12.6.21 Masses and reference densities in dimension threeD70657252686

Proof. Since the forms $\mathrm d\alpha _ j$ are holomorphic, we have

Indeed $\bar\partial (\alpha _ j\wedge \bar\alpha _ k) =-\alpha _ j\wedge \mathrm d\bar\alpha _ k$, and applying $\partial $ and multiplying by $(\mathrm i/(2\pi ))^2$ gives the identity in (12.6.24). Diagonalizing the positive coefficient matrix expresses the result as a sum of nonnegative multiples of $\beta \wedge \bar\beta $ for invariant $(2,0)$-forms $\beta $. In dimension three these are nonnegative $(2,2)$-forms: the wedge product with $\mathrm i\xi \wedge \bar\xi $, for a $(1,0)$-form $\xi $, has the squared modulus of $\xi \wedge \beta $ as coefficient.

By proposition 12.6.5, $\theta _ A$ is balanced, hence Gauduchon. The maximum case of theorem 12.16.3 therefore gives $v_+(\theta _ A)=\int _ X\theta _ A^3=H\det A$. The supremum of $\mathcal V_\nu (\theta _ A)$ over $\nu \in \mathscr M$ equals this upper mass by theorem 12.15.4, and (12.6.6) shows that it is attained at $\mu _0$. Finally, proposition 13.14.15 makes the finiteness of the upper mass a property of the manifold. $\square$

Proof. For $\varepsilon >0$, the same bounded $u$ is $\theta _{A+\varepsilon I}$-plurisubharmonic, so the preceding proposition bounds its mass by $H\det (A+\varepsilon I)$, which tends to zero with $\varepsilon $. By the multilinear construction of bounded products in proposition 5.28.2,

is a polynomial in $\varepsilon $ whose coefficients are mixed products of finite mass. Its total mass therefore converges to $\int _ X(\theta _ A+\mathrm{dd}^{\mathrm{c}}u)^3$ as $\varepsilon \downarrow 0$, so this mass is zero, and positivity makes the measure zero.

For unbounded $u$, we subtract a constant so that $u\leq 0$. Since $\theta _ A\geq 0$, the truncations $u_ k=\max (u,-k)$ are bounded and $\theta _ A$-plurisubharmonic, so their top products vanish by the bounded case. Restricting them to $\{ u>-k\} $ and passing to the limit that defines the non-pluripolar product gives zero as well. $\square$

Proof. The map $p$ is well defined because multiplication by lattice elements adds Gaussian integers to the first two coordinates. It is a submersion, and fixing representatives of those coordinates identifies each fiber with the central quotient $E$. Integrating first in the third coordinate over the fundamental cube proves the pushforward identity.

The closed equation on the base. Write the positive Hermitian matrix in block form and take its Schur complement:

On $B$, let

We apply theorem 14.2.1 to this closed background with the density $Q_ Bf\mu _ B$. Since $\omega _ S$ is closed and the density has total mass $Q_ B$, Stokes' theorem forces the constant in the equation to be one. We obtain a smooth real $v=v_{A,f}$, unique up to an additive constant, such that

If $K$ is the coefficient matrix of $\mathrm{dd}^{\mathrm{c}}v$ in the base coframe, this says $\det (S+K)=(\det S)f$. The form

has coefficient matrix $\left(\begin{smallmatrix}C+K& b\\ b^*& a\end{smallmatrix}\right)$ in the invariant coframe. Since $a>0$ and the Schur complement $S+K$ is positive, $\Omega >0$; its determinant is $a\det (S+K)=(\det A)(f\circ p)$. Consequently $\Omega ^3=(H\det A)\mu _ f$.

This solution form is also balanced. Put $\beta =\mathrm{dd}^{\mathrm{c}}(p^*v)$, which is closed and uses only $\alpha _1,\alpha _2$ and their conjugates. Since $\mathrm d\alpha _3=-\alpha _1\wedge \alpha _2$, every term of $\partial \theta _ A$ contains $\alpha _1\wedge \alpha _2$, and every conjugate term contains $\bar\alpha _1\wedge \bar\alpha _2$. Thus $\mathrm d\theta _ A\wedge \beta =0$. Together with $\mathrm d\theta _ A^2=0$, this proves

Theorem 12.2.3 now gives (12.6.27). Its characterization of the minimizers, the balancedness of $\Omega $ and the homogeneity of degree two of $D_{\mu _ f}$ give the unique normalized minimizer (12.6.28).

All maximizing densities. Let $Z_3=\partial /\partial z_3$ be dual to $\alpha _3$, and set $\tau _ j=(\mathrm i/(2\pi ))\alpha _ j\wedge \bar\alpha _ j$. Directly from (12.6.24),

Since $\theta _ A$ is Gauduchon, the identity (12.16.4) therefore becomes, for every smooth real $u$,

Here $\mathrm du\wedge \mathrm d^{\mathrm c}u=(\mathrm i/(2\pi ))\partial u\wedge \bar\partial u$, and wedging with $\tau _1\wedge \tau _2$ selects its coefficient on $\alpha _3\wedge \bar\alpha _3$; this also accounts for the factor $1/2$.

For an arbitrary $\nu \in \mathscr M$, let $u_\nu $ be the smooth solution of the equation for $\theta _ A$ and $\nu $. Its mass equals $\mathcal V_\nu (\theta _ A)$ by theorem 12.2.3. By (12.6.30), this mass equals $H\det A$ exactly when $Z_3u_\nu =0$. As $u_\nu $ is real, its derivatives in both real directions of each elliptic fiber then vanish, and connectedness of the fibers gives $u_\nu =p^*v$ for a smooth $v$ on $B$. The density of $(\theta _ A+\mathrm{dd}^{\mathrm{c}}u_\nu )^3$ relative to $\mu _0$ is then a function on $B$, so the equation makes $\nu $ fiber invariant. The converse was proved by the construction on the base.

Completion in the background parameters. For fixed $f$, formula (12.6.27) holds on the entire cone of positive matrices, and differentiating the logarithm of its cube root twice gives the stated Hessian metric. Since this is the same metric as in theorem 12.6.12, the matrix-path and coefficient-limit arguments of that proof apply unchanged on each fixed $W$. They give the distance bounds, completeness, properness and the Cauchy criterion; they do not use that the minimizing forms have constant coefficients. $\square$