10.29.13 Hermitian metrics whose $\mathrm{dd}^{\mathrm{c}}$ has a signD5CAAE8D5B60

Proof. Suppose $\widetilde\omega $ is such a form, with $\mathrm{dd}^{\mathrm{c}}\widetilde\omega \geq 0$ on every complex $2$-plane; the other case is analogous. Average the lift of $\widetilde\omega $ over the translations $(w,z)\mapsto (w+r,z+\zeta )$, $(r,\zeta )\in (\mathbb R^ s\times \mathbb C)/(\sigma _{\mathbb R},\sigma _{\mathbb C})(\mathcal O_ K)$, as in the definition of $Au$ above. The averaged form $\overline\omega $ is again $\Gamma $-invariant and Hermitian, its coefficients in the frame $\mathrm dw_1,\ldots ,\mathrm dw_ s,\mathrm dz$ depend only on $y$, and $\mathrm{dd}^{\mathrm{c}}\overline\omega \geq 0$ on complex $2$-planes, because translations are holomorphic and positivity on $2$-planes is preserved by averaging.

Let $h(y)>0$ be the coefficient of $\mathrm i\, \mathrm dz\wedge \mathrm d\overline z$ in $\overline\omega $. Invariance under a unit $\varepsilon $ gives $h(\sigma (\varepsilon )y)|\sigma _{s+1}(\varepsilon )|^2=h(y)$, that is, $h(\sigma (\varepsilon )y)=\sigma _1(\varepsilon )\cdots \sigma _ s(\varepsilon )\, h(y)$, since $\varepsilon $ has norm one. Hence $h/(y_1\cdots y_ s)$ is invariant under $U$, and since $\Lambda _ U$ is a lattice it is bounded between two positive constants:

Let $v$ be a nonzero vector in the span of $\partial /\partial w_1,\ldots ,\partial /\partial w_ s$, and let $P$ be the complex $2$-plane spanned by $v$ and $\partial /\partial z$. Write $\overline\omega =\sum _{a,b}g_{ab}(y)\, \mathrm i\, \xi _ a\wedge \overline\xi _ b$ with $\xi =(\mathrm dw_1,\ldots ,\mathrm dw_ s,\mathrm dz)$. Since the forms $\xi _ a$ are closed, $\mathrm{dd}^{\mathrm{c}}\overline\omega =\sum _{a,b}\mathrm{dd}^{\mathrm{c}}g_{ab}\wedge \mathrm i\, \xi _ a\wedge \overline\xi _ b$, and each $\mathrm{dd}^{\mathrm{c}}g_{ab}$ has only $w$-components. A term restricted to $P$ is nonzero only if it contains both $\mathrm dz$ and $\mathrm d\overline z$, which happens only for $\xi _ a=\xi _ b=\mathrm dz$. Hence the restriction of $\mathrm{dd}^{\mathrm{c}}\overline\omega $ to $P$ equals the restriction of $\mathrm{dd}^{\mathrm{c}}h\wedge \mathrm i\, \mathrm dz\wedge \mathrm d\overline z$, and its sign is that of the value of $\mathrm{dd}^{\mathrm{c}}h$ on $v$. Thus $\mathrm{dd}^{\mathrm{c}}h\geq 0$ in the $w$-variables, and $h$ is convex in $y$.

Let $R>1$ and put $A=(R,R^{-1},1,\ldots ,1)$, $B=(R^{-1},R,1,\ldots ,1)$ and $M=(A+B)/2$. By (10.29.15), $h(A),h(B)\leq C$, whereas $h(M)\geq c(R+R^{-1})^2/4$, which contradicts convexity for large $R$. In the seminegative case, $h$ is concave, so $t\mapsto h(t,\ldots ,t)$ grows at most linearly as $t\to \infty $; this contradicts the lower bound $h(t,\ldots ,t)\geq ct^ s$ with $s\geq 2$. $\square$