The Hodge conjecture for $H^2$ (Lefschetz $(1,1)$ theorem, Kodaira–Spencer)

Read-back: hodge_conjecture_divisors

Ambient objects. Fix a natural number $N$. Write $V := \mathbb{C}^{N+1}$ (functions $\{0,\dots,N\}\to\mathbb{C}$) and $\mathcal{A} :=$ the space of $(N+1)\times(N+1)$ complex matrices, viewed as the complex Hilbert space $\ell^2$ on index pairs $(i,j)$ (Frobenius inner product $\langle A,B\rangle=\sum_{ij}\overline{A_{ij}}B_{ij}$); as a real vector space $\mathcal{A}$ has dimension $2(N+1)^2$ and carries the induced Euclidean metric. Let $\mathcal{P} := \mathbb{C}[x_0,\dots,x_N]$.

The following auxiliary maps are used throughout.

• $\operatorname{proj}: V\to\mathcal{A}$, $\operatorname{proj}(v)_{ij} = v_i\,\overline{v_j}\,/\,\sum_k |v_k|^2$ (with the convention that division by $0$ gives $0$, so $\operatorname{proj}(0)=0$). For $v\neq 0$ this is the rank-one orthogonal projection onto $\mathbb{C}v$.
• $J_P(A) := i\,(AP - PA) = i[A,P]$ for $P,A\in\mathcal{A}$.
• $\operatorname{cmul}_P(z,A) := (\operatorname{Re} z)\,A + (\operatorname{Im} z)\,J_P(A)$ for $z\in\mathbb{C}$.
• $\langle a,b\rangle_{\mathbb{R}} := \operatorname{Re}\langle a,b\rangle$ on $\mathcal{A}$.
• For a set $S\subseteq\mathcal{P}$, $\operatorname{zeros}(S) := \{v\in V : Q(v)=0\ \forall Q\in S\}$.

Projective varieties. A ProjVariety $X$ (of ambient index $N$) is a set of polynomials $X.\mathrm{eqs}\subseteq\mathcal{P}$ together with a proof that each $Q\in X.\mathrm{eqs}$ is homogeneous of some degree $d_Q\in\mathbb{N}$ (degrees may differ; the zero polynomial qualifies; the set may be empty, or may contain a nonzero constant, in which case the cone below is empty). Two varieties with different equation sets but the same zero set are different objects. For such $X$:

• $\operatorname{cone}(X) := \{v\in V : v\neq 0,\ Q(v)=0\ \forall Q\in X.\mathrm{eqs}\}$ (nonzero vectors of the affine cone).
• $\operatorname{pts}(X) := \operatorname{proj}(\operatorname{cone}(X))\subseteq\mathcal{A}$.
• $\operatorname{van}(X) := \{Q\in\mathcal{P} : Q(v)=0\ \forall v\in\operatorname{cone}(X)\}$, all polynomials (homogeneous or not) vanishing on the cone.
• For $Q\in\mathcal{P}$ and $v\in V$, $dQ_v : V\to\mathbb{C}$ is the $\mathbb{C}$-linear map $u\mapsto\sum_i \partial_iQ(v)\,u_i$.
• $T_v X := \bigcap_{Q\in\operatorname{van}(X)}\ker(dQ_v)\subseteq V$, a complex subspace (defined for every $v\in V$, not only $v\in\operatorname{cone}(X)$; it is cut out by the full vanishing ideal, not just by $X.\mathrm{eqs}$).
• IsNonsingular$(X,n)$: for every $v\in\operatorname{cone}(X)$, $\dim_{\mathbb{C}} T_vX = n+1$. (Vacuous if the cone is empty.)
• $\operatorname{tan}_X(P)\subseteq\mathcal{A}$, a real subspace: the span (supremum of subspaces) over all $v\in\operatorname{cone}(X)$ with $\operatorname{proj}(v)=P$ of the image of $T_vX$ (viewed as a real subspace) under the real Fréchet derivative $D\operatorname{proj}(v):V\to\mathcal{A}$. If $P\notin\operatorname{pts}(X)$ the supremum is over an empty family and $\operatorname{tan}_X(P)=\{0\}$. (The Fréchet derivative is a total function equal to $0$ where $\operatorname{proj}$ is not differentiable; $\operatorname{proj}$ is differentiable at every $v\neq0$.)
• IsIrreducible$(X)$: $\operatorname{cone}(X)\neq\varnothing$, and for all sets $S_1,S_2\subseteq\mathcal{P}$ (arbitrary, not necessarily homogeneous), if $\operatorname{cone}(X)\subseteq\operatorname{zeros}(S_1)\cup\operatorname{zeros}(S_2)$ then $\operatorname{cone}(X)\subseteq\operatorname{zeros}(S_1)$ or $\operatorname{cone}(X)\subseteq\operatorname{zeros}(S_2)$.
• $\dim X := \Big(\inf_{v\in\operatorname{cone}(X)} \dim_{\mathbb{C}}T_vX\Big) \mathbin{\dot-} 1$, where $\dot-$ is truncated natural-number subtraction and the infimum of an empty set of naturals is $0$. So an empty cone gives $\dim X = 0$, and $\dim X = 0$ also whenever the infimum is $0$ or $1$.
• $\operatorname{reg}(X) := \operatorname{proj}\big(\{v\in\operatorname{cone}(X) : \dim_{\mathbb{C}}T_vX = \dim X + 1\}\big)$, the images of cone points where the tangent-space dimension attains the minimum.

