Pome

21.14 Hopf fibrations and toric potentials23660BEC4C1F

23660BEC4C1F

Let $n\geq 2$ and let $X_ n$ be the scalar Hopf manifold of subsection 19.8.12, which fibers over $\mathbb P^{n-1}$ with compact elliptic fibers. A potential in a pulled-back closed class is constant on each fiber and therefore descends to the base. For such a weight, the weighted section integrands also descend, and fiber integration identifies the section norms and Bergman kernels on the two spaces.

A potential of the standard Gauduchon form may vary along the fibers. For one that descends, the vertical part of this form supplies a single factor in the Hermitian Monge–Ampère measure, and fiber integration reduces the equation to the projective base. For bounded descended potentials, closedness on the base determines the normalizing constant and gives uniqueness up to additive constants. On the Hopf threefold, the descended masses are at most six, so unbounded masses require variation along the fibers.

On the base, invariance under separate rotations of the homogeneous coordinates gives a real convex description in logarithmic coordinates. Strict positivity of the curvature makes the gradient a parametrization of the relative interior of a simplex.

Write $\pi :X_ n\to \mathbb P^{n-1}$ for the projection. The forms $\alpha ,\zeta ,\omega $ and the probability volume form $\mathrm{d}V=\omega ^ n/(2n)$ are as in subsection 19.8.12. Write $\sigma $ for the Fubini–Study form on $\mathbb P^{n-1}$ normalized by $\int _{\mathbb P^{n-1}}\sigma ^{n-1}=1$, and $\mathrm{MA}^{\mathrm{ext}}$ for the extended non-pluripolar Monge–Ampère measure of theorem 13.14.3. The fiber calculations use the following identities, where $y\in \mathbb P^{n-1}$:

\[ \alpha =\pi ^*\sigma ,\qquad \omega =\alpha +\zeta ,\qquad \zeta ^2=0,\qquad \mathrm{d}V=\tfrac 12\alpha ^{n-1}\wedge \zeta ,\qquad \int _{\pi ^{-1}(y)}\zeta =2. \]

21.14.1 Weighted sections on the projective base10355FA4232C

For an integer $k\geq 0$, the line bundle $L_ k$ of subsection 19.8.12, with its metric $h_ k$, is the pullback of $\mathcal O(k)$ with its Fubini–Study metric. By the proof of proposition 19.8.13, sections of $L_ k^ m$ are represented by homogeneous polynomials $f$ of degree $km$ in the covering coordinates $z\in \mathbb C^ n\setminus \{ 0\} $, exactly as sections of $\mathcal O(km)$ on $\mathbb P^{n-1}$, and the pointwise squared norm $|f(z)|^2|z|^{-2km}$ is invariant under every nonzero complex scalar. If the weight also descends, the weighted integrand is a function on the base, and fiber integration identifies the norms and Bergman kernels of subsection 19.8.1 on the two spaces.

Theorem 21.14.2.DCED1F266D4F Let $k\geq 0$ be an integer and $u\in \mathrm{PSH}(X_ n,k\alpha )$. Then $u=\pi ^*v$ for a unique $v\in \mathrm{PSH}(\mathbb P^{n-1},k\sigma )$. For every positive integer $m$, pullback identifies the weighted section spaces isometrically: for a section $s$ in the space on the base,

\[ \begin{split} H^0\bigl (\mathbb P^{n-1},\mathcal O(km)\otimes \mathcal I(mv)\bigr ) & \simeq H^0\bigl (X_ n,L_ k^ m\otimes \mathcal I(mu)\bigr ),\\ \| \pi ^*s\| _{X_ n,m,u}^2& =\| s\| _{\mathbb P^{n-1},m,v}^2. \end{split} \]

Here the norm on the base is defined with the Fubini–Study metric and the volume form $\sigma ^{n-1}$. The kernels $K$ and, off their pole sets, $B$ on $X_ n$ are the pullbacks of the corresponding kernels on the base. Pullback also identifies the bounded weighted section algebras $\mathcal W_\bullet (v)$ and $\mathcal W_\bullet (u)$.

