Quantitative injectivity of the Fubini–Study map

Yoshinori Hashimoto Department of Mathematics, Osaka Metropolitan University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan. [email protected]

Abstract.

We prove a quantitative version of the injectivity of the Fubini–Study map that is polynomial in the exponent of the ample line bundle, and correct the arguments in the author’s previous papers [yhhilb, yhextremal].

1. Introduction

Suppose that we have a polarised smooth projective variety $(X,L)$ over $\mathbb{C}$ of complex dimension $n$ and a projective embedding $\iota_{k}:X\hookrightarrow\mathbb{P}(H^{0}(X,L^{k})^{\vee})$ , where $L$ is an ample line bundle over $X$ and $k$ is a large enough integers such that $L^{k}$ is very ample. We may moreover assume, by replacing $L$ by a higher tensor power if necessary, that the automorphism group $\mathrm{Aut}_{0}(X,L)$ acts on $(X,L)$ via the ambient linear action of $SL(H^{0}(X,L^{k}))$ fixing $\iota_{k}(X)$ . We fix a hermitian metric $h$ on $L$ which defines a Kähler metric $\omega_{h}>0$ on $X$ , where $\omega_{h}\in c_{1}(L)$ . This in turn defines a positive definite hermitian form $\mathrm{Hilb}(h^{k})$ on $H^{0}(X,L^{k})$ defined (see [donproj2]) as

\mathrm{Hilb}(h^{k})(s_{1},s_{2}):=\frac{N_{k}}{V}\int_{X}h^{k}(s_{1},s_{2})\frac{\omega^{n}_{h}}{n!}

for $s_{1},s_{2}\in H^{0}(X,L^{k})$ , where we wrote

N_{k}:=\dim_{\mathbb{C}}H^{0}(X,L^{k}),\quad V:=\int_{X}c_{1}(L)^{n}/n!.

Choosing a $\mathrm{Hilb}(h^{k})$ -orthonormal basis $\{s_{i}\}_{i=1}^{N_{k}}$ , we identify $\mathbb{P}(H^{0}(X,L^{k})^{\vee})$ with $\mathbb{P}^{N_{k}-1}$ , and also identify the set $\mathcal{B}_{k}$ of all positive definite hermitian forms on $H^{0}(X,L^{k})$ with positive definite $N_{k}\times N_{k}$ hermitian matrices, noting that $\mathrm{Hilb}(h^{k})$ is the identity matrix with respect to this basis. The set $\mathcal{B}_{k}$ in turn can be identified with the symmetric space $GL(N_{k},\mathbb{C})/U(N_{k})$ . The Fubini–Study map $\mathrm{FS}_{k}$ associates to $A\in\mathcal{B}_{k}$ a hermitian metric $\mathrm{FS}_{k}(A)$ , called the Fubini–Study metric, on $L^{k}$ . Recall that $\mathrm{FS}_{k}(A)$ is defined as a unique hermitian metric on $L^{k}$ which satisfies

\sum_{i=1}^{N_{k}}\mathrm{FS}_{k}(A)(s^{A}_{i},s^{A}_{i})=1

(1)

over $X$ , for an $A$ -orthonormal basis $\{s^{A}_{i}\}_{i=1}^{N_{k}}$ for $H^{0}(X,L^{k})$ . When we write $H_{k}:=\mathrm{Hilb}(h^{k})$ , the equation above immediately implies

\mathrm{FS}_{k}(A)=\left(\sum_{i,j=1}^{N_{k}}A^{-1}_{ij}\mathrm{FS}_{k}(H_{k})(s_{i},s_{j})\right)^{-1}\mathrm{FS}_{k}(H_{k}),

by noting that $\{\sum_{j}A^{-1/2}_{ij}s_{j}\}_{i=1}^{N_{k}}$ is an $A$ -orthonormal basis. For $A,B\in\mathcal{B}_{k}$ , we define a smooth real function $f_{k}(A,B;H_{k})$ on $X$ defined as

