Lemma 20.4.20721E66DF8C88

Proof. Set $L_\gamma f=\sigma _\gamma (\mathrm{dd}^{\mathrm{c}}f)$. We use the inverse $G_\gamma $ of (11.31.3), normalized with the volume form $\gamma ^ n/V_\gamma $. It acts on top-degree distributions of total mass zero and is continuous by the elliptic estimates for the operator and its transpose.

Step 1: construction. Define

The current $B_\gamma (T)$ is closed and has scalar trace $c_ T$. We now prove its smoothness and the finite-dimensionality assertion without using any Kähler identity.

Step 2: an elliptic system. Let $J$ be the complex structure acting on real covectors. On real $(1,1)$-forms consider the first-order operator

Its principal symbol at a nonzero real covector $\xi $ is $B\mapsto (\xi \wedge B,\xi \, \sigma _\gamma (B))$, and we claim that it is injective. If $\xi \wedge B=0$, then the $(1,1)$ condition forces $B$ to be a real multiple of $\xi \wedge J\xi $. To check this, choose unitary real coordinates with $\xi =\mathrm{d}x_1$: every term of $B$ contains $\mathrm{d}x_1$, and invariance under $J$ forces every term to contain $\mathrm{d}y_1$ as well. If moreover $\xi \, \sigma _\gamma (B)=0$, then $\sigma _\gamma (B)=0$; since the scalar trace of $\xi \wedge J\xi $ is nonzero, the multiple is zero.

Let $\mathscr P^*$ be the formal adjoint of $\mathscr P$ for the fixed metric and volume. Since the symbol of $\mathscr P$ is injective, $\mathscr P^*\mathscr P$ is elliptic. By elliptic regularity and compactness, its distributional kernel is a finite-dimensional space of smooth forms. Since $\mathscr P B_\gamma (T)=0$, the form $B_\gamma (T)$ belongs to this space. Conversely, the kernel consists exactly of the closed forms with constant scalar trace: integrating $\langle \mathscr P^*\mathscr P B,B\rangle $ gives $\mathscr P B=0$, and connectedness makes the trace constant. Denote this kernel by $\mathscr H_\gamma $.

Step 3: Bott–Chern representatives. The construction is linear and continuous on distributions. Convergence of the components $B_\gamma (T)$ therefore takes place in the finite-dimensional space $\mathscr H_\gamma $, hence in every smooth norm. If $T=\mathrm{dd}^{\mathrm{c}}f$ for a distribution $f$, then $c_ T=0$ and the inverse gives $v_\gamma (T)=f-V_\gamma ^{-1}\int _ Yf\gamma ^ n$, so $B_\gamma (T)=0$. Applied to smooth closed forms, this shows that $\mathscr H_\gamma $ contains exactly one representative of each Bott–Chern class. Uniqueness of the decomposition can also be seen directly: if two decompositions are given, $L_\gamma $ of the difference of the two potentials is a constant, integration forces this constant to be zero, and the normalized kernel of $L_\gamma $ is zero.

Step 4: positive currents. Suppose now that $T\geq 0$. The inequality $\mathrm{dd}^{\mathrm{c}}v_\gamma (T)\geq -B_\gamma (T)$ gives a quasi-plurisubharmonic representative of the distribution $v_\gamma (T)$. Let positive closed currents $T_ j$ converge weakly to $T$. Their components $B_\gamma (T_ j)$ converge smoothly, so the potentials $v_\gamma (T_ j)$ have a common smooth lower curvature bound. By theorem 5.25.2, after subtracting their suprema they form an $L^1$-precompact family. If $a_ j=\sup _ Yv_\gamma (T_ j)$, the zero-mean normalization gives

and the right-hand side is uniformly bounded. Hence the potentials $v_\gamma (T_ j)$ themselves are $L^1$-precompact. Every cluster point equals their distributional limit $v_\gamma (T)$, so the whole sequence converges in $L^1$.

Finally, the positive closed currents with trace mass $\int _ YT\wedge \gamma ^{n-1}=1$ form a weakly compact set: positivity bounds their coefficient measures, weak limits remain positive and closed, and testing with $\gamma ^{n-1}$ preserves the trace mass. The smooth and $L^1$ continuity just proved bounds both norms on this set, and positive homogeneity gives (20.4.21). A positive current of trace mass zero is zero, so this case is included. $\square$