Theorem 11.16.30A9CF0366804

Proof. Step 1: coordinates. An algebraic frame trivializes $E$ over a nonempty affine open subset $U\subset X$. Since $X$ is smooth and connected, it is irreducible, so $U$ is dense and connected. Write the fiber coordinates as $(w,t,s)\in \mathbb C\times \mathbb C\times \mathbb C^{r-2}$, and put $B=U\times \mathbb C^*\times \mathbb C^{r-2}$. We seek $\mathcal S$ as a graph $w=f(x,t,s)$ over $B$; such a graph avoids the zero section because $t\ne 0$.

Step 2: a local test for one metric. Fix a candidate metric $G$. Since its Kobayashi curvature is not semipositive, its horizontal Schur complement has a positive direction somewhere. This condition is open, so it holds at a point $v_0=(x_0,w_0,t_0,s_0)$ with $x_0\in U$, $t_0\neq 0$ and $G_ w(v_0)\neq 0$. Indeed, the first two conditions define a dense open set, and the zero set of $G_ w$ has empty interior, since vanishing of $G_ w$ on an open set would give $G_{w\bar w}=0$ there, contradicting the positivity of the vertical Hessian.

Let $H$ be the Levi-orthogonal complement of the vertical tangent space $V$ at $v_0$. The Euler identities in the proof of lemma 11.12.1 give

Indeed, $\partial G$ vanishes on $H$, and its value on the vertical radial vector is $G(v_0)>0$. The two summands are Levi orthogonal, the vertical summand is positive definite, and the horizontal summand has a positive direction. Thus the Levi form of $G$ restricted to $K$ has at least $r$ positive eigenvalues.

Since $G_ w(v_0)\neq 0$, the space $K$ is the graph of a linear map defined on $T_{b_0}^{1,0}B$, where $b_0=(x_0,t_0,s_0)$. Choose a local holomorphic function $g$ near $b_0$ with $g(b_0)=w_0$ whose graph has tangent space $K$ at $v_0$. Then $F=G\circ \operatorname {Graph}(g)$ has a critical point at $b_0$, and its Levi form there is the restriction of the Levi form of $G$ to $K$. Keep this first jet of $g$ fixed and vary its quadratic jet. The mixed second derivatives of $F$ do not change, while in local coordinates $z$ on $B$ the pure second derivatives satisfy

where $A$ is fixed. Since $G_ w(v_0)\ne 0$, we can prescribe the pure quadratic part of $F$ to be $\tau \operatorname {Re}\sum _\alpha z_\alpha ^2$, whose real Hessian has eigenvalues $\pm 2\tau $. If $\tau $ is larger than half the operator norm of the real Hessian of the fixed mixed part, the full real Hessian is invertible by the Neumann-series criterion. The critical point is then nondegenerate, with the same Levi form.

Step 3: stability of the test. Choose compact neighborhoods $C'\subset \operatorname {int}C$ of $b_0$ on which $g$ is defined, with $C$ holomorphically convex and $t(C)\subset \{ |t-t_0|\leq |t_0|/4\} $. A small closed Euclidean ball in a closed affine embedding of $B$ is such a set $C$, since linear functions separate points outside the ball from the ball. Fix a smooth Hermitian background metric on $E$ and its compact unit sphere bundle $\Sigma $. By homogeneity, the $C^2$ distance between metrics on $\Sigma $ controls their $C^2$ distance on each fixed compact subset of $E^\times $. Persistence of nondegenerate critical points and openness of the conditions on the Levi eigenvalues therefore give a neighborhood $\mathcal V_ G$ of $G|_\Sigma $ for the $C^2$ topology and a number $\delta _ G>0$ with the following property: for every metric whose restriction to $\Sigma $ lies in $\mathcal V_ G$ and every holomorphic function within $\delta _ G$ of $g$ uniformly on $C$, the restriction of the metric to the graph of the function still has a nondegenerate critical point over $C'$ with at least $r$ positive Levi eigenvalues. Here the Cauchy estimates on $C'$ turn uniform closeness on $C$ into $C^2$ closeness on $C'$.

Step 4: one hypersurface for all metrics. The restrictions of the candidate metrics to $\Sigma $ form a subspace of the separable metrizable space $C^2(\Sigma )$. This subspace is second countable, hence Lindelöf, so the cover by the neighborhoods $\mathcal V_ G$ has a countable subcover $(\mathcal V_ j)_{j\geq 1}$, with associated data $C_ j,g_ j,\delta _ j$. For $D_\lambda (x,w,t,s)=(x,\lambda w,\lambda t,\lambda s)$,

Dilation therefore preserves critical points of restrictions to graphs, their real nondegeneracy and their Levi inertia, for every candidate metric. It maps the graph of a function $g$ over $C$ to the graph of $\lambda g\circ d_\lambda ^{-1}$ over $d_\lambda (C)$, with $d_\lambda $ as in lemma 11.16.2. A function within $|\lambda |\delta _ j$ of $\lambda g_ j\circ d_\lambda ^{-1}$ on $d_\lambda (C_ j)$ pulls back under the dilation to a function within $\delta _ j$ of $g_ j$ on $C_ j$. In particular, the metric neighborhood $\mathcal V_ j$ needs no change.

Applying lemma 11.16.2, we obtain one function $f\in \mathcal O(B)$ that satisfies all the dilated tests. Its graph $\mathcal S$ is biholomorphic to the connected affine Stein manifold $B$, and it is smooth and closed in the open subset $\{ x\in U,\ t\neq 0\} $ of $E^\times $. Every candidate metric belongs to some $\mathcal V_ j$, so its restriction to $\mathcal S$ has the required nondegenerate critical point. Since $\dim _{\mathbb C}\mathcal S=n+r-1$, at least $r$ positive Levi eigenvalues leave at most $n-1$ negative ones. $\square$