	$\displaystyle f_{k}(A,B;H_{k})$	$\displaystyle:=\frac{\mathrm{FS}_{k}(H_{k})}{\mathrm{FS}_{k}(A)}-\frac{\mathrm{FS}_{k}(H_{k})}{\mathrm{FS}_{k}(B)}$
		$\displaystyle=\sum_{i,j=1}^{N_{k}}(A^{-1}_{ij}-B^{-1}_{ij})\mathrm{FS}_{k}(H_{k})(s_{i},s_{j}).$

Given $A,B\in\mathcal{B}_{k}$ , it is natural to conjecture that $A$ is close to $B$ in $\mathcal{B}_{k}$ if and only if $f_{k}(A,B;H_{k})$ is close to zero in some appropriate norm. Indeed, Lempert [Lem21, Theorem 1.1] proved that the Fubini–Study map is injective, which is equivalent to saying that $A=B$ if and only if $f_{k}(A,B;H_{k})\equiv 0$ , and also proved that its image is not closed [Lem21, Theorem 1.1, Example 6.2, Theorem 6.4]. In this paper, we consider the following quantitative version of injectivity.

Problem 1.1.

Can we bound an appropriately defined distance between $A$ and $B$ in $\mathcal{B}_{k}$ by $Ck^{l}\|f_{k}(A,B;H_{k})\|$ , where $C,l>0$ are some constants which depend only on $h$ ?

The problem here is that the constant appearing in the estimate is at most polynomial in $k$ , as well as the right choice of the norm for $f_{k}(A,B;H_{k})$ . In [yhhilb, Lemma 3.1], the author claimed that this problem can be solved with the $C^{0}$ -norm, but its proof does not hold since it is based on an erroneous claim that the Hilbert map is surjective [yhhilb, Theorem 1.1, Proposition 2.15]. Indeed, a counterexample was provided by Lempert in an email correspondence with the author (see [Finski22, Proposition 4.16] and also [Finski22s, Proposition 3.6]), and there is also a counterexample to the surjectivity of the Hilbert map by Sun [Sun22, §3, Example]. Both these counterexamples involve basis elements that are redundant in terms of the global generation of the line bundle (see the definition of a rather ample subspace in [Lem21, §6]). Sun [Sun22, Theorems 1.2 and 1.3] also described the closure of the image of the Hilbert map.

It is still plausible, however, that Problem 1.1 can be solved affirmatively once the right norm is chosen. The main result of this paper is the following.

Theorem 1.2.

There exist constants $k_{0}\in\mathbb{N}$ and $C_{h}>0$ , depending only on $h$ , such that we have

\|A^{-1}-B^{-1}\|^{2}_{\mathrm{HS}(H_{k})}\leq C_{h}k^{n}\|f_{k}(A,B;H_{k})\|^{2}_{W^{2,2}(\omega_{h})}

for any $A,B\in\mathcal{B}_{k}$ and for any $k\geq k_{0}$ , where $\|\cdot\|_{\mathrm{HS}(H_{k})}$ is the Hilbert–Schmidt norm with respect to $H_{k}=\mathrm{Hilb}(h^{k})$ and $\|\cdot\|_{W^{2,2}(\omega_{h})}$ is the Sobolev norm with respect to $\omega_{h}$ .

We stress here that $A,B\in\mathcal{B}_{k}$ are regarded as matrices with respect to the $H_{k}$ -orthonormal basis, when we consider the inverse matrices above.

The choice of the reference metric $H_{k}$ is important in Theorem 1.2; see (6) for the formula with respect to a different reference. Indeed, the counterexamples of Lempert and Sun mentioned above deal with the case when the reference metric is close to being degenerate.

Acknowledgements. The author thanks Siarhei Finski, László Lempert, and Jingzhou Sun for very helpful discussions and immensely valuable examples. This research is partially supported by JSPS KAKENHI (Grant-in-Aid for Scientific Research (C)), Grant Number JP23K03120, and JSPS KAKENHI (Grant-in-Aid for Scientific Research (B)) Grant Number JP24K00524.