Forms and cohomology. A $k$-form is any function $w:\mathcal{A}\to\operatorname{Alt}^k_{\mathbb{R}}(\mathcal{A};\mathbb{C})$ assigning to each point of the whole ambient space $\mathcal{A}$ a continuous alternating $\mathbb{R}$-multilinear map $\mathcal{A}^k\to\mathbb{C}$. Forms are ambient objects; $X$ enters only via the following restrictions.

• $w$ vanishes on $X$: for every $P\in\operatorname{pts}(X)$ and every $k$-tuple $v_1,\dots,v_k\in\operatorname{tan}_X(P)$, $w_P(v_1,\dots,v_k)=0$.
• $w$ is closed on $X$: $w$ is $C^\infty$ on all of $\mathcal{A}$ and its exterior derivative $dw$ (Mathlib's total exterior derivative of a form, a $(k+1)$-form) vanishes on $X$.
• $w$ is exact on $X$: for $k=0$, $w$ is $C^\infty$ and vanishes on $X$; for $k\geq1$, $w$ is $C^\infty$ and there exists a $C^\infty$ $(k-1)$-form $\alpha$ such that $w-d\alpha$ vanishes on $X$.
• $\mathcal{Z}^k(X) :=$ the $\mathbb{C}$-linear span of the closed $k$-forms; $\mathcal{B}^k(X) :=$ the $\mathbb{C}$-linear span of the exact $k$-forms.
• $H^k(X) :=$ the image of $\mathcal{Z}^k(X)$ in the quotient $\{\text{all }k\text{-forms}\}/\mathcal{B}^k(X)$, a complex vector space; $[w]\in H^k(X)$ denotes the class of $w\in\mathcal{Z}^k(X)$.
• $w$ has type $(p,q)$ on $X$: for every $P\in\operatorname{pts}(X)$, every tuple $v_1,\dots,v_k\in\operatorname{tan}_X(P)$, and every $z\in\mathbb{C}$,

$$w_P\big(\operatorname{cmul}_P(z,v_1),\dots,\operatorname{cmul}_P(z,v_k)\big) \;=\; z^{p}\,\overline{z}^{\,q}\; w_P(v_1,\dots,v_k).$$