Proof. The descent of the potential follows from proposition 5.43.5. By the proof of proposition 19.8.13, the unweighted section spaces on both manifolds consist of the same homogeneous polynomials of degree $km$, and the pointwise norms of corresponding sections agree because the metrics correspond. For every nonnegative Borel function $F$ on the base, fiber integration gives

\[ \int _{X_ n}(F\circ \pi )\, \mathrm{d}V =\frac12\int _{\mathbb P^{n-1}}F\sigma ^{n-1} \int _{\pi ^{-1}(y)}\zeta =\int _{\mathbb P^{n-1}}F\sigma ^{n-1}. \]

Applied to $F=|s|^2\mathrm{e}^{-mv}$, with arbitrary values on the pole set, this formula gives equality of the two possibly infinite integrals. Since finiteness of the weighted norm characterizes the multiplier-ideal section spaces, this proves the asserted isometry. An orthonormal basis therefore pulls back to an orthonormal basis, which proves the kernel identities. Finally, a nonnegative function on the base is bounded if and only if its pullback is bounded, so pullback also identifies the bounded weighted sections, compatibly with multiplication. $\square$

21.14.3 Descended measures and equations544154BAA081

The pulled-back curvature of a descended potential has no component in the fiber direction; the form $\zeta $ supplies it. Since $\zeta ^2=0$, the Hermitian top power of a descended potential consists of a single mixed term, and fiber integration of this term determines the mass and reduces the Monge–Ampère equation for descended potentials to the base.

For a positive Borel measure $\nu $ on $\mathbb P^{n-1}$, we write $\pi ^*\nu \wedge \zeta $ for the measure on $X_ n$ defined, on nonnegative Borel functions $F$, by

\[ \int _{X_ n}F\, \pi ^*\nu \wedge \zeta =\int _{\mathbb P^{n-1}} \left(\int _{\pi ^{-1}(y)}F\zeta \right)\mathrm{d}\nu (y). \]

When $\nu $ is given by a smooth top-degree form, this agrees with the pullback of the form wedged with $\zeta $.

Theorem 21.14.4.94243AEE3360 For a quasi-plurisubharmonic function $v$ on $\mathbb P^{n-1}$, the pullback $u=\pi ^*v$ belongs to $\mathrm{PSH}(X_ n,\omega )$ if and only if $v\in \mathrm{PSH}(\mathbb P^{n-1},\sigma )$. In this case

\[ \mathrm{MA}^{\mathrm{ext}}_\omega (u) =n\, \pi ^*\mathrm{MA}^{\mathrm{ext}}_\sigma (v)\wedge \zeta . \]

Consequently

\[ \int _{X_ n}\mathrm{MA}^{\mathrm{ext}}_\omega (u) =2n\int _{\mathbb P^{n-1}}\mathrm{MA}^{\mathrm{ext}}_\sigma (v) \leq 2n, \]

with equality for bounded $v$. For an integer $k\geq 0$ and $v\in \mathrm{PSH}(\mathbb P^{n-1},k\sigma )$,

\[ \mathrm{MA}^{\mathrm{ext}}_{k\alpha }(\pi ^*v)=0 \]

as a measure on the $n$-dimensional manifold $X_ n$.

Proof. If $v$ is $\sigma $-plurisubharmonic, then

\[ \omega +\mathrm{dd}^{\mathrm{c}}u=\pi ^*(\sigma +\mathrm{dd}^{\mathrm{c}}v)+\zeta \geq 0. \]

Conversely, suppose that the left-hand side is positive. Wedge it with the smooth semipositive form $\zeta $ and push the result forward to the base. Since $\zeta ^2=0$ and the fiber integral of $\zeta $ is two,

\[ \pi _*\bigl ((\omega +\mathrm{dd}^{\mathrm{c}}u)\wedge \zeta \bigr ) =2(\sigma +\mathrm{dd}^{\mathrm{c}}v)\geq 0; \]

this identity of currents follows by testing against smooth forms on the base and integrating over the fibers. It proves the converse.

For the measure formula, first let $v\in \mathrm{PSH}(\mathbb P^{n-1},\sigma )$ be bounded. By lemma 5.28.4, its bounded products commute with $\pi ^*$. Since $\zeta ^2=0$ and the base has dimension $n-1$, expanding the power gives

\[ (\omega +\mathrm{dd}^{\mathrm{c}}u)^ n =n\, \pi ^*(\sigma +\mathrm{dd}^{\mathrm{c}}v)^{n-1}\wedge \zeta , \]

and fiber integration identifies the right-hand side with the measure defined above.

