• $\pi(v)\in\mathcal A_N$ for $v\in\mathbb C^{N+1}$: the matrix with entries $\pi(v)_{ij}=v_i\overline{v_j}/\sum_k|v_k|^2$, i.e. the orthogonal projector onto the line $\mathbb Cv$. (For $v=0$ the division is by $0$, which evaluates to $0$, so $\pi(0)=0$; this value is never used below because $0$ is excluded from every cone.) $\pi(\lambda v)=\pi(v)$ for $\lambda\ne0$.
• $J_P(A)=i\,(AP-PA)=i[A,P]$ for $P,A\in\mathcal A_N$ (a real-linear map of $\mathcal A_N$ depending on $P$).
• $z\cdot_P A := (\operatorname{Re}z)\,A+(\operatorname{Im}z)\,J_P(A)$ for $z\in\mathbb C$ — "multiplication by $z$" using $J_P$ as the complex structure.
• $\mathcal Z(S)=\{v\in\mathbb C^{N+1}: Q(v)=0\ \forall Q\in S\}$ for any set $S$ of polynomials (not necessarily homogeneous, possibly empty or infinite).

Projective varieties. A projective variety $X$ (of "ambient index" $N$) is a set $E_X$ of polynomials in $\mathbb C[x_0,\dots,x_N]$ together with the property that each $Q\in E_X$ is homogeneous of some degree $d_Q\in\mathbb N$ (degrees may differ from one $Q$ to another; degree $0$, i.e. nonzero constants, and the zero polynomial are allowed; $E_X$ may be empty or infinite). Derived notions:

• Cone: $C_X=\{v\in\mathbb C^{N+1}: v\ne0,\ Q(v)=0\ \forall Q\in E_X\}$. If $E_X=\varnothing$ then $C_X=\mathbb C^{N+1}\setminus\{0\}$; if $E_X$ contains a nonzero constant or $\{x_0,\dots,x_N\}$ then $C_X=\varnothing$.
• Points: $\mathrm{pts}_X=\pi(C_X)\subseteq\mathcal A_N$ (a set of rank-one Hermitian projectors).
• Vanishing ideal: $I_X=\{Q\in\mathbb C[x_0,\dots,x_N]: Q(v)=0\ \forall v\in C_X\}$ (all polynomials, not only homogeneous ones). $0\in I_X$ always.
• Tangent cone at $v\in\mathbb C^{N+1}$ (defined for every $v$, not only $v\in C_X$): the complex subspace

$$T^{\mathrm{aff}}_vX=\bigcap_{Q\in I_X}\ker\Big(u\mapsto\sum_{i=0}^N\frac{\partial Q}{\partial x_i}(v)\,u_i\Big)\subseteq\mathbb C^{N+1},$$

the common kernel of the differentials at $v$ of all polynomials vanishing on $C_X$.

• Nonsingular of dimension $n$: $X$ is nonsingular of dimension $n$ iff for every $v\in C_X$, $\dim_{\mathbb C}T^{\mathrm{aff}}_vX=n+1$. This is vacuously true when $C_X=\varnothing$. (When $C_X\ne\varnothing$ it forces $n+1\le N+1$.)
• Tangent space at $P\in\mathcal A_N$: the real subspace

$$T_PX=\sum_{\substack{v\in C_X\\ \pi(v)=P}}\ D\pi_v\big(T^{\mathrm{aff}}_vX\big)\ \subseteq\mathcal A_N,$$

the real span of the images of the tangent cones under the real Fréchet derivative $D\pi_v:\mathbb C^{N+1}\to\mathcal A_N$ of $\pi$ at $v$ (Mathlib's fderiv, which is the zero map at any point where $\pi$ is not differentiable; on $C_X$, i.e. away from $0$, $\pi$ is smooth). If $P\notin\mathrm{pts}_X$ the sum is empty and $T_PX=\{0\}$. Since $\pi$ is constant on $\mathbb C^\times v$, $D\pi_v$ kills $\mathbb Cv$; for $X$ nonsingular of dimension $n$ this gives $\dim_{\mathbb R}T_PX\le 2n$. Nothing in the definitions asserts that $J_P$ preserves $T_PX$.