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

Chains, periods, rationality. A $k$-simplex is any function $\sigma:\mathbb{R}^{k+1}\to\mathcal{A}$ (no regularity assumed at this stage); a $k$-chain is a finite formal $\mathbb{Z}$-combination $\sum_\sigma m_\sigma\,\sigma$ of simplices. The $i$-th face of a $(k+1)$-simplex is $t\mapsto\sigma(t \text{ with a }0\text{ inserted in slot } i)$; $\partial\big(\sum m_\sigma\sigma\big) := \sum_\sigma m_\sigma\sum_{i=0}^{k+1}(-1)^i\,\operatorname{face}_i\sigma$. A chain is a cycle if $k=0$ (always) or $\partial c=0$ (for $k\ge1$). With $\Delta_k := \{t\in\mathbb{R}^k : t_i\ge0,\ \sum t_i\le1\}$ and $e(t) := (1-\sum_i t_i,\,t_1,\dots,t_k)\in\mathbb{R}^{k+1}$,

$$\int_\sigma w \;:=\; \int_{\Delta_k} w_{\sigma(e(t))}\Big(D(\sigma\circ e)(t)\,\varepsilon_1,\ \dots,\ D(\sigma\circ e)(t)\,\varepsilon_k\Big)\,dt ,$$

a Bochner integral against Lebesgue measure on $\mathbb{R}^k$, where $\varepsilon_i$ are the standard basis vectors, $D$ is the (total) real Fréchet derivative, taken to be $0$ where $\sigma\circ e$ is not differentiable, and the integral is $0$ if the integrand is not integrable. For $k=0$, $\Delta_0$ is a single point of mass one and $\int_\sigma w = w_{\sigma(1)}$. The period is $\operatorname{per}(w,c) := \sum_\sigma m_\sigma\int_\sigma w\in\mathbb{C}$.

A simplex $\sigma$ is in $X$ if $\sigma$ is $C^\infty$ on all of $\mathbb{R}^{k+1}$ and $\sigma(t)\in\operatorname{pts}(X)$ for every $t$ in the standard simplex $\{t_i\ge0,\ \sum t_i=1\}$; a chain is in $X$ if every simplex in its support is.

• IsRational$(X,c)$ for $c\in H^k(X)$: for every $w\in\mathcal{Z}^k(X)$ with $[w]=c$ and every $k$-chain $ch$ that is in $X$ and is a cycle, there exists $q\in\mathbb{Q}$ with $\operatorname{per}(w,ch)=q$.
• IsHodgeClass$(X,p,c)$ for $c\in H^{2p}(X)$: IsRational$(X,c)$ and $c$ has type $(p,p)$.

Frames, measures, pairing. For a real subspace $T\subseteq\mathcal{A}$, a point $P\in\mathcal{A}$ and $u:\{0,\dots,m-1\}\to\mathcal{A}$, $u$ is a $J$-frame in $T$ at $P$ if 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 $i,j$. The frame $\operatorname{fr}(T,P,d)\in\mathcal{A}^d$ is built as follows: let $u\in\mathcal{A}^{\lfloor d/2\rfloor}$ be some $J$-frame of length $\lfloor d/2\rfloor$ in $T$ at $P$ chosen by Hilbert's $\varepsilon$ (Classical.epsilon) — if none exists, $u$ is an arbitrary, unspecified tuple with no properties — and set

$$\operatorname{fr}(T,P,d)_k := \begin{cases} u_{\lfloor k/2\rfloor} & k \text{ even},\ \lfloor k/2\rfloor<\lfloor d/2\rfloor,\\ J_P\,u_{\lfloor k/2\rfloor} & k \text{ odd},\ \lfloor k/2\rfloor<\lfloor d/2\rfloor,\\ 0 & \text{otherwise (only when } d \text{ is odd and } k=d-1). \end{cases}$$

The measure $\mu_d := \dfrac{\pi^{\lfloor d/2\rfloor}}{\lfloor d/2\rfloor!\;2^{d}}\cdot \mathcal{H}^{d}$ on $\mathcal{A}$, where $\mathcal{H}^d$ is Mathlib's $d$-dimensional Hausdorff measure for the Euclidean metric of $\mathcal{A}$. For a $k$-form $w$, an $l$-form $\eta$, a point $P$ and $f\in\mathcal{A}^{k+l}$,

