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12.25 Integer orders in fixed projective conesE6D8A97F4D2C

E6D8A97F4D2C

In the examples of section 12.21 the order is not an integer because a coefficient of the background vanishes at a point. We now consider backgrounds that are combinations, with strictly positive coefficient functions, of finitely many fixed closed semipositive generators on products of projective spaces and their linear blowups. Such backgrounds have integral orders even when they are not closed. The tool is a comparison with a closed semipositive form $B$ of constant rank. If $\theta $ is bounded above by a multiple of $B$ and, after the addition of a possibly singular potential, below by another multiple, then the Gauduchon quotients of $\theta +\epsilon h$ and of the closed form $B+\epsilon \Omega $ are comparable, and the latter have an explicit order. On a blown-up factor the pulled-back hyperplane form loses rank along the exceptional divisor, and the logarithmic potential of that divisor supplies the lower bound.

Lemma 12.25.1.DAA29135206D Let $X$ be a compact Kähler manifold of dimension $n$, let $\Omega $ be a Kähler form, and let $B$ be a smooth closed semipositive $(1,1)$-form. Suppose a smooth nef background $\theta $ satisfies

\[ \theta \leq CB,\qquad \theta +\mathrm{dd}^{\mathrm{c}}\varphi \geq cB \]

as currents, for constants $c,C>0$ and a real function $\varphi \in L^1(X)$. For every Hermitian form $h$ and smooth strictly positive probability form $\mu $, there are constants $k,K>0$ such that, for $0<\epsilon \leq 1$,

ED15CA2DA3FC 12.25.2
\begin{equation} k\lambda _\mu (B+\epsilon \Omega ) \leq \lambda _\mu (\theta +\epsilon h) \leq K\lambda _\mu (B+\epsilon \Omega ). \end{equation}

If $B$ has constant rank $R$, there are moreover constants $L,U>0$ with

\[ L\epsilon ^{n-R}\leq \mathcal V_\mu (\theta +\epsilon h) \leq U\epsilon ^{n-R},\qquad 0<\epsilon \leq 1. \]

In particular, $\theta $ then has common order $n-R$.

Proof. Choose $c_ h,C_ h>0$ with $c_ h\Omega \leq h\leq C_ h\Omega $. For every Gauduchon metric $\gamma $, distributional Stokes gives

\[ \int _ X\mathrm{dd}^{\mathrm{c}}\varphi \wedge \gamma ^{n-1} =\langle \varphi ,\mathrm{dd}^{\mathrm{c}}\gamma ^{n-1}\rangle =0. \]

The lower bound for $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $ therefore gives $\int _ X(\theta +\epsilon h)\wedge \gamma ^{n-1}\geq k\int _ X(B+\epsilon \Omega )\wedge \gamma ^{n-1}$ with $k=\min (c,c_ h)$, and the upper bound for $\theta $ gives the reverse inequality with $K=\max (C,C_ h)$. Dividing by $D_\mu (\gamma )>0$ and taking infima over all Gauduchon metrics proves (12.25.2); no rank hypothesis has been used.

Now suppose that $B$ has constant rank $R$. Its $R$ positive eigenvalues relative to $\Omega $ are then bounded above and away from zero, uniformly on $X$. With $A_\epsilon =B+\epsilon \Omega $, compactness gives constants $m_0,M_0>0$ with

022F404D6957 12.25.3
\begin{equation} m_0\epsilon ^{n-R}\mu \leq A_\epsilon ^ n \leq M_0\epsilon ^{n-R}\mu ,\qquad 0<\epsilon \leq 1. \end{equation}

For every Gauduchon metric $\gamma $, the pointwise arithmetic–geometric mean inequality gives

\[ A_\epsilon \wedge \gamma ^{n-1} \geq \left(\frac{A_\epsilon ^ n}{\gamma ^ n}\right)^{1/n}\gamma ^ n, \]

and hence $\lambda _\mu (A_\epsilon )\geq m_0^{1/n}\epsilon ^{(n-R)/n}$. In the other direction, the positive closed form $\gamma =A_\epsilon $ is itself an admissible test. Writing $A_\epsilon ^ n=f_\epsilon \mu $, we have

\[ \begin{aligned} D_\mu (A_\epsilon )& =\int _ X f_\epsilon ^{(n-1)/n}\mu \geq m_0^{(n-1)/n}\epsilon ^{(n-R)(n-1)/n},\\ \int _ X A_\epsilon ^ n& \leq M_0\epsilon ^{n-R}, \end{aligned} \]

so the quotient of this test is at most $M_0m_0^{-(n-1)/n}\epsilon ^{(n-R)/n}$. Raising both bounds to the $n$th power and using (12.25.2) gives the two-sided power bounds. The argument includes $R=0$; when $n=1$, $D_\mu =1$. Fixed positive factors do not affect the order, since their logarithms are divided by $\log \epsilon $. $\square$

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.

12.25.10 Equality of orders under modificationsA34399DFCD46

By proposition 12.22.5, pulling back by a modification can only increase the orders. For the fixed cone, the orders are preserved.

