12.34 Current tests for smooth positivity509A5021C77D

Proof. If $\theta +\mathrm{dd}^{\mathrm{c}}u\geq ah$ with $a>0$ and $u$ smooth, then integration by parts gives $\int _ X\theta \wedge T\geq a$ for every $T\in \mathscr G_ h$. Since a smooth positive representative satisfies such a bound for some $a>0$ by compactness, this proves one implication.

For the converse, work in the real Fréchet space of smooth real $(1,1)$-forms. Let $\mathcal K$ be its open convex cone of strictly positive forms, and let $\mathcal L=\mathrm{dd}^{\mathrm{c}}C^\infty (X,\mathbb R)$. If the affine subspace $\theta +\mathcal L$ is disjoint from $\mathcal K$, the Hahn–Banach separation theorem for an open convex set and a disjoint affine subspace gives a nonzero continuous real linear functional $T$ with

Replacing $\ell $ by arbitrary real multiples shows that $T(\mathcal L)=0$. Scaling a fixed $\eta \in \mathcal K$ to zero shows that $T(\theta )\leq 0$, while scaling it to infinity shows that $T(\eta )\geq 0$. Since every smooth semipositive form is a limit of strictly positive forms, $T$ is a positive current, and positivity makes it a current of order zero. The equality $T(\mathcal L)=0$ says exactly that $\mathrm{dd}^{\mathrm{c}}T=0$. Since $T\ne 0$ and $h>0$, the trace $T(h)$ is positive, and normalizing $T$ gives an element of $\mathscr G_ h$ with nonpositive pairing against $\theta $. This contradicts $m_ h(\theta )>0$ and proves the converse.

Applying the equivalence just proved to $\theta -ah$, and using $m_ h(\theta -ah)=m_ h(\theta )-a$, shows that the set in the supremum is exactly $(-\infty ,m_ h(\theta ))$. Finally, $\theta $ is nef if and only if $\theta +\varepsilon h$ has a smooth positive representative for every $\varepsilon >0$, that is, if and only if $m_ h(\theta )+\varepsilon >0$ for all such $\varepsilon $, or equivalently $m_ h(\theta )\geq 0$. $\square$

Proof. In dimension one the hypotheses cannot hold, since $\int _ X\theta \wedge T=\lambda _\mu (\theta )/\int _ Xh>0$. Suppose $n\geq 2$, and write $D_\mu (T)$ for the total mass of this determinant-weighted measure; for $T=\gamma ^{n-1}$ it is the denominator $D_\mu (\gamma )$ of definition 12.2.1. If $\eta \in \mathscr P$, its solution form $\Omega $ satisfies $\Omega ^ n=\lambda _\mu (\eta )^ n\mu $, so $\Omega /\lambda _\mu (\eta )$ is an admissible test in the infimum of lemma 12.7.2. Together with Stokes' theorem, this gives

Apply this to $\eta =\theta +\varepsilon h$. The left side equals $\varepsilon $, while theorem 12.11.1 gives $\lambda _\mu (\eta )\to \lambda _\mu (\theta )>0$. Hence $D_\mu (T)=0$. The determinant-weighted measure is positive and has total mass zero, so it vanishes. $\square$

Proof. Call a time $t\geq 0$ admissible if $\theta _ t$ has a smooth positive representative. The admissible times form an interval containing zero, since convex combinations of positive representatives at two times are positive representatives at the intermediate times, and the set is open because positivity is an open condition. We show that the flow exists on $[0,t_1)$ for every admissible $t_1>0$ and that it cannot exist beyond $\tau $.

Step 1: the scalar equation. Fix an admissible $t_1>0$, and choose a smooth real function $a$ with $\theta _{t_1}+\mathrm{dd}^{\mathrm{c}}a>0$. The reference forms

are uniformly positive, with all derivatives uniformly bounded. Put

Then $\dot\gamma _ t=-\operatorname {Ric}(\Omega )$, where the Ricci form of a volume form is defined by the same local expression. Consider the scalar equation

Applying $\mathrm{dd}^{\mathrm{c}}$ shows that $\omega (t)=\gamma _ t+\mathrm{dd}^{\mathrm{c}}u(t)$ solves the metric flow. Conversely, for a solution $\omega (t)$ of the metric flow, the function $u(t)=2\pi \int _0^ t\log (\omega (s)^ n/\Omega )\, \mathrm ds$ satisfies $\omega (t)=\gamma _ t+\mathrm{dd}^{\mathrm{c}}u(t)$ and solves (12.34.5). The linearization of (12.34.5) is $\partial _ t-2\pi \operatorname {tr}_{\omega (t)}\mathrm{dd}^{\mathrm{c}}$, a strictly parabolic operator, so the classical short-time existence theory for smooth strictly parabolic equations gives a solution. The difference of two solutions satisfies a linear parabolic equation whose principal coefficients, the averages of the inverse metrics along the segment between the two solutions, are positive; the maximum principle therefore proves uniqueness.