2. Proof of Theorem 1.2

In what follows, we write $\omega_{h,k}\in kc_{1}(L)$ for the Kähler metric on $X$ associated to $\mathrm{FS}(H_{k})$ , and $\tilde{\omega}_{h,k}$ for the Fubini–Study metric on $\mathcal{O}_{\mathbb{P}^{N_{k}-1}}(1)$ over $\mathbb{P}^{N_{k}-1}=\mathbb{P}(H^{0}(X,L^{k})^{\vee})$ defined by $H_{k}$ . Recall $\omega_{h,k}=\iota^{*}_{k}\tilde{\omega}_{h,k}$ , and that we have

\omega_{h,k}=k\omega_{h}-\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\log\rho_{k}(\omega_{h})

by a result due to Rawnsley [rawnsley], where we wrote $\rho_{k}(\omega_{h})$ for the $k$ -th Bergman function with respect to $\omega_{h}$ . We also use the re-scaled version $\bar{\rho}_{k}(\omega_{h}):=\frac{V}{N_{k}}\rho_{k}(\omega_{h})$ . We recall the asymptotic expansion of the Bergman function (see e.g. [mm])

\rho_{k}(\omega_{h})=k^{n}+O(k^{n-1}),\quad\text{or}\quad\bar{\rho}_{k}(\omega_{h})=1+O(1/k),

(2)

where $N_{k}=Vk^{n}+O(k^{n-1})$ by the asymptotic Riemann–Roch theorem. We also recall that the expansion above holds in $C^{m}$ -norm for any $m\in\mathbb{N}$ . We thus have

\omega_{h,k}=k\omega_{h}+O(1/k).

We start with the following lemma concerning the second fundamental form of the projective embedding.

Lemma 2.1.

Suppose $H_{k}=\mathrm{Hilb}(h^{k})$ , and let $A_{h,k}$ be the second fundamental form of $TX$ in $\iota^{*}T\mathbb{P}^{N_{k}-1}$ with respect to $\tilde{\omega}_{h,k}$ . Then, there exist $k_{0}\in\mathbb{N}$ and a constant $\lambda_{\mathrm{min}}(h)=\lambda_{\mathrm{min}}>0$ depending only on $h$ , such that the minimum eigenvalue of $-A_{h,k}^{*}\wedge A_{h,k}$ with respect to a $g_{h,k}$ -orthonormal basis can be bounded below by $\lambda_{\mathrm{min}}(h)$ for all $k\geq k_{0}$ and all $x\in X$ . In particular, $A_{h,k}$ is nondegenerate everywhere on $X$ for all large enough $k$ .

Proof.

We recall that the curvature of the Fubini–Study metrics on the ambient projective space and $X$ can be related by

-A_{h,k}^{*}\wedge A_{h,k}=\pi_{T}\circ(\tilde{F}_{h,k}|_{TX})-F_{h,k}

as given in [grihar, page 78] or [PS04, equation (5.27)], where we wrote $\tilde{F}_{h,k}$ for the curvature form of $\tilde{\omega}_{h,k}$ on the ambient projective space and $F_{h,k}$ for that of $\omega_{h,k}$ on $X$ . The adjoint above is with respect to $\tilde{\omega}_{h,k}$ . We recall that the curvature tensor of the Fubini–Study metric on $\mathbb{P}^{N_{k}-1}$ can be written as

(\tilde{F}_{h,k})_{i\bar{j}l\bar{m}}=(\tilde{g}_{h,k})_{i\bar{j}}(\tilde{g}_{h,k})_{l\bar{m}}+(\tilde{g}_{h,k})_{i\bar{m}}(\tilde{g}_{h,k})_{l\bar{j}}

where $\tilde{g}_{h,k}$ is the metric tensor of $\tilde{\omega}_{h,k}$ written with respect to its orthonormal basis.

By the Bergman kernel expansion (2), we find that the restriction of $\tilde{g}_{h,k}$ to $TX$ satisfies the asymptotic expansion

g_{h,k}=kg_{h}+O(1/k),

where $g_{h}$ stands for the Kähler metric associated to $h$ . Thus we have

{{(\pi_{T}\circ(\tilde{F}_{h,k}|_{TX})-F_{h,k})_{i}}^{j}}_{l\bar{m}}=\delta_{i}^{j}\delta_{l\bar{m}}+\delta_{l}^{j}\delta_{i\bar{m}}-{{(F_{h})_{i}}^{j}}_{l\bar{m}}+O(1/k)

with respect to any $kg_{h}$ -orthonormal basis for $TX$ (note that the curvature tensor $F_{h}$ is of order 1 with respect to a $kg_{h}$ -orthonormal basis, which is of order $k^{-1}$ with respect to a $g_{h}$ -orthonormal basis). Since this agrees with $-A_{h,k}^{*}\wedge A_{h,k}$ , the tensor above is positive semidefinite.

