Effective Geometric Hermitian Positivstellensatz

Abstract

We consider positive Hermitian algebraic functions on holomorphic line bundles over compact complex manifolds. In particular, we consider the tensor product of positive powers of a positive Hermitian algebraic function satisfying the strong global Cauchy–Schwarz condition on a holomorphic line bundle with another fixed positive Hermitian algebraic function on another holomorphic line bundle. Our main result is to give an effective estimate (in terms of certain geometric data) on the smallest power needed to be taken so that the resulting tensor product is a maximal sum of Hermitian squares, or equivalently, the induced Hermitian metric on the resulting line bundle is the pull-back (via some holomorphic map) of the standard Hermitian metric on the universal line bundle over some complex projective space. This result is an effective version of Catlin–D’Angelo’s Hermitian Positivstellensatz.

Appendix: An Example

In this appendix, we consider an example of a one-parameter family of bihomogeneous polynomials constructed by Catlin and D’Angelo [2, p. 160] (see also [25, p. 182, Example VI.1.6] and [8, Example 5.1.1]). For this example, we will illustrate how the constant $m_o$ (and some of the geometric data involved in its definition) in Theorem 2.1 varies with the parameter asymptotically. We also examine how the minimum value of m (for the conclusion of Theorem 2.1 to hold) varies with the parameter.

Let R be the bihomogeneous polynomial in ${\mathbb {C}}^2$ given by

$$\begin{aligned} R(z_1,z_2):=|z_1|^2+|z_2|^2\quad \text {for }z=(z_1,z_2)\in {\mathbb {C}}^2. \end{aligned}$$

(5.121)

For each real number $0<\alpha <2$, we consider the bihomogeneous polynomial $P_\alpha$ in [2, p. 160] given by

$$\begin{aligned} P_\alpha (z_1, z_2)&:=(|z_1|^2 -|z_2|^2)^2+ \alpha |z_1 z_2|^2 \nonumber \\&=|z_1^2|^2 + |z_2^2|^2 + (\alpha -2 ) |z_1 z_2|^2\quad \text {for }z=(z_1,z_2)\in {\mathbb {C}}^2. \end{aligned}$$

(5.122)

From the discussion in Sect. 2.1, one sees that R (resp. $P_\alpha$) defines a positive Hermitian algebraic function on the line bundle $L:={\mathcal {O}}_{{\mathbb {P}}^1}(1)$ (resp. $E:={\mathcal {O}}_{{\mathbb {P}}^1}(2)$) over $X:={\mathbb {P}}^1$, and R satisfies the strong global Cauchy–Schwarz condition. Let $m_o=m_o(\alpha )$ be the constant in (2.26) associated to R and $P=P_\alpha$ as defined above, and let $m_1=m_1(\alpha )$ be defined by

$$\begin{aligned} m_1=m_1(\alpha )&:=\min \big \{m\in {\mathbb {N}}\,\big |\, R^mP_\alpha \text { is a maximal sum} \nonumber \\&\quad \text { of Hermitian squares}\big \}. \end{aligned}$$

(5.123)

By Theorem 2.1, one has $m_o\ge m_1$. We are going to consider the asymptotic behaviors of $m_o$ and $m_1$ as $\alpha \rightarrow 0$. In our present case, the geometric data (n, $c_L$, $\lambda _{P_\alpha }$, $\kappa _{P_\alpha }$, $\kappa _{R}$, $\eta _R$) that determine $m_o$ are such that $n=1$, $c_L=1$, and $\kappa _{R}$ and $\eta _R$ (as well as R itself) do not depend on $\alpha$. Some considerations similar to those in [8, Example 5.1.1] show that $\lambda _{P_\alpha }$ and $\kappa _{P_\alpha }$ have growth orders $\lambda _{P_\alpha }\sim 1/{\alpha }^2$ and $\kappa _{P_\alpha }\sim 1$ as $\alpha \rightarrow 0$. Here for two functions $a=a(\alpha )$ and $b=b(\alpha )$, we write $a\sim b$ as $\alpha \rightarrow 0$ if there exist constants $C_1,\, C_2,\,\alpha _o>0$ such that $C_1 b(\alpha )\le a(\alpha )\le C_2b(\alpha )$ for all $0<\alpha \le \alpha _o$. Together with (2.26), it follows that $m_o$ has growth order

$$\begin{aligned} m_o\sim \dfrac{1}{\alpha ^{\frac{3}{2}}}\quad \text {as }\alpha \rightarrow 0. \end{aligned}$$

(5.124)

Finally it was shown in [8, Example 5.1.1] that when m is even (resp. odd), $R^mP_\alpha$ is a maximal sum of Hermitian squares if and only if $m>\dfrac{4}{\alpha }-2$ (resp. $m>\dfrac{4}{\alpha }-3$). It follows that $m_1$ has growth order

$$\begin{aligned} m_1\sim \dfrac{1}{\alpha }\quad \text {as }\alpha \rightarrow 0. \end{aligned}$$

(5.125)

From (5.124) and (5.125), one sees that Theorem 2.1 is not far from being optimal asymptotically in the example under consideration, while it also applies to much more general situations.