9.11.63 Nonclosed equations on cuspidal products25867983D80D

Proof. Choose real smooth functions $A$ on $C$, supported away from $q$, and $B$ on $E_2$. Let

The form $\alpha $ is real and satisfies $\mathrm{dd}^{\mathrm{c}}\alpha =0$. At a point where $\partial A=0$, $\partial \bar\partial A\ne 0$ and $\partial B\ne 0$,

Thus $\mathrm d\alpha \ne 0$. The form $\alpha $ vanishes near the singular locus. Its operator norm relative to $H_0$ is bounded on its support, and its norm relative to $K_0$ is bounded everywhere. Choose $|\varepsilon |$ so small that both norms, multiplied by $|\varepsilon |$, are less than $1/2$. This proves positivity of $h$ in the ambient sense, semipositivity of $H$, and the inequality $K_0+\varepsilon \pi ^*\alpha \geq \tfrac 12K_0$.

The equation $\partial _ z\partial _{\bar z}v=\lambda -\rho $ on $E_1$, whose right-hand side has mean zero, has a smooth real periodic solution, obtained by dividing its nonzero Fourier coefficients. Its pullback $v_0$ satisfies $\mathrm{dd}^{\mathrm{c}}v_0=(\lambda -\rho )\kappa _1$, which proves the formula for $k$. The forms $H_0$ and $K_0$ are closed. Hence $\mathrm dk$ and $\mathrm d^{\mathrm c}k$ come from the complex surface $E_1\times E_2$, so that $\mathrm dk\wedge \mathrm d^{\mathrm c}k=0$ for degree reasons, and $\mathrm{dd}^{\mathrm{c}}k=0$. The same identities hold for $H$. The Leibniz rule now gives $\mathrm{dd}^{\mathrm{c}}(k^ r)=\mathrm{dd}^{\mathrm{c}}(H^ r)=0$ for all $r$; this uses both torsion identities, not only pluriclosedness. Expanding $(H+\mathrm{dd}^{\mathrm{c}}v_0)^ n$ and integrating by parts shows that every term containing $\mathrm{dd}^{\mathrm{c}}v_0$ integrates to zero. Hence the masses agree, and positivity of $k$ makes them positive. $\square$

Proof. Set $b_1=\lambda $ and $b_ j=1$ for $j\geq 2$. The operator

has Fourier eigenvalues $\ell (m)=-\pi ^2\sum _ jb_ j^{-1}(m_{2j-1}^2+m_{2j}^2)$. It has an inverse $H^ s_0\to H^{s+2}_0$ of norm

Let $L_\varepsilon w=nk^{n-1}\wedge \mathrm{dd}^{\mathrm{c}}w/\Omega $. It preserves zero mean because $\mathrm{dd}^{\mathrm{c}}(k^{n-1})=0$. Fix $\varepsilon _0\leq 1$ within the positivity range. The finite expansion of $k^{n-1}$ gives

Choose bounds $e_ j$ for the norms $\| P_ j\| _{H^{s+2}\to H^ s}$, which are finite since multiplication by smooth coefficients is bounded on $H^ s$, and set $B=\sum _ j\varepsilon _0^{j-1}e_ j$. If $|\varepsilon |\leq \varepsilon _0$ and $|\varepsilon |A_0B\leq 1/2$, the Neumann series gives $\| L_\varepsilon ^{-1}\| \leq A:=2A_0$.