• Irreducible: $C_X\ne\varnothing$, and for all sets $S_1,S_2$ of (arbitrary) polynomials, $C_X\subseteq\mathcal Z(S_1)\cup\mathcal Z(S_2)\Rightarrow C_X\subseteq\mathcal Z(S_1)$ or $C_X\subseteq\mathcal Z(S_2)$.
• Dimension: $\dim X=\big(\min_{v\in C_X}\dim_{\mathbb C}T^{\mathrm{aff}}_vX\big)-1$ with the minimum taken in $\mathbb N$ and the subtraction truncated at $0$. If $C_X=\varnothing$ the infimum of the empty set of naturals is $0$ and $\dim X=0-1=0$. (For $v\in C_X$ one has $v\in T^{\mathrm{aff}}_vX$, so the minimum is $\ge1$ when $C_X\neq\varnothing$ and the subtraction is then genuine.)
• Regular points: $\mathrm{reg}_X=\pi\big(\{v\in C_X:\dim_{\mathbb C}T^{\mathrm{aff}}_vX=\dim X+1\}\big)$, the points where the tangent-cone dimension attains its minimum.

Forms. A $k$-form is any function $\omega:\mathcal A_N\to\operatorname{Alt}^k_{\mathbb R}(\mathcal A_N;\mathbb C)$ assigning to every point $P$ of the whole ambient space $\mathcal A_N$ a continuous, $\mathbb R$-multilinear, alternating map $\omega_P:(\mathcal A_N)^k\to\mathbb C$. (No regularity is built into the type.) Forms form a $\mathbb C$-vector space. "Smooth" below means $C^\infty$ as a map between normed spaces (ContDiff ℝ ∞, where $\infty$ is $C^\infty$, not analytic). $d\omega$ denotes Mathlib's exterior derivative of $\omega$ as a form on the normed space $\mathcal A_N$ (alternatisation of the Fréchet derivative; it takes the junk value $0$ at points where $\omega$ is not differentiable).

• $\omega$ vanishes on $X$ iff for all $P\in\mathrm{pts}_X$ and all $v_1,\dots,v_k\in T_PX$, $\omega_P(v_1,\dots,v_k)=0$. (For $k=0$: $\omega_P=0$ for all $P\in\mathrm{pts}_X$.)
• $\omega$ is closed on $X$ iff $\omega$ is smooth and $d\omega$ vanishes on $X$.
• $\omega$ is exact on $X$: for $k=0$, iff $\omega$ is smooth and vanishes on $X$; for $k\ge1$, iff $\omega$ is smooth and there is a smooth $(k-1)$-form $\alpha$ (on all of $\mathcal A_N$, otherwise unconstrained) such that $\omega-d\alpha$ vanishes on $X$.
• $\mathcal C^k_X$ = the $\mathbb C$-linear span of the $k$-forms closed on $X$; $\mathcal E^k_X$ = the $\mathbb C$-linear span of the $k$-forms exact on $X$. (The definitions do not by themselves assert $\mathcal E^k_X\subseteq\mathcal C^k_X$.)
• Cohomology: $H^k(X)$ := the image of $\mathcal C^k_X$ in the quotient $\{\text{all }k\text{-forms}\}/\mathcal E^k_X$, a $\mathbb C$-vector space isomorphic to $\mathcal C^k_X/(\mathcal C^k_X\cap\mathcal E^k_X)$. For $\omega\in\mathcal C^k_X$, $[\omega]\in H^k(X)$ is its class; a representative of $c\in H^k(X)$ is any $\omega\in\mathcal C^k_X$ with $[\omega]=c$.
• $\omega$ has type $(p,q)$ on $X$ iff for all $P\in\mathrm{pts}_X$, all $v_1,\dots,v_k\in T_PX$ and all $z\in\mathbb C$,

$$\omega_P(z\cdot_Pv_1,\dots,z\cdot_Pv_k)=z^p\,\bar z^{\,q}\,\omega_P(v_1,\dots,v_k)$$

(with $0^0=1$).