Proposition 12.25.11.57BC641DA62E Replacing some plain factors by linear blowups as above preserves the order of every background in the fixed cone. More generally, for every modification $\pi :Y\to X$ with $Y$ a compact Kähler manifold, the pullback $\pi ^*\theta $ has common order $n-R(\theta )$, and satisfies two-sided power bounds with this exponent for every smooth positive probability form and every Hermitian perturbation on $Y$.

Proof. For a product of linear blowups, pull back the coefficient functions and take the new coefficients of the forms $Q_ i$ to be zero. A form $H_ i$ that occurs on a plain factor becomes a form $H_ i$ that occurs on a blown-up factor, with the same rank $m_ i$, and factors in which nothing occurs remain so. Hence theorem 12.25.8, applied on both products, gives the same order.

For a general compact Kähler manifold $Y$, pull back the semipositive smooth remainders and the divisor identities used in the proof above. Each nonzero holomorphic section remains nonzero under the surjective map $\pi $, and the logarithm of the squared norm of its pullback is locally integrable, with $\mathrm{dd}^{\mathrm{c}}$ equal to the current of its effective zero divisor minus the pulled-back curvature. Hence

\[ \pi ^*\theta \leq C\pi ^*B_\theta ,\qquad \pi ^*\theta +\mathrm{dd}^{\mathrm{c}}(\varphi \circ \pi )\geq c\pi ^*B_\theta . \]

Fix a Kähler form $\Omega _ Y$ on $Y$, and apply the first part of lemma 12.25.1, which needs no rank hypothesis; the pulled-back background is semipositive and hence nef. Put $R=R(\theta )$. Pullback does not increase rank, so $(\pi ^*B_\theta )^{R+1}=0$, while

\[ \int _ Y(\pi ^*B_\theta )^ R\wedge \Omega _ Y^{n-R}>0: \]

for $R>0$ the integrand is positive on the nonempty set where $\pi $ is biholomorphic, and for $R=0$ the integral is the Kähler volume. For any smooth positive probability form $\mu _ Y$, the closed nef formula following theorem 12.11.1 gives

\[ \mathcal V_{\mu _ Y}(\pi ^*B_\theta +\epsilon \Omega _ Y) =\sum _{k=0}^{R}\binom nk\epsilon ^{n-k} \int _ Y(\pi ^*B_\theta )^ k\wedge \Omega _ Y^{n-k}. \]

All coefficients are nonnegative and the coefficient for $k=R$ is positive, so this polynomial is bounded above and below by positive multiples of $\epsilon ^{n-R}$ for $0<\epsilon \leq 1$. The comparison of quotients gives the same bounds for $\pi ^*\theta +\epsilon h_ Y$. In particular, the order upstairs equals the order downstairs. $\square$

12.25.12 Subtracting effective divisorsA528EDFC5F96

We now subtract effective divisors from members of the fixed cone. Let $\mathcal O(H_ i)$ and $\mathcal O(Q_ i)$ be the pullbacks of $\mathcal O(1)$ under the projections defining $H_ i$ and $Q_ i$, with their Fubini–Study metrics, whose curvature forms are $H_ i$ and $Q_ i$. Consider line bundles that are tensor products of integer powers of these bundles, negative powers included, and finite nonnegative real combinations of zero divisors of their nonzero holomorphic sections $s_\nu $:

\[ \begin{gathered} D=\sum _\nu t_\nu \operatorname {div}(s_\nu ),\quad t_\nu \geq 0,\\ \eta _ D=\sum _ i(\delta _ iH_ i+\rho _ iQ_ i),\qquad \Phi _ D=\sum _\nu t_\nu \log |s_\nu |^2. \end{gathered} \]

Here $\eta _ D$ is the corresponding combination of the curvature forms of the fixed metrics, so that $[D]=\eta _ D+\mathrm{dd}^{\mathrm{c}}\Phi _ D$. The terms involving $Q_ i$ are omitted for plain factors.

Lemma 12.25.13.5DAA9B186CEE The coefficients $(\delta _ i,\rho _ i)$ that occur for divisors $D$ of this form are exactly those satisfying

\[ \delta _ i\geq 0\text{ on plain factors};\qquad \delta _ i\geq 0,\quad \delta _ i+\rho _ i\geq 0 \text{ on blown-up factors}. \]

Proof. It suffices to treat one nonzero holomorphic section $s$, since nonnegative real combinations preserve the inequalities. Choose a point at which $s$ is nonzero and which lies outside all exceptional divisors, and fix the coordinates of the other factors. A projective line through that point inside a fiber of $q_ i$ has $H_ i$-degree one and $Q_ i$-degree zero. The restriction of $s$ to it is a nonzero section, so the exponent of $\mathcal O(H_ i)$ is nonnegative. On a plain factor, a line through the point gives the same inequality.