Suppose that $-A_{h,k}^{*}\wedge A_{h,k}$ is degenerate at $x\in X$ . We then perturb $h$ locally around $x$ so that the perturbation $h_{\epsilon}$ satisfies

h(x)=h_{\epsilon}(x),\quad g(x)=g_{\epsilon}(x)

and the curvature tensor is perturbed by $-\epsilon\delta_{i}^{j}(kg_{h})_{l\bar{m}}$ at $x$ , which is of order $k\epsilon$ locally at $x$ , and such that $\mathrm{Hilb}(h_{\epsilon}^{k})=\mathrm{Hilb}(h^{k})+O(\epsilon^{2})$ . This is possible by considering the perturbation of the Kähler potential by $\epsilon|z_{i}|^{2}|z_{l}|^{2}$ where $(z_{1},\dots,z_{n})$ are holomorphic normal coordinates around $x$ , multiplied by a cutoff function which decreases very rapidly away from $x$ so that the perturbation introduced in the integral inside $\mathrm{Hilb}$ is small.

Applying the argument above to $h_{\epsilon}$ and writing with respect to a $g_{h}$ -orthonormal basis, we find

	$\displaystyle-A_{h,k,\epsilon}^{*}\wedge A_{h,k,\epsilon}=\pi_{T}\circ(\tilde{F}_{h,k,\epsilon}\|_{TX})-F_{h,k,\epsilon}$	$\displaystyle=-A_{h,k}^{*}\wedge A_{h,k}-\epsilon\delta_{i}^{j}(kg_{h})_{l\bar{m}}+O(\epsilon^{2})+O(1/k)$
		$\displaystyle=-A_{h,k}^{*}\wedge A_{h,k}-\epsilon\delta_{i}^{j}\delta_{l\bar{m}}+O(\epsilon^{2})+O(1/k),$

where $\tilde{F}_{h,k,\epsilon}$ is the curvature of the Fubini–Study metric on $\mathbb{P}^{N_{k}-1}$ with respect to $\mathrm{Hilb}(h^{k}_{\epsilon})$ , $F_{h,k,\epsilon}$ is the one with respect to its restriction to $TX$ , and $A_{h,k,\epsilon}$ is the second fundamental form that is associated to it. This yields a contradiction for a sufficiently small $\epsilon>0$ and for all large enough $k$ , if we assume that $-A_{h,k}^{*}\wedge A_{h,k}$ is degenerate at $x\in X$ .

The second claim is a consequence of the fact that, with respect to a $kg_{h}$ -orthonormal basis, $\pi_{T}\circ(\tilde{F}_{h,k}|_{TX})-F_{h,k}$ converges to a positive definite form on $X$ by the previous argument and the Bergman kernel expansion $g_{h,k}=kg_{h}+O(1/k)$ , together with the compactness of $X$ . ∎

Given $A,B\in\mathcal{B}_{k}$ , we define a (not necessarily positive) hermitian matrix $\Lambda$ by

\Lambda:=A^{-1}-B^{-1},

and also define

f(\Lambda;H_{k}):=f(A,B;H_{k}).

We write $\xi_{\Lambda}$ for the holomorphic vector field on $\mathbb{P}^{N_{k}-1}$ generated by the linear action of $\Lambda$ . We recall that, when $[Z_{1}:\cdots:Z_{N_{k}}]$ is a set of homogeneous coordinates for $\mathbb{P}^{N_{k}-1}$ consisting of $H_{k}$ -orthonormal basis for $H^{0}(X,L^{k})$ , the Hamiltonian function for $\xi_{\Lambda}$ with respect to $\tilde{\omega}_{h,k}$ is given by

\tilde{f}(\Lambda;H_{k}):=\sum_{i,j=1}^{N_{k}}\Lambda_{ij}\frac{Z_{i}\bar{Z}_{j}}{\sum_{l=1}^{N_{k}}|Z_{l}|^{2}},

which is a well-defined smooth function on $\mathbb{P}^{N_{k}-1}$ . By the definition of $\mathrm{FS}_{k}(H_{k})$ , we have

f(\Lambda;H_{k})=\iota^{*}_{k}\tilde{f}(\Lambda;H_{k}).