For general $v$, normalize $v\leq 0$ and set $v_ j=\max (v,-j)$. Since $\sigma $ is positive, these are bounded $\sigma $-plurisubharmonic functions, and their pullbacks are exactly $u_ j=\max (u,-j)$. Multiply the bounded identity by the common indicator function $\mathbf1_{\{ u>-j\} }=\pi ^*\mathbf1_{\{ v>-j\} }$ and pass to the increasing limits defining the extended measures. Fiber integration of nonnegative functions justifies the limit on the right-hand side, including over the poles.

The mass formula follows because each fiber has $\zeta $-mass two. The extended mass on the base is at most one: on $\{ v>-j\} $, its truncated measure is bounded above by the full bounded product, whose mass is $\int \sigma ^{n-1}=1$ by Stokes' theorem for the closed form $\sigma $. For bounded $v$ nothing is removed, and equality holds.

For the last assertion, take bounded truncations of $v\in \mathrm{PSH}(\mathbb P^{n-1},k\sigma )$. By lemma 5.28.4, the $n$th powers of their pulled-back currents vanish. Restricting to the truncation levels and taking the increasing limit shows that the extended measure is zero; this includes the case $k=0$. $\square$

If an $\omega $-plurisubharmonic potential is constant on the fibers, the local-section argument in proposition 5.43.5 shows that it descends to a quasi-plurisubharmonic function on the base. The theorem therefore bounds its Hermitian mass by $2n$. On the Hopf threefold this bound is six, so the unbounded masses of section 23.1 require potentials that vary along the fibers.

For the equation, let $p>1$ and let $f\geq 0$ belong to $L^ p(\mathbb P^{n-1},\sigma ^{n-1})$, with $\int f\sigma ^{n-1}>0$. For potentials $u=\pi ^*v$ with bounded $v\in \mathrm{PSH}(\mathbb P^{n-1},\sigma )$, the equation

D50C510DA3A9 21.14.5
\begin{equation} \index{Monge--Ampere equation@Monge--Amp\` ere equation!Hopf reduction} (\omega +\mathrm{dd}^{\mathrm{c}}u)^ n=c(f\circ \pi )\, \mathrm{d}V,\qquad c>0, \end{equation}

is equivalent to the base equation

\[ (\sigma +\mathrm{dd}^{\mathrm{c}}v)^{n-1}=\frac{c}{2n}f\sigma ^{n-1}. \]

Indeed, the base equation implies (21.14.5) by theorem 21.14.4 and the formula for $\mathrm{d}V$. Conversely, pushing (21.14.5) forward to the base gives $2n(\sigma +\mathrm{dd}^{\mathrm{c}}v)^{n-1}=cf\sigma ^{n-1}$.

Integrating the base equation determines the constant:

\[ c=\frac{2n}{\int _{\mathbb P^{n-1}}f\sigma ^{n-1}}. \]

Since $\sigma $ is closed, it satisfies the Guan–Li condition (14.25.1). Applied to $(\mathbb P^{n-1},\sigma )$, 14.25.2 gives a continuous solution of the base equation with this constant, unique up to additive constants among bounded $\sigma $-plurisubharmonic functions. Consequently (21.14.5) has a continuous solution, unique up to additive constants among bounded descended potentials. For smooth positive $f$, theorem 14.2.1 on the base gives a smooth solution $v$, and the constant is again determined by Stokes' theorem.

For a real parameter $\lambda $, the same calculation shows that the equation with right-hand side $c\mathrm{e}^{\lambda u}(f\circ \pi )\mathrm{d}V$ is equivalent to the base equation with right-hand side $(c/(2n))\mathrm{e}^{\lambda v}f\sigma ^{n-1}$.

21.14.6 Logarithmic coordinates and the gradient simplex11FD5CD723A9

For potentials invariant under rotations of the individual coordinates, the complex Hessian becomes the real Hessian of a function of the logarithmic coordinates. Invariance under complex scaling gives this Hessian a null direction; transverse to it, positivity becomes convexity, and boundedness of the potential confines the gradient to a simplex. Logarithmic growth makes every interior point of the simplex a gradient value, and strict positivity on the base makes the gradient injective modulo the null direction.

