Theorems Pointwise Hodge-Riemann Bilinear Relations for Primitive Forms Attributions

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Let $(V,h)$ be a Hermitian complex [vector space](/page/Vector%20Space) of complex dimension $n$. Let $V_{\mathbb R}$ be the underlying real vector space, and let

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\begin{align*} \Lambda^\bullet V^*:=\Lambda^\bullet(V_{\mathbb R}^*)\otimes_{\mathbb R}\mathbb C \end{align*}

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denote the complexified exterior algebra. Let $V^{1,0*}$ and $V^{0,1*}$ denote the $(1,0)$ and $(0,1)$ cotangent subspaces determined by the complex structure, and let the induced type decomposition be

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\begin{align*} \Lambda^\bullet V^*=\bigoplus_{p,q}\Lambda^{p,q}V^*. \end{align*}

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Let $\omega\in\Lambda^{1,1}V^*$ be the Kähler form determined by $h$, and let

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\begin{align*} dV:=\frac{\omega^n}{n!} \end{align*}

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be the positive volume form. Let $*: \Lambda^jV^*\to\Lambda^{2n-j}V^*$ be the Hodge star associated to $h$ and $dV$, characterized by

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\begin{align*} \eta\wedge *\overline{\xi}=(\eta,\xi)_h dV \end{align*}

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for all $\eta,\xi\in\Lambda^jV^*$, where $(\cdot,\cdot)_h$ is linear in the first argument and conjugate-linear in the second. Define the Lefschetz operator $L:\Lambda^mV^*\to\Lambda^{m+2}V^*$ by

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\begin{align*} L(\eta)=\omega\wedge\eta. \end{align*}

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Let $\Lambda$ be the adjoint of $L$ with respect to the Hermitian [inner product](/page/Inner%20Product) on exterior powers induced by $h$. For $0\le p,q\le n$, define the primitive subspace

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\begin{align*} P^{p,q}:=\{\alpha\in\Lambda^{p,q}V^*:\Lambda\alpha=0\}. \end{align*}

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Assume the pointwise primitive Hodge star identity: whenever $k=p+q\le n$ and $\alpha\in P^{p,q}$,

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\begin{align*} *\overline{\alpha}=i^{p-q}(-1)^{k(k-1)/2}\frac{\omega^{n-k}}{(n-k)!}\wedge\overline{\alpha}. \end{align*}

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Assume also the finite-dimensional Lefschetz decomposition in each degree:

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\begin{align*} \Lambda^\ell V^*=\bigoplus_{r,p,q}L^rP^{p,q},\qquad p+q=\ell-2r, \end{align*}

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where the sum ranges over integers $r,p,q$ satisfying $0\le\ell\le 2n$, $0\le r\le\lfloor \ell/2\rfloor$, $m:=\ell-2r=p+q\le n$, and $n-\ell+r\ge0$. For every such admissible triple $(r,p,q)$, assume that the restricted map $L^r:P^{p,q}\to L^rP^{p,q}$ is injective.

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Let $k=p+q\le n$. For $\alpha,\beta\in P^{p,q}$, define $Q_{p,q}(\alpha,\beta)$ to be the unique scalar satisfying

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\begin{align*} i^{p-q}(-1)^{k(k-1)/2}\alpha\wedge\overline{\beta}\wedge\frac{\omega^{n-k}}{(n-k)!}=Q_{p,q}(\alpha,\beta)\frac{\omega^n}{n!}. \end{align*}

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Then $Q_{p,q}$ is positive definite: for every nonzero $\alpha\in P^{p,q}$,

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\begin{align*} Q_{p,q}(\alpha,\alpha)>0. \end{align*}

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More generally, fix $0\le \ell\le 2n$. For every integer $r$ with $0\le r\le \lfloor \ell/2\rfloor$, set $m=\ell-2r$. Assume $m\le n$ and $n-\ell+r\ge 0$. For every decomposition $m=p+q$, define a Hermitian form on the Lefschetz summand $L^rP^{p,q}\subseteq\Lambda^\ell V^*$ by declaring, for $\alpha,\beta\in P^{p,q}$, that $Q_\ell(L^r\alpha,L^r\beta)$ is the unique scalar satisfying

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\begin{align*} i^{p-q}(-1)^{m(m-1)/2}\frac{r!}{(n-\ell+r)!}\alpha\wedge\overline{\beta}\wedge\omega^{n-\ell+2r}=Q_\ell(L^r\alpha,L^r\beta)\frac{\omega^n}{n!}. \end{align*}

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This definition is made using the assumed injectivity of $L^r:P^{p,q}\to L^rP^{p,q}$. Declare distinct Lefschetz summands $L^rP^{p,q}$ and $L^sP^{p',q'}$ to be $Q_\ell$-orthogonal whenever $(r,p,q)\ne(s,p',q')$. With respect to the finite-dimensional Lefschetz decomposition

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\begin{align*} \Lambda^\ell V^*=\bigoplus_{r,p,q}L^rP^{p,q},\qquad p+q=\ell-2r, \end{align*}

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the form $Q_\ell$ is orthogonal across the displayed summands and is positive definite on every nonzero summand.

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