We first consider the case when the trace of $\Lambda$ is zero, i.e. when $\Lambda\in\mathfrak{sl}(N_{k},\mathbb{C})$ . Following [PS04, §5], we consider a holomorphic vector bundle $\iota^{*}T\mathbb{P}^{N_{k}-1}$ over $X$ , which we decompose as

\iota^{*}T\mathbb{P}^{N_{k}-1}=TX\oplus\mathcal{N},

(3)

where the direct sum is orthogonal with respect to the Fubini–Study metric on $\iota^{*}T\mathbb{P}^{N_{k}-1}$ with respect to $H_{k}$ . We write $\pi_{T}$ and $\pi_{\mathcal{N}}$ for the projection to $TX$ and $\mathcal{N}$ . By the defining equation

\iota(\xi_{\Lambda})\tilde{\omega}_{h,k}=-d\tilde{f}(\Lambda;H_{k})

for the Hamiltonian function for $\xi_{\Lambda}$ , together with $\omega_{h,k}=\iota^{*}_{k}\tilde{\omega}_{h,k}$ and (2), we find that $\pi_{T}\xi_{\Lambda}$ is $\omega_{h,k}$ -metric dual to $df_{k}(\Lambda;H_{k})$ as pointed out in [Fine10, Lemma 20].

Recall that we have a faithful representation $\mathrm{Aut}_{0}(X,L)\hookrightarrow SL(N_{k},\mathbb{C})$ by the linearisation of the action of $\mathrm{Aut}_{0}(X,L)$ , which is unique up to an overall multiplicative constant (by replacing $L$ by a higher tensor power if necessary). We further decompose $\Lambda=\alpha+\beta$ according to $\mathfrak{sl}(N_{k},\mathbb{C})=\mathfrak{aut}(X,L)\oplus\mathfrak{aut}(X,L)^{\perp}$ , where the orthogonality is defined with respect to the $L^{2}$ -metric defined by $\omega_{h,k}$ which is the Fubini–Study metric on $\iota^{*}T\mathbb{P}^{N_{k}-1}$ with respect to $H_{k}$ . We then have

\|\xi_{\Lambda}\|^{2}_{L^{2}(\omega_{h,k})}=\|\xi_{\alpha+\beta}\|^{2}_{L^{2}(\omega_{h,k})}=\|\xi_{\alpha}\|^{2}_{L^{2}(\omega_{h,k})}+\|\xi_{\beta}\|^{2}_{L^{2}(\omega_{h,k})}

by definition. Note also that, according to the orthogonal decomposition (3), we have

\|\xi_{\beta}\|^{2}_{L^{2}(\omega_{h,k})}=\|\pi_{T}\xi_{\beta}\|^{2}_{L^{2}(\omega_{h,k})}+\|\pi_{\mathcal{N}}\xi_{\beta}\|^{2}_{L^{2}(\omega_{h,k})}

because of the pointwise orthogonality, and also note $\xi_{\alpha}=\pi_{T}\xi_{\alpha}$ .

Lemma 2.2.

Suppose $\iota_{k}:X\hookrightarrow\mathbb{P}(H^{0}(X,L^{k})^{\vee})$ , and that we consider the Fubini–Study metric $\mathrm{FS}(H_{k})$ where $H_{k}=\mathrm{Hilb}(h^{k})$ . Then there exists $C_{1}>0$ depending only on $h$ such that

\|\pi_{\mathcal{N}}\xi_{\beta}\|^{2}_{L^{2}(\omega_{h,k})}\leq C_{1}\|\bar{\partial}(\pi_{T}\xi_{\beta})\|^{2}_{L^{2}(\omega_{h,k})}

holds for any $\beta\in\mathfrak{aut}(X,L)^{\perp}$ and for all large enough $k$ .

Proof.