• A class $c\in H^k(X)$ has type $(p,q)$ iff there exists a representative $\omega\in\mathcal C^k_X$ of $c$ with $\omega$ of type $(p,q)$ on $X$.

Chains and rationality. A $k$-simplex is any function $\sigma:\mathbb R^{k+1}\to\mathcal A_N$; a $k$-chain is a finitely supported formal $\mathbb Z$-combination $\sum_\sigma m_\sigma\,\sigma$ of $k$-simplices. The $i$-th face ($i=0,\dots,k+1$) of a $(k+1)$-simplex $\sigma$ is $t\mapsto\sigma(t\text{ with a }0\text{ inserted at position }i)$; $\partial\big(\sum m_\sigma\sigma\big)=\sum_\sigma m_\sigma\sum_{i=0}^{k+1}(-1)^i\,\mathrm{face}_i\sigma$. A chain is a cycle iff $k=0$ (every $0$-chain counts) or $\partial c=0$. A $k$-simplex $\sigma$ lies in $X$ iff $\sigma$ is smooth on all of $\mathbb R^{k+1}$ and $\sigma(t)\in\mathrm{pts}_X$ for every $t$ in the standard simplex $\{t_i\ge0,\ \sum t_i=1\}$; a chain lies in $X$ iff every simplex in its support does. The period of a $k$-form $\omega$ over a chain is

$$\Big\langle\omega,\sum m_\sigma\sigma\Big\rangle=\sum_\sigma m_\sigma\int_{\Delta_k}\omega_{\sigma(\beta(t))}\Big(\partial_1(\sigma\circ\beta)(t),\dots,\partial_k(\sigma\circ\beta)(t)\Big)\,dt,$$

where $\Delta_k=\{t\in\mathbb R^k: t_i\ge0,\ \sum t_i\le1\}$, $\beta(t)=(1-\sum_it_i,\ t_1,\dots,t_k)$, $\partial_i$ are the coordinate partial derivatives obtained from the Fréchet derivative (junk value $0$ where not differentiable), and the integral is a Bochner integral against Lebesgue measure on $\mathbb R^k$ (value $0$ if the integrand is not integrable; for $k=0$, $\mathbb R^0$ carries the unit point mass and the "integral" is $\omega_{\sigma(1)}$).

A class $c\in H^k(X)$ is rational iff for every representative $\omega\in\mathcal C^k_X$ of $c$ and every $k$-chain lying in $X$ that is a cycle, the period $\langle\omega,\text{chain}\rangle\in\mathbb C$ equals some rational number $q$ (i.e. is real and rational). (The zero chain always qualifies with period $0$, so this hypothesis is never vacuous; it says nothing about non-cycles.)

Hodge class. $c\in H^{2p}(X)$ is a Hodge class of level $p$ iff $c$ is rational (as above) and $c$ has type $(p,p)$ (as above, an existential over representatives).

Frames. For a real subspace $T\subseteq\mathcal A_N$, a point $P\in\mathcal A_N$ and $m\in\mathbb N$, a family $u=(u_0,\dots,u_{m-1})$ in $\mathcal A_N$ is a $J$-frame for $(T,P)$ iff every $u_i\in T$, $\langle u_i,u_j\rangle_{\mathbb R}=\delta_{ij}$, and $\langle u_i,J_P(u_j)\rangle_{\mathbb R}=0$ for all $i,j$. (Nothing requires $J_P(u_j)\in T$, $|J_P(u_j)|=1$, or that the family spans $T$; for $m=0$ the empty family is a $J$-frame.) For $d\in\mathbb N$, the frame $F(T,P,d)=(f_0,\dots,f_{d-1})$ is built as follows: let $u$ be some $J$-frame for $(T,P)$ of size $\lfloor d/2\rfloor$, chosen by Hilbert's choice operator (Classical.epsilon) — if no such $J$-frame exists, $u$ is an arbitrary, unspecified family of $\lfloor d/2\rfloor$ vectors; then

