Lemma 11.29.7944943C60132

Proof. For nesting, multiply the sections in $V_ m$ by each $t_\alpha $ and express the products in the basis of $V_{m+1}$. The identity $\sum _\alpha \mathrm e^{-\phi }|t_\alpha |^2=1$ writes $G_{m,B}$ as the sum of the corresponding weighted quadratic functions at level $m+1$, which combine into one Hermitian coefficient matrix at that level. Surjectivity of the multiplication map is not needed.

For density, write $p_{m_0}(v)=P_0v$ in a local frame of $F$, and put $L=\mathrm e^{-m_0\phi /2}P_0$. Global generation of $E\otimes A^{m_0}$ means that $L$ is injective, so $M=L^*L$ is positive definite. If a $C^2$ Hermitian form has matrix $H$, define

Then $L^*CL=H$. A change of frame of $F$ cancels in this expression; a change of frame of $A$ multiplies $L$ by a scalar of modulus one, which also cancels. Thus $C$ is a globally defined $C^2$ Hermitian matrix function on $X$.

Applying lemma 11.29.6 to its real coordinates in the vector space of Hermitian matrices, we obtain Hermitian matrix polynomials in the entries of $Y$ that converge to $C$ in $C^2$; their degrees may increase along the sequence. Let $C_ q(Y)$ be such a polynomial, of degree at most $q$. Every monomial of degree $k$ in $Y$ has the form $t^ I\bar t^ J/\rho ^ k$, with $|I|=|J|=k$. For a polynomial of degree at most $q$, multiply each numerator by $\rho ^{q-k}$ to use the common denominator $\rho ^ q$. Consequently

is a quadratic expression in the product sections $\sigma _ i t^ I$ of $E\otimes A^{m_0+q}$, with weight $\mathrm e^{-(m_0+q)\phi }$. Express these products in the basis of $V_{m_0+q}$ and replace the coefficient matrix by its Hermitian part, which does not change the real function. The result is an element of $\mathcal H_{m_0+q}$. Since $L^*CL=H$ and multiplication by the fixed smooth matrix $L$ is continuous in $C^2$, these elements converge to $H$. $\square$