20.3.46 The full trace envelope

Let $w_ a=\mathcal T_{a\kappa }(U;B)$. The closed-background construction of theorem 20.3.33 applies with $\Omega _0=C\kappa $. It gives

The lower bound comes from the trace majorant in the half-sized background, and the upper bound from closed mass monotonicity. In particular, the mass is positive, as required for convergence of the divisorial Lelong numbers of the decreasing base models.

Theorem 20.3.48.63EC1D93ABC1 For the projective pullback data above,

E6C82B883427 20.3.49

\begin{equation} \mathcal T_{\Omega _{a,b}}(u;Y)=q^*w_ a. \end{equation}

For every Hermitian form $\Omega \geq \Omega _{a,b}$, every smooth compact modification $\pi :Z\to Y$, and every prime divisor $D\subset Z$,

\[ \nu \bigl (\mathcal T_\Omega (u;Y)\circ \pi ,D\bigr )=\tau _{Y,u}(D). \]

Proof. Apply lemma 20.3.44 to $f=q\circ \pi $. If $D$ dominates $B$, both the trace order and the Lelong number of $q^*w_ a$ are zero. Otherwise the ideal identity (20.3.42) and the extracted divisor $F$ give

\[ \tau _{Y,u}(D)=e\tau _{B,U}(F) =e\nu (w_ a\circ \sigma ,F) =\nu (q^*w_ a\circ \pi ,D). \]

Thus the same potential $q^*w_ a$, independent of the choice of $D$, is an exact trace competitor on every compact model.

The compact elliptic group $\mathbb C^*/\langle \mathrm e^{-1}\rangle $ acts by scalar multiplication on $Y$. It preserves $\alpha $, $\beta $ and all ideals in (20.3.42). It also preserves the full competitor family: for a translation $\rho $, the constraint for $v\circ \rho $ on $\pi $ is the constraint for $v$ on the modification $\rho \circ \pi $, with the same ideal order. By theorem 20.3.27, the envelope is the largest member of this family, so it is invariant under the group. The action is transitive on each fiber; local holomorphic sections of $q$ therefore show that the envelope descends to a genuine quasi-plurisubharmonic function $v$ on $B$.

For a basic potential, positivity is equivalent to positivity on the base:

3B4C37A0F595 20.3.50

\begin{equation} \Omega _{a,b}+\mathrm{dd}^{\mathrm{c}}q^*v\geq 0 \quad \Longleftrightarrow \quad a\kappa +\mathrm{dd}^{\mathrm{c}}v\geq 0. \end{equation}

Indeed, wedge the left-hand current with the smooth semipositive form $\beta $ and push forward by $q$. Since $\beta ^2=0$ and its fiber integral is two, the resulting positive current is $2(a\kappa +\mathrm{dd}^{\mathrm{c}}v)$. Conversely, pull back the base inequality and add $b\beta \geq 0$.

Every smooth compact modification $\sigma :B'\to B$ lifts to the smooth compact modification $Y\times _ B B'\to Y$. The inverse image of a prime divisor $F\subset B'$ is prime, because the fibers are connected, and has generic multiplicity one. The trace constraint on this divisor says exactly that $v$ satisfies the constraint at $F$ on $B'$. Thus $v\leq w_ a$. The opposite inequality follows from the exact competitor $q^*w_ a$, proving (20.3.49).

For $\Omega \geq \Omega _{a,b}$, this competitor lies below $T_\Omega =\mathcal T_\Omega (u;Y)$. The defining lower bound and the order of the two potentials give

\[ \tau _{Y,u}(D)\leq \nu (T_\Omega \circ \pi ,D) \leq \nu (q^*w_ a\circ \pi ,D)=\tau _{Y,u}(D), \]

which proves exactness in every such background. $\square$

20.3.46 The full trace envelope`286FEC8F6AC1`

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20.3.46 The full trace envelope286FEC8F6AC1

Comments

20.3.46 The full trace envelope`286FEC8F6AC1`