Pome852EFF5927B9Corollary 10.18.4.852EFF5927B9 Assume $b>0$ and that $D_ p$ has a smooth maximizing sequence $(T_ j)$ with uniformly integrable densities $p k_{T_ j}$. Let $E$ be an effective real divisor whose class has a smooth semipositive representative $\beta _ E$. If
then no threshold subsolution $u$ for $f$ satisfies $\omega +\mathrm{dd}^{\mathrm{c}}u\geq [E]$.
Proof. Suppose such a subsolution exists. By proposition 10.18.1, there is a maximizing current $T\in \mathscr G_\omega $ with $\int _ X\beta _ E\wedge T=0$. We use the matrix-density notation of section 12.7: $\sigma _ T=\omega \wedge T$ is the trace measure of $T$, $P_ T$ is its matrix density and $B$ is the coefficient matrix of $\beta _ E$. Since both matrices are positive semidefinite, the vanishing pairing gives $\operatorname {tr}(BP_ T)=0$ at $\sigma _ T$-almost every point. Wherever $B\ne 0$, this forces $P_ T$ to be singular: a positive definite $P_ T$ would have strictly positive trace against the nonzero positive semidefinite matrix $B$. The determinant measure $\Gamma _0(T)$ vanishes where $P_ T$ is singular, so it is concentrated on $\{ \beta _ E=0\} $. By theorem 12.9.1, $\Gamma _0(T)=k_ T\, \mathrm{d}V$, and the hypothesis on $f$ gives $p k_ T=0$ almost everywhere. This contradicts $D_ p(T)=\rho _\omega (p)>0$. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.