$$f_{2j}=u_j,\qquad f_{2j+1}=J_P(u_j)\qquad(0\le j<\lfloor d/2\rfloor),$$

i.e. the order is $(u_0,\,J_Pu_0,\,u_1,\,J_Pu_1,\dots)$, and if $d$ is odd the last vector $f_{d-1}$ is $0$. The same (fixed but arbitrary) choice is used everywhere below.

Measure. For $d\in\mathbb N$, $\mu_d$ is the measure on $\mathcal A_N$

$$\mu_d=\frac{\pi^{\lfloor d/2\rfloor}}{\lfloor d/2\rfloor!\;2^{d}}\cdot\mathcal H^{d},$$

where $\mathcal H^d$ is Mathlib's $d$-dimensional Hausdorff measure for the Hilbert–Schmidt metric on $\mathcal A_N$ (defined with $\operatorname{diam}^d$ and no normalising constant). For even $d$ the prefactor is $\alpha_d/2^d$ with $\alpha_d=\pi^{d/2}/(d/2)!$ the volume of the unit ball in $\mathbb R^d$, which is the factor converting Mathlib's $\mathcal H^d$ into the standard $d$-dimensional Hausdorff measure (hence $d$-dimensional Lebesgue/surface measure on $d$-dimensional Euclidean pieces); for odd $d$ the prefactor uses $\lfloor d/2\rfloor$. For $d=0$, $\mu_0=\mathcal H^0$ is counting measure. In the theorem only even $d$ occur.

Wedge value. For a $k$-form $\omega$, an $l$-form $\eta$, $P\in\mathcal A_N$ and $f=(f_0,\dots,f_{k+l-1})$,

$$(\omega\wedge\eta)_P(f)=\frac{1}{k!\,l!}\sum_{\sigma\in S_{k+l}}\operatorname{sgn}(\sigma)\ \omega_P\big(f_{\sigma(0)},\dots,f_{\sigma(k-1)}\big)\ \eta_P\big(f_{\sigma(k)},\dots,f_{\sigma(k+l-1)}\big).$$

Pairing and integral.

$$\langle\omega,\eta\rangle_X=\int_{P\in\mathrm{pts}_X}(\omega\wedge\eta)_P\big(F(T_PX,P,k+l)\big)\ d\mu_{k+l}(P),\qquad \int_Z\eta=\int_{P\in\mathrm{reg}_Z}\eta_P\big(F(T_PZ,P,l)\big)\ d\mu_{l}(P),$$

both Bochner integrals of a complex-valued function against the restriction of $\mu_\bullet$ to the indicated set (value $0$ if the integrand is not integrable, e.g. if it is not measurable or the set has infinite measure with non-summable integrand). Note the pairing integrates over all points of $X$ using frames in $T_PX$ of $k+l$ vectors, whereas $\int_Z$ integrates over the regular points of $Z$ using frames in $T_PZ$ of $l$ vectors.

Cycle class. For projective varieties $X,Z$, naturals $n,p$, and $c\in H^{2p}(X)$: $c$ is a cycle class of $Z$ (relative to $X,n,p$) iff for every representative $\omega\in\mathcal C^{2p}_X$ of $c$ and every $(2n-2p)$-form $\eta$ that is closed on $X$ (smooth with $d\eta$ vanishing on $X$; no condition relative to $Z$),

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

Here $2n-2p$ is natural-number subtraction (truncated at $0$; genuine when $p\le n$), the pairing uses $2p+(2n-2p)=2n$ frame vectors in $T_PX$ and the measure $\mu_{2n}$, and $\int_Z$ uses $2n-2p$ frame vectors in $T_PZ$ and the measure $\mu_{2n-2p}$.