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

Then

$$\langle w,\eta\rangle_X := \int_{P\in\operatorname{pts}(X)} (w\wedge\eta)_P\big(\operatorname{fr}(\operatorname{tan}_X(P),P,k+l)\big)\,d\mu_{k+l}(P),\qquad \int_Z\eta := \int_{P\in\operatorname{reg}(Z)} \eta_P\big(\operatorname{fr}(\operatorname{tan}_Z(P),P,l)\big)\,d\mu_l(P),$$

both Bochner integrals over the indicated sets (restriction of the measure to a possibly non-measurable set, via outer measure), each equal to $0$ if the integrand fails to be integrable. Note the pairing integrates over $\operatorname{pts}(X)$ while $\int_Z$ integrates over $\operatorname{reg}(Z)$ only.

• IsCycleClass$(X,Z,n,p,c)$ for $c\in H^{2p}(X)$: for every $w\in\mathcal{Z}^{2p}(X)$ with $[w]=c$ and every $(2n\mathbin{\dot-}2p)$-form $\eta$ that is closed on $X$,

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

Here $2n\mathbin{\dot-}2p$ is truncated subtraction. The wedge in the pairing is a $(2p + (2n\dot-2p))$-form evaluated on a frame of that length in $\operatorname{tan}_X(P)$.

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The theorem. Let $N,n\in\mathbb{N}$ with $1\le n$, let $X$ be a ProjVariety of ambient index $N$ satisfying IsNonsingular$(X,n)$ (every cone point has $\dim_{\mathbb{C}}T_vX=n+1$), let $c\in H^{2}(X)$ — the degree is written $2\cdot1$ — and assume IsHodgeClass$(X,1,c)$, i.e.

1. every closed representative $w$ of $c$ has a rational period over every smooth $2$-chain in $X$ that is a cycle, and
2. some closed representative $w$ of $c$ satisfies $w_P(\operatorname{cmul}_P(z,v_1),\operatorname{cmul}_P(z,v_2)) = z\bar z\,w_P(v_1,v_2) = |z|^2 w_P(v_1,v_2)$ for all $P\in\operatorname{pts}(X)$, all $v_1,v_2\in\operatorname{tan}_X(P)$, all $z\in\mathbb{C}$.

Then there exist a natural number $m$, a family of ProjVarieties $Z_0,\dots,Z_{m-1}$ of ambient index $N$, rationals $a_0,\dots,a_{m-1}\in\mathbb{Q}$, and classes $\mathrm{cl}_0,\dots,\mathrm{cl}_{m-1}\in H^2(X)$ such that

• for every $i<m$:
  • $\operatorname{cone}(Z_i)\subseteq\operatorname{cone}(X)$;
  • IsIrreducible$(Z_i)$ (in particular $\operatorname{cone}(Z_i)\ne\varnothing$);
  • $\dim Z_i = n\mathbin{\dot-}1$, i.e. $\big(\inf_{v\in\operatorname{cone}(Z_i)}\dim_{\mathbb{C}}T_vZ_i\big)\mathbin{\dot-}1 = n-1$ (genuine subtraction since $n\ge1$);
  • IsCycleClass$(X,Z_i,n,1,\mathrm{cl}_i)$: for every closed $2$-form $w$ on $X$ with $[w]=\mathrm{cl}_i$ and every $(2n-2)$-form $\eta$ closed on $X$, $\langle w,\eta\rangle_X = \int_{Z_i}\eta$, where the left side uses a frame of length $2n$ in $\operatorname{tan}_X(P)$ over $\operatorname{pts}(X)$ with measure $\mu_{2n}$, and the right side uses a frame of length $2n-2$ in $\operatorname{tan}_{Z_i}(P)$ over $\operatorname{reg}(Z_i)$ with measure $\mu_{2n-2}$;
• and, in the complex vector space $H^2(X)$,

$$c = \sum_{i<m} a_i\cdot \mathrm{cl}_i ,$$

with each $a_i\in\mathbb{Q}$ coerced to $\mathbb{C}$.