Suppose that the maximal time $t_*$ of this solution satisfies $t_*<t_1$. We derive estimates independent of $t<t_*$ that allow us to continue the solution.

Step 2: bounds for $u$ and $\dot u$. For a large constant $C$, the functions $Ct$ and $-Ct$ are a supersolution and a subsolution of (12.34.5), respectively: at a spatial extremum of $u$ the complex Hessian has the appropriate sign, and $\gamma _ t^ n/\Omega $ is bounded above and away from zero. The maximum principle gives $|u|\leq Ct_1$.

Set $v=\dot u$ and $\mathcal L=2\pi \operatorname {tr}_{\omega (t)}\mathrm{dd}^{\mathrm{c}}$. Differentiation gives

Since $\gamma _ t$ is affine in time, these identities imply

By the maximum principle, the first expression is bounded above by its initial value zero, and the second is bounded below by $\min _ Xt_1v(0)$. The bound for $u$ and $t_1-t\geq t_1-t_*>0$ give a lower bound for $v$. The first inequality gives an upper bound for $v$ on $[t_*/2,t_*)$, and the smooth solution bounds it on $[0,t_*/2]$. In particular, $\omega (t)^ n/\Omega =\mathrm{e}^{v/(2\pi )}$ is bounded above and away from zero.

Step 3: a parabolic trace inequality. We next bound the trace, keeping track of the time derivative of the reference metric. In the coordinates of lemma 14.5.1, write $g(t)$ for the matrix of $\gamma _ t$ and, as in that lemma, $h=g+(u_{i\bar j})$ for the matrix of $\omega (t)$. Put

so that $\log (\det h/\det g)=v/(2\pi )+F$. The family $g(t)$ has uniformly bounded Chern torsion and curvature, so at each fixed time lemma 14.5.1 applies with uniform constants. Direct differentiation of $S$ gives

In $(\partial _ t-\mathcal L)\log S$, the term $\Delta _ gv/S$ cancels the corresponding term in the lower bound for $\mathcal L\log S$ given by the lemma. With $\tau _ k=\sum _ iT_{ik}^ i$ the torsion trace of $g(t)$, as in the lemma, we obtain on $\{ S\geq 1\} $

Indeed, $\Delta _ gF$ and $\operatorname {tr}_ g\dot g$ are uniformly bounded, $S^{-1}\leq B/n^2$, and $-Cg\leq \dot g\leq Cg$ bounds the last quotient above. These bounds account for all the terms coming from the time dependence of the reference metric.

Step 4: the trace bound. Choose a constant $C_0$ such that $1\leq s=u+C_0\leq D$ on the whole time interval, and maximize

on $X\times [0,t_0]$, where $t_0<t_*$ and the constant $A\geq 1$ will be fixed independently of $t_0$. If the maximum is attained at time zero, the initial data bound $Q$. At a maximum with positive time, $\partial \log S=(A+s^{-2})\partial u$. Since $|\tau |_ h^2\leq CB$, Young's inequality bounds the absolute value of the first term on the right of (12.34.6) by

If $S^2\leq (A+1)^2D^3$, the trace is already bounded at the maximum. Otherwise the last coefficient is at most $C$, and

Choosing $A$ with $2\pi A>C+1$ bounds $B$ at the maximum. In terms of the eigenvalues, $S\leq n(\det h/\det g)B^{n-1}$ for $n\geq 2$, and $S=\det h/\det g$ for $n=1$, so the determinant bound of Step 2 bounds $S$ at the maximum in every case. The bound for $u$ transfers this estimate from $Q$ to $S$ everywhere, uniformly in $t_0$, and the lower bound for the determinant then bounds the smallest eigenvalue away from zero. Thus $\omega (t)$ and $\gamma _ t$ are uniformly equivalent for $t<t_*$.

Step 5: higher estimates and continuation. Uniform ellipticity now gives the higher estimates by classical local parabolic regularity. In a coordinate ball, write the logarithmic determinant as a concave function of the complex part of the real Hessian. On the compact set of positive matrices just obtained, its supporting affine functions have uniformly positive and bounded slopes, and their infimum extends the operator to a concave, uniformly elliptic function of all real symmetric matrices. The dependence on space and time enters only through the smooth family $g(t)$ and the fixed density $\Omega $. The parabolic Evans–Krylov estimate gives uniform $C^{2+\alpha ,1+\alpha /2}$ bounds away from the initial time, and the smooth solution covers a fixed initial interval. Differentiating the equation and applying parabolic Schauder estimates then bounds all spatial and time derivatives uniformly. This is the parabolic counterpart of the local argument used in corollary 14.5.8.

The bounds for the time derivatives show that $u(t)$ converges in every spatial $C^ k$ norm as $t\uparrow t_*$. By uniform ellipticity, the smooth limit still has a strictly positive solution form, and short-time existence with this limit as initial datum extends the solution beyond $t_*$, a contradiction. Hence the flow exists up to every admissible time $t_1$, and by uniqueness these solutions patch together on $[0,\tau )$.

