11.29 Projective first jets and positive bundle metricsA1CB99CAE47B

Proof. On vertical tangent vectors the Levi form is $h$ itself, so strict positivity forces $h>0$. At a fixed base point, choose a holomorphic frame in which the matrix $H$ of $h$ is the identity and has vanishing first derivatives. Then $R_ h(u,\bar u)=-H_{u\bar u}$ at this point. A tangent vector to $F$ has horizontal and vertical components $(u,w)$, and

For $v\ne 0$, this is positive for every nonzero $(u,w)$ precisely when the curvature evaluation is negative for every nonzero $u$. The curvature of the dual metric is the negative transpose of the original curvature, with respect to the dual Hermitian pairing, which gives the last assertion. $\square$

Proof. The real vector space of Hermitian $N_ m\times N_ m$ matrices has dimension $N_ m^2$. Carathéodory's theorem expresses every point of the convex hull as a convex combination of at most $N_ m^2+1$ points of the compact image $\Psi _ m(K)$, so the hull is compact. If it contains zero, the coefficients of such an expression define the required atomic probability measure $\mu $, and then $\int _ K\operatorname {tr}(B\Psi _ m)\, \mathrm d\mu =0$ excludes the first alternative.

If the hull misses zero, let $Z_0$ be its point of least Hilbert–Schmidt norm. Differentiating the squared norm along the segment from $Z_0$ to any $Z$ in the hull gives

Taking $B=Z_0$ proves the first alternative. $\square$

Proof. The normalized rank-one projector $Y$ determines the point $[t]$ of projective space, and its differential vanishes only in directions in which $\mathrm dt$ is proportional to $t$. Since $A$ is very ample, $Y$ is therefore injective with injective differential, and compactness makes it an embedding. A $C^2$ function on this image extends to a $C^2$ function on an ambient cube: use local coordinates in which the image is a graph, extend constantly in the normal coordinates, and combine these extensions with a finite smooth partition of unity and a cutoff. Rescale the cube to $[0,1]^ q$.

For such an extension $f$, its multivariate Bernstein polynomial is

where $j=(j_1,\ldots ,j_ q)$. For a multi-index $\alpha $ with $|\alpha |\leq 2$, differentiation expresses $\partial ^\alpha B_ Nf$ as an average of $\Delta _{1/N}^\alpha f(j/N)$, with prefactor $\prod _\ell (N)_{\alpha _\ell }$ and Bernstein degrees $N-\alpha _\ell $. Here $(N)_ r=N(N-1)\cdots (N-r+1)$, and $\Delta ^\alpha _{1/N}$ is the corresponding iterated forward difference with step $1/N$. Repeated use of the fundamental theorem of calculus expresses $N^{|\alpha |}\Delta _{1/N}^\alpha f(j/N)$ as an average of $\partial ^\alpha f$ at points within $2/N$ in each affected coordinate.

The remaining Bernstein weights are the law of independent binomial variables of degrees $N-\alpha _\ell $, divided by $N$. Their means differ from $y_\ell $ by at most $2/N$ and their variances are at most $1/(4N)$. Uniform continuity of $\partial ^\alpha f$, splitting the average into a fixed small neighborhood of $y$ and its complement, and Chebyshev's inequality give uniform convergence to $\partial ^\alpha f$. The prefactor divided by $N^{|\alpha |}$ tends to one. This proves convergence for every derivative of order at most two, including the function itself. Restrict to the embedded image and use the chain rule in a finite collection of charts to obtain the asserted $C^2$ approximation. $\square$

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 $H$. $\square$

Proof. Given a metric as in the first assertion, its dual metric $h$ on $F$ satisfies $Dh\geq \delta b$ on $S$ for some $\delta >0$, by lemma 11.29.1 and compactness. The $C^2$ density in lemma 11.29.7 and the continuity of $D$ give a form $h_{m,B}$ with $D h_{m,B}\geq \delta b/2$, which proves the second assertion. Conversely, if $G_{m,B}$ is strictly plurisubharmonic, its vertical Levi form shows that $h_{m,B}$ is positive, and lemma 11.29.1 shows that its dual metric is strictly Griffiths-positive. The second and third assertions are equivalent by proposition 11.29.4.

If the first assertion holds, every nonzero positive matrix-valued measure $T$ has positive trace mass, and

so such a measure cannot satisfy $D^*T=0$. Conversely, suppose that the third assertion fails at every level. By proposition 11.29.4, choose at each level $m$ an atomic probability measure $\mu _ m$ on $K$ with $\int _ K\Psi _ m\, \mathrm d\mu _ m=0$, so that $\mu _ m$ annihilates $Dh$ for every $h\in \mathcal H_ m$. A subsequence converges weakly to a probability measure $\mu $ on the compact set $K$. Since the spaces $\mathcal H_ m$ are nested, the limit annihilates $Dh$ for every $h$ in each fixed $\mathcal H_ j$, and the $C^2$ density and the continuity of $D$ extend this to every smooth Hermitian form.

Define

This is a positive matrix-valued measure on $W\to S$ with $\langle T,b\rangle =1$ and $D^*T=0$. Its entries in local frames are represented by scalar measures: positivity bounds the functional on Hermitian forms of uniform norm at most one by the trace mass, so the Riesz representation theorem applies to the coordinate functionals, and a finite partition of unity combines these local representations. This contradicts the fourth assertion and completes the proof. $\square$

Proof. At $m=0$, one has $\Psi _0=\mathrm d\Phi (\xi )\mathrm d\Phi (\xi )^*$, where $\Phi $ is written in the coordinates given by the chosen basis. If $\mathrm d\Phi $ is injective off the zero section, $B=\operatorname {Id}$ gives a strictly positive Levi form. If $\mathrm d\Phi $ has a nonzero kernel vector there, fiber dilation and normalization of the tangent vector place such a vector in $K$, where $\Psi _0=0$, and no Hermitian matrix can separate the image from zero. $\square$