No nonsingularity, no relation between $\mathrm{cl}_i$ and any specific form, and no distinctness of the $Z_i$ is required; the only link between $Z_i$ and $\mathrm{cl}_i$ is the IsCycleClass condition. The parameter $n$ enters IsCycleClass independently of $\dim X$ (which is never referenced for $X$).

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Edge and degenerate cases visible in the statement.

• Trivial witness. With $m=0$ the family conditions are vacuous and the sum is $0$; so the conclusion holds for $c=0$ regardless of hypotheses. More generally any $\mathrm{cl}_i$ that is the zero class of $H^2(X)$ satisfies IsCycleClass with respect to some $Z_i$ exactly when $\langle w,\eta\rangle_X=\int_{Z_i}\eta$ for all exact-on-$X$ representatives $w$ of $0$ and closed $\eta$.
• Empty variety. Nothing forbids $\operatorname{cone}(X)=\varnothing$ (e.g. a nonzero constant among the equations). Then IsNonsingular is vacuous, $\operatorname{pts}(X)=\varnothing$, every form vanishes on $X$, every $C^\infty$ form of positive degree is exact (take $\alpha=0$), so $H^2(X)=0$, $c=0$, and the conclusion is satisfied with $m=0$. Irreducibility of the $Z_i$ forces $\operatorname{cone}(Z_i)\neq\varnothing$, so with $m\ge1$ the subvarieties would have to be nonempty.
• Hypothesis strength. IsRational quantifies over all representatives and all smooth cycles in $X$ (the empty chain is always such a cycle, with period $0$); HasTypeClass asks for one representative of type $(1,1)$. IsCycleClass quantifies over all representatives $w$ of $\mathrm{cl}_i$ and all ambient $(2n-2)$-forms $\eta$ that are closed on $X$.
• Junk values. The pairing and $\int_Z$ are Bochner integrals returning $0$ when the integrand is not integrable (or not a.e. strongly measurable) with respect to the scaled Hausdorff measure restricted to $\operatorname{pts}(X)$ or $\operatorname{reg}(Z)$; the exterior derivative and Fréchet derivatives are total functions returning $0$ at non-differentiable points; the frame is chosen by Classical.epsilon and is an arbitrary tuple whenever no $J$-frame of length $\lfloor d/2\rfloor$ exists in the given tangent space (e.g. when $\operatorname{tan}(P)$ has too small a real dimension or is not $J_P$-stable), in which case the integrands are evaluated on unspecified vectors; for odd frame length the last vector is $0$ (here the lengths $2n$ and $2n-2$ are even).
• Natural subtraction. $n\mathbin{\dot-}1$ and $2n\mathbin{\dot-}2$ are genuine because $1\le n$; for $n=1$ the test forms $\eta$ are $0$-forms (functions $\mathcal{A}\to\mathbb{C}$), the frame on $Z_i$ has length $0$, and $\int_{Z_i}\eta = \int_{\operatorname{reg}(Z_i)}\eta\,d\mu_0$ with $\mu_0 = \mathcal{H}^0$ (counting measure). In $\dim Z_i$, if the infimum of tangent dimensions were $0$ or $1$ the truncated result is $0$, which meets the requirement $\dim Z_i = n-1$ exactly when $n=1$; the infimum over an empty cone is excluded by irreducibility.
• Scalars. Cohomology is a $\mathbb{C}$-vector space; the coefficients $a_i$ are rationals cast into $\mathbb{C}$, so the conclusion asserts membership of $c$ in the $\mathbb{Q}$-span of the classes $\mathrm{cl}_i$.
• Tangent-space conventions. $\operatorname{tan}_X(P)$ is a real subspace of the ambient matrix space obtained by pushing the Zariski tangent space of the affine cone (computed from the full vanishing ideal) through $D\operatorname{proj}$; "closed", "exact", "type $(p,q)$", and "vanishes" are all tested only against these subspaces at points of $\operatorname{pts}(X)$, while smoothness of forms and simplices is demanded on the entire ambient space $\mathcal{A}$, resp. all of $\mathbb{R}^{k+1}$.