The remainder is

For $s>n$, Fourier convolution and the Cauchy–Schwarz inequality, together with $\sum _ m(1+|m|^2)^{-s}<\infty $, make $H^ s$ an algebra. Multiplication by smooth coefficients and the finite expansion therefore give a constant $C_ Q$ such that, on the unit ball of $H^{s+2}$,

Each term of $Q(w)$ has zero integral against $\Omega $: for smooth $w$, its integral is $\int w\, \mathrm{dd}^{\mathrm{c}}(k^{n-r})\wedge (\mathrm{dd}^{\mathrm{c}}w)^{r-1}=0$, and Sobolev approximation gives the general case. The Sobolev embedding also gives a constant $D$ such that the pointwise operator norm of $\mathrm{dd}^{\mathrm{c}}w$ with respect to $K_0$ is at most $D\| w\| _{H^{s+2}}$. Choose

The equation is equivalent to $L_\varepsilon w+Q(w)=g$. The map $w\mapsto L_\varepsilon ^{-1}(g-Q(w))$ takes the closed ball of radius $R$ in $H^{s+2}_0$ into the ball of radius $3R/4$ and has Lipschitz constant at most $1/2$. Its fixed point gives existence and uniqueness in that ball. Since $k\geq K_0/2$ and the operator norm of $\mathrm{dd}^{\mathrm{c}}w$ is at most $1/4$, the form $k+\mathrm{dd}^{\mathrm{c}}w$ is at least $K_0/4$. The embedding $H^{s+2}\subset C^2$ follows from $s\geq n+3$. $\square$

Proof. The products are the bounded products of proposition 5.28.2. For $r\geq 1$, the bounded local recursion and Stokes' theorem on the compact manifold $Y$ give

Alternatively, one may convolve $U$ on the torus, add a fixed local positive quadratic function, and pass to the limit in the decreasing bounded expansion; every smooth integral vanishes by the same identity. Expanding the top power proves the assertion. The proof for $H$ is identical, since its powers are also $\mathrm{dd}^{\mathrm{c}}$-closed and its mass is $M$. $\square$

Proof. Convolve $U$, $V$ and $k$ with the same nonnegative radial kernel $\varrho _ j$ on the torus, of radius $r_ j\downarrow 0$, and write

Translations preserve positivity, so $A_ j\geq 0$ and $B_ j\geq \beta K_0$ with $\beta =1/4$. The forms $k_ j$ converge smoothly, and $V_ j$ converges in $C^2$. On lifted coordinate balls, choose a common positive quadratic function $q$ with $\mathrm{dd}^{\mathrm{c}}q\geq k$. Convolution gives $q*\varrho _ j=q+b r_ j^2$ for a fixed $b>0$. The radial averages of the bounded plurisubharmonic functions $U+q$ and $V+q$ decrease as the radii decrease. Hence $U_ j+b r_ j^2+q$ and $V_ j+b r_ j^2+q$ decrease to $U+q$ and $V+q$; the identical constants cancel in the difference.

Put $\phi _ j=U_ j-V_ j$ and $T_ j=\sum _{r=0}^{n-1}A_ j^ r\wedge B_ j^{n-1-r}$. The identities $\mathrm{dd}^{\mathrm{c}}k_ j=0$ and $\mathrm dk_ j\wedge \mathrm d^{\mathrm c}k_ j=0$ persist under convolution, since the torsion is still pulled back from $E_1\times E_2$. Differentiating the monomials therefore gives $\mathrm{dd}^{\mathrm{c}}T_ j=0$. Also $T_ j\geq B_ j^{n-1}\geq \beta ^{n-1}K_0^{n-1}$. Stokes' theorem applied to $\mathrm{dd}^{\mathrm{c}}(\phi _ j^2)$ yields

Only the $\mathrm{dd}^{\mathrm{c}}$-closedness of $T_ j$ is needed here.

We treat the four terms on the left separately. Expand $A_ j^ n$ with respect to the local plurisubharmonic potential $U_ j+b r_ j^2+q$ and the smoothly converging coefficient $k_ j-\mathrm{dd}^{\mathrm{c}}q$. The weighted limit lemma 5.27.2 gives $\int U_ jA_ j^ n\to \int U(k+\mathrm{dd}^{\mathrm{c}}U)^ n$. Decreasing convergence and the uniform convergence of $V_ j$ give the corresponding limit for $\int V_ jA_ j^ n$. Since $B_ j^ n$ has uniformly converging continuous densities, bounded pointwise convergence and dominated convergence give the limit of $\int U_ jB_ j^ n$. The term $\int V_ jB_ j^ n$ converges by $C^2$ convergence. A finite partition of unity gives the global statements. Since the two limiting measures are equal, the left-hand side of (9.11.70) tends to zero.

