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12.25.4 Linear blowups and the fixed generators6EBDFF04B7CA

The manifolds are products of factors of two kinds. A plain factor is $Z_ i=\mathbb P^{m_ i}$ with $m_ i\geq 1$; let $H_ i$ be its Fubini–Study form normalized to have integral one on lines, and set $\kappa _ i=H_ i$. A blown-up factor is

\[ Z_ i=\operatorname {Bl}_{\mathbb P^{\ell _ i}}\mathbb P^{m_ i}, \qquad m_ i\geq 2,\quad 0\leq \ell _ i\leq m_ i-2, \qquad r_ i=m_ i-\ell _ i-1, \]

the blowup of $\mathbb P^{m_ i}$ along a linear subspace of dimension $\ell _ i$. We describe one such factor and suppress the index for the moment. In homogeneous coordinates $[z_0:\cdots :z_\ell :w_0:\cdots :w_ r]$ on $\mathbb P^ m$, with the center $\{ w=0\} $, the blowup has the incidence model

\[ Z=\{ (x,[t])\in \mathbb P^ m\times \mathbb P^ r: w_ at_ b=w_ bt_ a\text{ for all }a,b\} . \]

Let $p:Z\to \mathbb P^ m$ and $q:Z\to \mathbb P^ r$ be the projections. In the chart $t_ j=1$, the equations $w_ a=t_ aw_ j$ are the usual charts of the blowup along $\{ w_0=\cdots =w_ r=0\} $. Writing $w=\lambda t$ shows that $q$ is locally a $\mathbb P^{\ell +1}$-bundle, with fiber coordinates $[z:\lambda ]$. Thus $Z$ is smooth, $q$ is a submersion, and the incidence inclusion is an embedding. With $\omega _ m$ and $\omega _ r$ the normalized Fubini–Study forms on $\mathbb P^ m$ and $\mathbb P^ r$, set

\[ H=p^*\omega _ m,\qquad Q=q^*\omega _ r,\qquad \kappa =H+Q. \]

The form $\kappa $ is Kähler, as the restriction of a product Kähler form, whereas $Q$ is semipositive of constant rank $r$.

Let $E=\{ w=0\} $ be the exceptional divisor. The coordinate section $w_ j$ of $p^*\mathcal O(1)$ vanishes exactly on $E+q^{-1}\{ t_ j=0\} $, so $\mathcal O(E)=p^*\mathcal O(1)\otimes q^*\mathcal O(-1)$. Give this line bundle the product metric, whose curvature is $H-Q$. For its canonical section $s_ E$, the function $\psi _ E=\log |s_ E|^2$ is locally integrable, and the Poincaré–Lelong formula, as in section 12.35, gives

00C606303647 12.25.5
\begin{equation} \mathrm{dd}^{\mathrm{c}}\psi _ E=[E]-H+Q,\qquad H+\mathrm{dd}^{\mathrm{c}}(\psi _ E/2)=\tfrac 12(\kappa +[E])\geq \kappa /2. \end{equation}

On $X=\prod _ i Z_ i$, put $n=\sum _ i m_ i$, and pull back all forms from the factors. Fix smooth strictly positive functions $f_ i$ on $X$, and, for each blown-up factor, a smooth strictly positive function $g_ i$ on $X$; these functions may depend on all coordinates of the product. The fixed cone consists of the backgrounds

DE6D982F2E21 12.25.6
\begin{equation} \theta =\sum _ i\bigl ((a_ i+b_ if_ i)H_ i+(c_ i+d_ ig_ i)Q_ i\bigr ), \qquad a_ i,b_ i,c_ i,d_ i\geq 0, \end{equation}

where the terms involving $Q_ i$, and the parameters $c_ i,d_ i$, are omitted for plain factors. All coefficient functions and generators are fixed, and only the parameters vary. Every member of the cone is smooth and semipositive, hence nef. We say that $H_ i$ occurs in $\theta $ if $a_ i+b_ i>0$, and that $Q_ i$ occurs if $c_ i+d_ i>0$. Define

EFE1F66045F4 12.25.7
\begin{equation} R_ i(\theta )= \begin{cases} m_ i,& \text{if }H_ i\text{ occurs},\\ r_ i,& \text{if }H_ i\text{ does not occur and }Q_ i\text{ occurs},\\ 0,& \text{if neither occurs}, \end{cases} \qquad R(\theta )=\sum _ i R_ i(\theta ). \end{equation}

The middle case can only occur on blown-up factors.

Theorem 12.25.8.3F58257FA940 Every background $\theta $ in the fixed cone (12.25.6) has common order $n-R(\theta )$. More precisely, for every smooth positive probability form $\mu $ and Hermitian form $h$ there are constants $L,U>0$, depending on $\theta ,\mu ,h$, such that

79FA42084398 12.25.9
\begin{equation} L\epsilon ^{n-R(\theta )} \leq \mathcal V_\mu (\theta +\epsilon h) \leq U\epsilon ^{n-R(\theta )},\qquad 0<\epsilon \leq 1. \end{equation}

Proof. Let $B_\theta $ be the sum of the forms $\kappa _ i$ over the factors in which $H_ i$ occurs and of the forms $Q_ i$ over the factors in which only $Q_ i$ occurs. It is closed and semipositive of constant rank $R(\theta )$. Since the coefficients are bounded and $H_ i,Q_ i\leq \kappa _ i$, we have $\theta \leq CB_\theta $ for some $C>0$.

For the lower bound, let $s_ i=\min _ X(a_ i+b_ if_ i)>0$ for each blown-up factor in which $H_ i$ occurs, and put

\[ \varphi =\sum _{\substack {i\text{ blown up}\\ H_ i\text{ occurs}}} (s_ i/2)\psi _{E_ i}. \]

By (12.25.5), $s_ iH_ i+\mathrm{dd}^{\mathrm{c}}(s_ i\psi _{E_ i}/2)=(s_ i/2)(\kappa _ i+[E_ i])\geq (s_ i/2)\kappa _ i$. The remaining terms $(a_ i+b_ if_ i-s_ i)H_ i$ and all terms involving $Q_ i$ are semipositive. On plain factors in which $H_ i$ occurs and on factors in which only $Q_ i$ occurs, the relevant coefficient is bounded below by a positive constant. Taking the minimum of these constants gives $\theta +\mathrm{dd}^{\mathrm{c}}\varphi \geq cB_\theta $ for some $c>0$. For $\theta =0$ take $\varphi =0$ and any $c>0$. Now apply lemma 12.25.1 with $\Omega =\sum _ i\kappa _ i$. $\square$

Members of the fixed cone need not be closed: by construction

\[ \mathrm d\theta =\sum _ i b_ i\, \mathrm df_ i\wedge H_ i +\sum _{i\text{ blown up}}d_ i\, \mathrm dg_ i\wedge Q_ i. \]

If only one of these terms has a nonzero coefficient and the corresponding form, say $\mathrm df_ i\wedge H_ i$, is nonzero, then $\theta $ is not closed. On a blown-up factor, for instance, if $g$ is a small positive perturbation of a constant that varies along a fiber of $q$, then $\mathrm dg\wedge Q\ne 0$, since $r\geq 1$. On $\mathbb P^ p\times \mathbb P^ q$ with $p,q\geq 1$, a positive function $f$ that varies on the second factor gives the nonclosed background $fH_1$ of order $q$. Adding $\mathrm{dd}^{\mathrm{c}}$ of a smooth function does not change the exterior derivative.

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