In this section, $k>0$ is a real number. Let $v$ be a smooth real function on $\mathbb P^{n-1}$ that is invariant under the torus action

\[ [z_1:\cdots :z_ n]\longmapsto [\mathrm e^{\mathrm it_1}z_1:\cdots :\mathrm e^{\mathrm it_ n}z_ n], \qquad t\in \mathbb R^ n, \]

and put $u=\pi ^*v$. On the locus where all covering coordinates $z_ i$ are nonzero, set $x_ i=\log |z_ i|^2$. By this invariance, $u$ is a function of $x=(x_1,\ldots ,x_ n)$, which we denote by $u(x)$, and we set

\[ F(x)=k\log \left(\sum _{i=1}^ n \mathrm{e}^{x_ i}\right)+u(x). \]

Write $\nabla F=(F_ i)$ and $D^2F=(F_{ij})$ for its Euclidean gradient and Hessian, and let $\mathbf1=(1,\ldots ,1)$. Since $u$ is invariant under complex scaling, for every $t\in \mathbb R$,

5DB0E5246A09 21.14.7
\begin{equation} F(x+t\mathbf1)=F(x)+kt. \end{equation}

Proposition 21.14.8.F23340CADBE0 On this locus, $k\alpha +\mathrm{dd}^{\mathrm{c}}u\geq 0$ if and only if $F$ is convex. If $F$ is convex, its gradient lies in

\[ \Sigma _ k=\{ p\in \mathbb R^ n:p_ i\geq 0,\ \textstyle \sum _ i p_ i=k\} , \]

and $D^2F\, \mathbf1=0$. If $k\sigma +\mathrm{dd}^{\mathrm{c}}v$ is positive definite on $\mathbb P^{n-1}$, the gradient induces a bijection from $\mathbb R^ n/\mathbb R\mathbf1$ onto the relative interior of $\Sigma _ k$.

Proof. On the covering space, $k\alpha +\mathrm{dd}^{\mathrm{c}}u=\mathrm{dd}^{\mathrm{c}}F$. In the coordinates $z_ i$, the Hermitian coefficient matrix of this form is a positive constant times

\[ \left(\frac{F_{ij}}{z_ i\overline z_ j}\right)_{i,j}. \]

This matrix is congruent to the real Hessian $(F_{ij})$ through the invertible diagonal matrix with entries $1/z_ i$, which proves the first equivalence. Differentiating (21.14.7) gives $\sum _ i F_ i=k$ and $D^2F\, \mathbf1=0$.

Since $v$ is bounded on the compact base, there is a constant $C$ such that

\[ \left|F(x)-k\log \sum _ i \mathrm{e}^{x_ i}\right|\leq C. \]

Fix $x$ and a coordinate vector $e_ i$. Since $n\geq 2$, the convex function $t\mapsto F(x+te_ i)$ is bounded above as $t\to -\infty $. Its derivative cannot be negative anywhere: otherwise the supporting line at such a point would tend to $+\infty $ as $t\to -\infty $. Thus $F_ i\geq 0$, which proves $\nabla F\in \Sigma _ k$.

Now assume that $k\sigma +\mathrm{dd}^{\mathrm{c}}v$ is positive definite. The kernel of the differential of the projection $\mathbb C^ n\setminus \{ 0\} \to \mathbb P^{n-1}$ is exactly the complex scalar direction, so the Hessian of $F$ is positive definite transverse to $\mathbf1$. In particular $F$ is strictly convex on $H=\{ x:\sum _ i x_ i=0\} $. Along each line $x+te_ i$, the derivative of $F$ is strictly increasing and nonnegative everywhere, hence positive. Thus every gradient lies in the relative interior of $\Sigma _ k$.

Conversely, take $p\in \mathbb R^ n$ with $p_ i>0$ and $\sum _ i p_ i=k$. The function $F(x)-p\cdot x$ is invariant under translation along $\mathbf1$, and we claim that it tends to $+\infty $ on $H$ as $|x|\to \infty $. Indeed,

\[ k\max _ i x_ i-p\cdot x =\sum _ i p_ i(\max _ j x_ j-x_ i) \]

is strictly positive at every nonzero vector of $H$, and hence has a positive minimum on the unit sphere of $H$. Since $F$ differs from $k\log \sum _ i \mathrm{e}^{x_ i}$ by a bounded function and $\log \sum _ i \mathrm{e}^{x_ i}\geq \max _ i x_ i$, the claim follows. Hence $F(x)-p\cdot x$ has a unique minimizer on $H$. At this point $\nabla F-p$ is a multiple of $\mathbf1$; both vectors have coordinate sum $k$, so the multiple is zero. This proves surjectivity, and strict convexity on $H$ proves injectivity. $\square$

Cite this page

Comments

Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.

Read this page’s discussion on GitHubPost a comment on GitHub