On a blown-up factor, write the chosen point as $([z:w],[w])$ with $w\ne 0$, and choose a linear map $A$ with $Aw=z$. The section of $q_ i$ given by $[t]\mapsto ([At:t],[t])$ passes through the point, and both $H_ i$ and $Q_ i$ have degree one on each projective line in its image. Such a line through the point exists because $r_ i\geq 1$, and $s$ restricts nontrivially to it. Its degree shows that the sum of the two exponents is nonnegative.

Conversely, let $F_ i$ be the pullback under $q_ i$ of a hyperplane. An admissible pair $(\delta _ i,\rho _ i)$ is realized by $\delta _ iE_ i+(\delta _ i+\rho _ i)F_ i$, since $E_ i$ has curvature $H_ i-Q_ i$, and coefficients on plain factors are realized by hyperplanes. These are allowed nonnegative combinations and give every coefficient tuple in the stated range. $\square$

Theorem 12.25.14.F79036F002E1 Let $\theta $ be in the fixed cone, with parameters $a_ i,b_ i,c_ i,d_ i$, and let $D$ be a divisor of the above form. Assume

\[ a_ i\geq \delta _ i\text{ for all }i,\qquad c_ i\geq \rho _ i\text{ for every blown-up factor}. \]

Then the residual $\beta =\theta -\eta _ D$ lies in the fixed cone, and the common orders $a(\beta )$ and $a(\theta )$ satisfy

\[ a(\beta )-a(\theta )=\sum _ i\bigl (R_ i(\theta )-R_ i(\beta )\bigr )\geq 0. \]

Equality holds if and only if, for every factor,

\begin{align*} a_ i-\delta _ i+b_ i& >0& & \text{whenever }a_ i+b_ i>0,\\ c_ i-\rho _ i+d_ i& >0& & \text{whenever }a_ i=b_ i=0\text{ and }c_ i+d_ i>0; \end{align*}

the second condition concerns only blown-up factors.

Proof. The residual parameters $(a_ i-\delta _ i,b_ i,c_ i-\rho _ i,d_ i)$ are nonnegative, so $\beta $ lies in the fixed cone; in particular it is smooth, semipositive and nef. Also

\[ (\theta -\beta )+\mathrm{dd}^{\mathrm{c}}\Phi _ D=[D]\geq 0, \]

which is the positive-current comparison of proposition 12.24.1. Apply theorem 12.25.8 to $\theta $ and to $\beta $. If $H_ i$ occurs in $\theta $ but not in $\beta $, the rank $R_ i$ drops from $m_ i$ to $r_ i<m_ i$ or to zero. If $H_ i$ does not occur in $\theta $, then $a_ i=b_ i=0$, and the range of lemma 12.25.13 together with $a_ i\geq \delta _ i$ forces $\delta _ i=0$ and hence $\rho _ i\geq 0$; thus $Q_ i$ cannot appear in $\beta $ if it was absent from $\theta $. Therefore no rank $R_ i$ increases, each difference $R_ i(\theta )-R_ i(\beta )$ is nonnegative, and the sum vanishes exactly when the generator that determines $R_ i(\theta )$ still occurs in $\beta $ for every $i$. This is the stated pair of conditions. $\square$

The parameter inequalities guarantee that the residual stays in the fixed cone; they do not describe all divisors whose residual is nef.

Example 12.25.15.EB3698E8A068 On $Z=\operatorname {Bl}_{\mathbb P^\ell }\mathbb P^ m$, choose a smooth function $g>0$ with $\mathrm dg\wedge Q\ne 0$. For $t>0$, the nonclosed positive background $\theta _ t=tH+gQ$ has order zero. Removing $sE$, with its fixed curvature form $s(H-Q)$ and $0\leq s\leq t$, gives

\[ \beta _ s=(t-s)H+(g+s)Q,\qquad \theta _ t+\mathrm{dd}^{\mathrm{c}}(s\psi _ E)=s[E]+\beta _ s. \]

Every residual $\beta _ s$ is nef and not closed. Its order equals that of $\theta _ t$ exactly for $s<t$, and at $s=t$ the order jumps by $\ell +1$. These examples realize each integer jump $1,\ldots ,m-1$.

Proof. The lower bound $\theta _ t\geq \min (t,\min g)\kappa >0$ shows that $\theta _ t$ is positive. For $s<t$ the form $H$ occurs in $\beta _ s$ and the rank is $m$; at $s=t$ only $Q$ occurs, with rank $r=m-\ell -1$. The order formula gives zero and $m-r=\ell +1$, respectively. The exterior derivative of every residual is $\mathrm dg\wedge Q\ne 0$. Finally, let $\ell $ range from zero to $m-2$. $\square$

At the endpoint $s=t$, the current $t[E]+\beta _ t$ in the class of $\theta _ t$ has a divisorial part, although the minimal multiplicities of $\theta _ t$ vanish because $\theta _ t$ is nef, as explained before corollary 12.24.5. The construction uses three hypotheses in an essential way: the coefficient functions are strictly positive, the subtracted divisor keeps the residual in the fixed cone, and, for general modifications, the source manifold is Kähler. Without such hypotheses the question in 12.33.4 is open.

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