Lemma 12.19.13BD9C00A32F59

Proof. Write the cofactor matrix of $\widehat\Theta _ j$ in block form, with a $2\times 2$ Hermitian block $A_ j$ in the $(x,y)$-directions and a column $\kappa _ j\in \mathbb C^2$:

The cofactor matrix of $T_{j,0}$ has the form $\operatorname {diag}(B_ j,0)$. Let $I_2$ denote the $2\times 2$ identity matrix. The lower bound for $K_ f$ and the definition of $\rho _ j$ give a constant $c_0>0$ such that $B_ j\geq c_0\delta _ j^{-2}I_2$ on $V_ j$. Since $Q_ j>0$, the Schur complement $A_ j-\kappa _ j\kappa _ j^*/\widehat F_ j$ is positive definite, and

by monotonicity of the determinant on positive Hermitian matrices. Combining the root formula with (12.19.8), we obtain

The smaller region $W_ j=\{ |x|,|y|<1/8,\ |\zeta |<\delta _ j/8\} $ has $\mathrm{d}V_0$-volume equal to a fixed positive constant times $\delta _ j^2$. Consequently

For this fixed $j$, lemma 12.19.10 gives smooth convergence of $T_{j,\varepsilon }$ to $T_{j,0}$ on the closure of $W_ j$. The cofactor matrix of $\widehat\Theta _ j+T_{j,0}$ is positive definite there, so the integrand of the root volume depends continuously on these smooth coefficients. We can therefore choose $0<\varepsilon _ j\leq 1/16$ so small that

Set $\Gamma _ j=\widehat\Theta _ j+T_{j,\varepsilon _ j}$. It is smooth, strictly positive and $\mathrm{dd}^{\mathrm{c}}$-closed. Its degree is bounded by (12.19.7) and lemma 12.19.10, whereas its root volume is at least $c_1j^2/2$. $\square$