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The theorem. Let $N,n,p\in\mathbb N$ with $p\le n$. Let $X$ be a projective variety of ambient index $N$ which is nonsingular of dimension $n$ (every $v\in C_X$ has $\dim_{\mathbb C}T^{\mathrm{aff}}_vX=n+1$). Let $c\in H^{2p}(X)$ be a Hodge class of level $p$ (rational, and having a representative of type $(p,p)$). Then there exist

• a natural number $m$ (possibly $0$),
• projective varieties $Z_0,\dots,Z_{m-1}$ of ambient index $N$,
• rational numbers $a_0,\dots,a_{m-1}$,
• classes $\mathrm{cl}_0,\dots,\mathrm{cl}_{m-1}\in H^{2p}(X)$ (classes on $X$, not on $Z_i$),

such that for every $i$:

1. $C_{Z_i}\subseteq C_X$ (containment of cones; no relation between $E_{Z_i}$ and $E_X$ is required);
2. $Z_i$ is irreducible (in particular $C_{Z_i}\ne\varnothing$);
3. $\dim Z_i=n-p$ (with $\dim$ as defined above via the minimal tangent-cone dimension; $n-p$ is a genuine subtraction since $p\le n$);
4. $\mathrm{cl}_i$ is a cycle class of $Z_i$ relative to $(X,n,p)$;

and

$$c=\sum_{i=0}^{m-1}a_i\cdot\mathrm{cl}_i\quad\text{in }H^{2p}(X),$$

with each $a_i$ regarded as a complex scalar.

Edge and degenerate cases surfaced by the statement.

• $m=0$ is allowed: then conditions 1–4 are vacuous and the sum is $0$, so the conclusion holds outright whenever $c=0$. Nothing requires $m\ge1$, $a_i\ne0$, the $Z_i$ to be distinct, or $\mathrm{cl}_i\ne0$.
• If $C_X=\varnothing$, the nonsingularity hypothesis is vacuous, $\mathrm{pts}_X=\varnothing$, every smooth form is both closed and exact on $X$, $H^{2p}(X)=0$, so $c=0$ and the conclusion holds with $m=0$ (and $m$ must then be $0$, since an irreducible $Z_i$ needs $\varnothing\ne C_{Z_i}\subseteq C_X$).
• The hypothesis "rational" is universal over representatives and cycles-in-$X$; "type $(p,p)$" is existential over representatives; "cycle class" is universal over representatives of $\mathrm{cl}_i$ and over all forms closed on $X$.
• $p=0$: $c\in H^0(X)$, the type-$(0,0)$ condition is automatic ($z^0\bar z^0=1$), and $\dim Z_i=n$. $p=n$: $\eta$ ranges over $0$-forms (functions), $\int_{Z_i}\eta$ is the sum of $\eta$ over $\mathrm{reg}_{Z_i}$ with counting measure (junk $0$ if not summable), and the frame in $\int_{Z_i}$ is empty.
• Junk values that can enter the conclusion: both sides of the cycle-class equation are Bochner integrals (value $0$ when non-integrable); the frames are chosen by a choice operator among all $J$-frames of the required size, with no spanning or orientation requirement beyond the order $(u_0,J_Pu_0,u_1,J_Pu_1,\dots)$, and are arbitrary vectors if no $J$-frame of that size exists in $T_PX$ (resp. $T_PZ$); $D\pi$, partial derivatives and $d$ are $0$ where undefined; $\dim$ uses $\inf\varnothing=0$ and truncated subtraction; $\mathcal H^d$ is Mathlib's unnormalised Hausdorff measure scaled by the constant displayed above.
• $n$ appears both in the nonsingularity hypothesis and inside the cycle-class condition (degrees $2n-2p$ and $2n$); $p$ appears in $H^{2p}(X)$, in the type $(p,p)$, in $\dim Z_i=n-p$ and in the cycle-class degrees. $N$ only fixes the ambient spaces.