We prove

\|\pi_{\mathcal{N}}\xi_{\beta}\|^{2}_{L^{2}(\omega_{h,k})}\leq C_{1}\|\bar{\partial}(\pi_{\mathcal{N}}\xi_{\beta})\|^{2}_{L^{2}(\omega_{h,k})}=C_{1}\|\bar{\partial}(\pi_{T}\xi_{\beta})\|^{2}_{L^{2}(\omega_{h,k})}.

(4)

The equality in (4) is straightforward, following [PS04, page 705], since

\bar{\partial}_{\mathbb{P}^{N_{k}-1}}\xi_{\beta}=0=\bar{\partial}_{\mathbb{P}^{N_{k}-1}}(\pi_{\mathcal{N}}\xi_{\beta})+\bar{\partial}_{\mathbb{P}^{N_{k}-1}}(\pi_{T}\xi_{\beta}),.

which in turn implies

\bar{\partial}\xi_{\beta}=0=\bar{\partial}(\pi_{\mathcal{N}}\xi_{\beta})+\bar{\partial}(\pi_{T}\xi_{\beta}),

where $\bar{\partial}$ makes sense as a $\bar{\partial}$ -operator on the holomorphic vector bundle $\iota^{*}T\mathbb{P}^{N_{k}-1}$ .

Thus it suffices to prove the inequality in (4), which is the reverse direction of [PS04, equation (5.19)], and we prove it by using the lower bound of the second fundamental form given in Lemma 2.1.

We reproduce parts of the argument in [PS04, §5] here for completeness. Fix a point $x\in X$ and a local holomorphic frame $\{e_{1},\dots,e_{n},f_{1},\dots,f_{m}\}$ of $\iota^{*}T\mathbb{P}^{N_{k}-1}$ (with $N-1=n+m$ ) in a neighbourhood of $x$ such that it is an orthonormal basis for $\iota^{*}T_{x}\mathbb{P}^{N_{k}-1}$ with respect to $\tilde{g}_{h,k}$ and $\{e_{1},\dots,e_{n}\}$ form a local holomorphic frame of $TX$ near $x$ . We then write

\xi_{\beta}=\sum_{i=1}^{n}a_{i}e_{i}+\sum_{j=1}^{m}b_{j}f_{j}

where $a_{1},\dots,a_{n},b_{1},\dots,b_{m}$ are (local) holomorphic functions. We then have

\pi_{\mathcal{N}}\xi_{\beta}=\sum_{j=1}^{m}b_{j}\left(f_{j}-\sum_{i=1}^{n}\phi_{ij}e_{i}\right)

for some smooth functions $\phi_{ij}$ vanishing at $x$ . Then we have

\bar{\partial}(\pi_{\mathcal{N}}\xi_{\beta})=\sum_{j=1}^{m}b_{j}\left(-\sum_{i=1}^{n}(\bar{\partial}\phi_{ij})e_{i}.\right)

It is explained in [PS04, §5] that $\bar{\partial}\phi_{ij}$ is the second fundamental form of $TX$ with respect to $\tilde{\omega}_{k,h}$ . By Lemma 2.1, there exists a constant $\lambda_{\mathrm{min}}(h)=\lambda_{\mathrm{min}}>0>0$ depending only on $h$ , such that

\sum_{i=1}^{n}\left|\sum_{j=1}^{N-1-n}b_{j}\bar{\partial}\phi_{ij}\right|^{2}(x)\geq\lambda_{\mathrm{min}}\sum_{j=1}^{N-1-n}|b_{j}|^{2}(x)

holds for any $x\in X$ . Integrating both sides over $X$ with respect to the volume form $\omega^{n}_{h,k}$ , we get the required estimate (4).∎

Proof of Theorem 1.2.

We first recall that there exists $C_{2}>0$ depending only on $h$ (and independent of $k$ as long as it is large enough)

\|\Lambda\|^{2}_{\mathrm{HS}(H_{k})}\leq C_{2}k\|\xi_{\Lambda}\|^{2}_{L^{2}(\omega_{h,k})}=C_{2}k\left(\|\xi_{\alpha}\|^{2}_{L^{2}(\omega_{h,k})}+\|\xi_{\beta}\|^{2}_{L^{2}(\omega_{h,k})}\right)

by [PS04, (5.7)], where we note that the assumption (see [PS04, (5.1)]) for this estimate is satisfied for $\mathrm{FS}_{k}(H_{k})$ since the associated centre of mass is the identity matrix up to an error of order $1/k$ (in the operator norm, after passing to a unitarily equivalent basis in which $D_{k}$ and $E_{k}$ are both diagonal) by the Bergman kernel expansion; note that $\Lambda$ is hermitian, and the Lie algebra $\mathfrak{su}(N)$ is isomorphic to the vector space consisting of hermitian matrices.