The flat Fourier Poincaré inequality now shows that the $L^2$ norm of $\phi _ j$ minus its average tends to zero. Bounded convergence gives $\phi _ j\to U-V$ in $L^2$, so $U-V$ is constant almost everywhere. Local smooth corrections and the canonical plurisubharmonic representatives extend this equality to every point. $\square$

Proof. The function $g=cf-f_\varepsilon $ has zero mean. Let $w$ be the solution given by theorem 9.11.66. The potential $v_0+w$ is bounded and $H$-plurisubharmonic. Descent through the resolution $\pi $ (theorem 9.11.5) gives

which satisfies the disk condition with the background $h$ itself. Since the normalization is a homeomorphism, $u$ is continuous. The definition of the extended measure and theorem 9.11.26 give the equation. Any other bounded solution pulls back to a bounded $H$-plurisubharmonic potential. Since $\pi $ is a Borel homeomorphism, equality of the pushforward measures means equality on $Y$. Subtracting $v_0$ and applying proposition 9.11.69 gives uniqueness. The mass identity of lemma 9.11.68 forces the scalar of every bounded solution to be $M/\nu (X)=c$.

The singular locus pulls back to the divisor $\{ p\} \times E_2\times \cdots \times E_ n$, which has zero smooth volume. If a set is locally pluripolar, a local plurisubharmonic function equal to $-\infty $ on it pulls back to a plurisubharmonic function, not identically $-\infty $, away from that divisor; its pole set has zero smooth volume. Thus $\nu $ is non-pluripolar. On the regular locus, the change of variables formula gives the density $f/J$. Near the cusp $\alpha =0$, so

The ratio gives the density $\lambda f/\rho $; the generic degree of $\pi $ and both factorials have been accounted for.

For the Hölder assertion, put $d=z-z'$ and $a=z+z'$. Then

If $|a|\geq |d|/4$, the first modulus is at least $|d|^2/4$, which controls $|d|^3$ on a bounded chart. Otherwise the second modulus is at least $13|d|^3/64$. Hence $|z-z'|\leq C(|z^2-z'^2|+|z^3-z'^3|)^{1/3}$. The $C^1$ regularity of $v_0+w$ in $z$ and in the other product coordinates gives the asserted ambient estimate.

Every resolution $\sigma :Z\to X$ factors through $Y$: in the cusp chart, the quotient $y/x$ is bounded off the exceptional set and extends holomorphically to the smooth source, giving the normalization coordinate. Write $\sigma =\pi \circ \mu $. The map $\mu $ is a proper modification. The bounded pullback products agree off its exceptional locus, and they give this analytic set zero mass by proposition 5.28.2. Hence their pushforwards agree on all of $Y$. This proves the equation on every resolution, and the same argument applies to further common refinements. The full potential is exactly $u\circ \sigma $, which is constant on every fiber, and surjectivity preserves all ordinary comparisons. $\square$

Proof. The torus $Y$ is Kähler, so it has bounded mass by lemma 19.9.12. Since $k>0$, zero is an ordinary model for $k$: $P_ k[0]=0$. Its lower volume is $M>0$ by lemma 9.11.68, because the potentials of the same ordinary type as zero are exactly the bounded ones. Hence theorem 10.31.3, applied on $Y$ with background $k$ and reference zero, gives a bounded $k$-plurisubharmonic solution for the measure $f\Omega $, and its scalar is $M/\int f\Omega $. Adding $v_0$ and using theorem 9.11.5, theorem 9.11.26 gives the normalized equation on $X$. The datum is non-pluripolar as in theorem 9.11.71. Neither smallness nor strict positivity of $f$ is needed for this existence statement. $\square$