11.1.4 The mass polynomialCA97FF13B1BB

Proof. Expand $(\omega +\mathrm{dd}^{\mathrm{c}}u)^ n$ by the binomial theorem. The term with $k=n$ integrates to zero, since it is exact:

For $1\leq k\leq n-1$, apply lemma 11.1.2 with $f=u$ and $T=(\mathrm{dd}^{\mathrm{c}}u)^{k-1}\wedge \omega ^{n-k}$. Since both $\mathrm{d}$ and $\mathrm d^{\mathrm c}$ annihilate $\mathrm{dd}^{\mathrm{c}}u$,

which gives the $k$th term of (11.1.6). When $n=1$, the sum is empty and the mass is constant. $\square$

Proof. Step 1: the first condition implies the second. Let $u$ be a smooth real function. For $|t|$ small, $tu$ belongs to $\mathcal H_\omega $ for both signs of $t$, so the polynomial $t\mapsto \int _ X(\omega +t\mathrm{dd}^{\mathrm{c}}u)^ n$ is constant on an interval and its nonconstant coefficients vanish. For $1\leq j\leq n-1$ this gives

Replace $u$ by a linear combination of $n-j$ smooth real functions $u_1,\ldots ,u_{n-j}$ and compare the coefficients of the monomial in which every parameter appears once. We obtain

Integrating by parts in the first factor and using that $u_1$ is arbitrary, we obtain the pointwise identity

for every choice of the remaining functions. At a given point, the complex Hessians of these functions can be prescribed arbitrarily: multiply a quadratic polynomial in a coordinate chart by a cutoff function equal to one near that point. Since the identity is multilinear, we may complexify it and use the quadratic monomials $z_ a\bar z_ b$. Wedge products of their complex Hessians give all coordinate forms of the complementary bidegree, and pairing with these forms shows that every coefficient of $\mathrm{dd}^{\mathrm{c}}(\omega ^ j)$ vanishes. When $j=n-1$ there are no remaining factors, and the same conclusion follows directly from the arbitrariness of $u_1$. This proves the second condition.

Step 2: the remaining equivalences for smooth potentials. The second condition implies the first by theorem 11.1.5. For $n\geq 3$, the identities for $j=1,2$ and lemma 11.1.3 give $\mathrm{dd}^{\mathrm{c}}\omega =0$ and

so the second condition implies the third. For $n=2$, the second condition is $\mathrm{dd}^{\mathrm{c}}\omega =0$, while $\mathrm{d}\omega \wedge \mathrm d^{\mathrm c}\omega $ vanishes for degree reasons; the implication is again immediate. Conversely, lemma 11.1.3 derives all the identities of the second condition from the third.

Step 3: bounded potentials. Assume the first condition and let $u$ be a bounded $\omega $-plurisubharmonic function. By theorem 5.32.1, there are uniformly bounded functions $u_ j\in \mathcal H_\omega $ decreasing to $u$. Their masses all equal $\int _ X\omega ^ n$, and the continuity of the bounded products of section 5.28 along such sequences passes this common value to the limit. Conversely, every element of $\mathcal H_\omega $ is a bounded $\omega $-plurisubharmonic function, so constancy for bounded potentials implies the first condition. The zero potential shows that the common mass is $\int _ X\omega ^ n$. $\square$

Proof. Since $\mathrm M_\omega (u+tv)$ is a polynomial in $t$, its derivative at $t=0$ is

and lemma 11.1.2 turns this into the right-hand side of (11.1.9). For the second assertion, fix $u\in \mathcal H_\omega $. For every smooth real direction $v$, the segment $u+tv$ lies in $\mathcal H_\omega $ when $|t|$ is sufficiently small, so the right-hand side of (11.1.9) vanishes for all smooth real $v$. Hence the smooth top-degree form $\mathrm{dd}^{\mathrm{c}}(\omega _ u^{n-1})$ vanishes. $\square$

Proof. The formula is the case $n=2$ of (11.1.6), and it shows that $\mathrm{dd}^{\mathrm{c}}\omega =0$ implies constancy. Conversely, if the mass is constant, proposition 11.1.8 at $u=0$ gives $\mathrm{dd}^{\mathrm{c}}\omega =0$. $\square$