Step 6: maximality. Integrating the metric flow gives

Hence every time at which a smooth positive solution exists is admissible, which proves maximality. $\square$

Proof. For $t\geq 0$, the normalization of the test currents gives the identity

Theorem 12.34.1, Theorem 12.34.4 give the formula for $\tau $. When $b>0$, the maximum defining $b$ is attained by weak compactness, and a maximizer $T_*$ annihilates $\theta _\tau $. The same identity gives $m_{\omega _0}(\theta _\tau )=0$, hence nefness. The last assertion is proposition 12.34.2. $\square$

Proof. The initial metric is strictly positive, and

With our normalization, the Ricci forms of the normalized Fubini–Study metrics are $2\pi n\omega $ and $4\pi w$. Consequently

Write $s$ for the flow time and put $r=2\pi s$. The affine background is

For $0\leq r<1$, adding $\mathrm{dd}^{\mathrm{c}}$ of the smooth function $-r\log (a+2)$ gives a strictly positive representative: the horizontal coefficient is at least one and the vertical coefficient is at least $2(1-r)>0$. At $r=1$ we obtain the identity $\alpha _\tau =\theta +\mathrm{dd}^{\mathrm{c}}\log (a+2)$. Since the logarithm is a smooth function on $X$, the bijection $u\mapsto u-\log (a+2)$ between the potentials of the two backgrounds preserves nefness and bigness, and Stokes' theorem for Gauduchon metrics shows that the variational volume is unchanged. Theorem 12.14.6 therefore shows that the endpoint background is nef, not big, and of volume $n$.

For $r>1$, choose a nonzero smooth nonnegative function $f$ on the base supported in $\{ a+2-2r<0\} $, a neighborhood of $p$. Its pullback $S=f\omega ^ m$ is a closed semipositive $(n-1,n-1)$-form. For every quasi-plurisubharmonic function $v$, distributional Stokes gives

so $\alpha _ s$ is not even pseudoeffective for $r>1$. At $r=1$, a smooth positive representative would contradict nonbigness. Thus $r=1$ is the supremum of the admissible times, and theorem 12.34.4 gives the smooth flow on $[0,\tau )$ with exactly this lifetime; the reference representatives used above need not solve the flow equation. Finally, proposition 12.15.1 gives positivity of the volume for every smooth positive probability form. $\square$

Proof. Only smooth forms need to be restricted to $Z$, and their restrictions satisfy $\omega _0|_ Z=2w$ and $\alpha _\tau |_ Z=0$. This proves that $[Z]/2$ is normalized and annihilates $\alpha _\tau $.

Let $R$ be a general annihilator. Since $\alpha _\tau $ and $\theta $ differ by $\mathrm{dd}^{\mathrm{c}}$ of a smooth function, Stokes' theorem gives $\int _ X\theta \wedge R=0$. The measure $\theta \wedge R$ is nonnegative, so it vanishes. On every compact set $K\subset X\setminus Z$, the explicit form $\theta $ satisfies $\theta \geq c_ K\omega _0$ for some $c_ K>0$. Thus $\sigma _ R(K)=0$, and a countable exhaustion by compact sets shows that $\sigma _ R$ is supported on $Z$.

Relative to $\omega _0$, the matrix of $R$ is characterized, for smooth test forms $\beta $ with matrix $B_\beta $, by

At points of $Z$, the form $\theta $ is positive on the horizontal $m$-plane and vanishes on the vertical line. The vanishing of $\theta \wedge R$ therefore forces all horizontal diagonal entries of $P_ R$ to vanish, and the inequality $|(P_ R)_{ij}|^2\leq (P_ R)_{ii}(P_ R)_{jj}$ for positive matrices removes their rows and columns. The remaining vertical entry is $n$, which proves the formula and the vanishing of the determinant. $\square$

Proof. Each of these metrics has constant coefficients, so its Chern–Ricci form vanishes; differentiation in $t$ proves the flow equations, and positivity of the coefficients gives the stated existence intervals. In the first family, pair a hypothetical positive representative at a time $t\geq 1$ with the closed positive form $\eta _1\wedge \eta _3\wedge \cdots \wedge \eta _ n$, where the empty product is $1$ when $n=2$. The potential term integrates to zero, and the pairing is a positive fixed multiple of $1-t$, so it cannot be strictly positive. Integrating the flow equation shows that every smooth solution differs from the affine background by $\mathrm{dd}^{\mathrm{c}}$ of a smooth function, so no smooth positive continuation beyond $t=1$ is possible.

Projection to the first elliptic factor does not increase lengths once the metric of that factor is scaled by $1+Mt$. Taking two points that differ only in the first factor proves the lower bound for the diameter. For the second flow, every path length is multiplied by $\sqrt{1+t}$, which proves the exact formula for the diameter. $\square$