Since $\xi_{\alpha}=\pi_{T}\xi_{\alpha}$ is $L^{2}$ -orthogonal to $\xi_{\beta}$ , and pointwise orthogonal to $\pi_{\mathcal{N}}\xi_{\beta}$ , $\pi_{T}\xi_{\alpha}$ is $L^{2}$ -orthogonal to $\pi_{T}\xi_{\beta}$ . We thus find

\|\pi_{T}\xi_{\alpha}\|^{2}_{L^{2}(\omega_{h,k})}+\|\pi_{T}\xi_{\beta}\|^{2}_{L^{2}(\omega_{h,k})}=\|\pi_{T}\xi_{\alpha}+\pi_{T}\xi_{\beta}\|^{2}_{L^{2}(\omega_{h,k})}=\|\pi_{T}\xi_{\alpha+\beta}\|^{2}_{L^{2}(\omega_{h,k})}

by the linearity of $\pi_{T}$ . Thus, combining these equalities we get

	$\displaystyle\\|\xi_{\alpha}\\|^{2}_{L^{2}(\omega_{h,k})}+\\|\xi_{\beta}\\|^{2}_{L^{2}(\omega_{h,k})}$	$\displaystyle=\\|\pi_{T}\xi_{\alpha}\\|^{2}_{L^{2}(\omega_{h,k})}+\\|\pi_{T}\xi_{\beta}\\|^{2}_{L^{2}(\omega_{h,k})}+\\|\pi_{\mathcal{N}}\xi_{\beta}\\|^{2}_{L^{2}(\omega_{h,k})}$
		$\displaystyle=\\|\pi_{T}\xi_{\alpha+\beta}\\|^{2}_{L^{2}(\omega_{h,k})}+\\|\pi_{\mathcal{N}}\xi_{\beta}\\|^{2}_{L^{2}(\omega_{h,k})}.$

By Lemma 2.2, we get

\|\Lambda\|^{2}_{\mathrm{HS}(H_{k})}\leq C_{3}k\left(\|\pi_{T}\xi_{\Lambda}\|^{2}_{L^{2}(\omega_{h,k})}+\|\bar{\partial}(\pi_{T}\xi_{\Lambda})\|^{2}_{L^{2}(\omega_{h,k})}\right),

for some $C_{3}>0$ which depends only on $h$ , where we used

\bar{\partial}(\pi_{T}\xi_{\beta})=\bar{\partial}(\xi_{\alpha}+\pi_{T}\xi_{\beta})=\bar{\partial}(\pi_{T}\xi_{\Lambda}).

Recalling that $\pi_{T}\xi_{\Lambda}$ is $\omega_{h,k}$ -metric dual to $df_{k}(\Lambda;H_{k})$ , we get

\left\|\pi_{T}\xi_{\Lambda}\right\|^{2}_{L^{2}(\omega_{h,k})}=\left\|df_{k}(\Lambda;H_{k})\right\|^{2}_{L^{2}(\omega_{h,k})}\leq 2k^{n-1}\left\|df_{k}(\Lambda;H_{k})\right\|^{2}_{L^{2}(\omega_{h})}

for all large enough $k$ , by noting $\omega_{h,k}=k\omega_{h}+O(1/k)$ which follows from the Bergman kernel expansion (2); we note that we used the natural dual metric of $\omega_{h,k}$ on $T^{*}X$ , which contributes to the factor $k^{-1}$ above. Similarly, we get

\left\|\bar{\partial}(\pi_{T}\xi_{\Lambda})\right\|^{2}_{L^{2}(\omega_{h,k})}\leq 2k^{n-1}\left\|\nabla^{0,1}df_{k}(\Lambda;H_{k})\right\|^{2}_{L^{2}(\omega_{h})},

where we wrote $\nabla$ is the covariant derivative of $\omega_{h}$ and $\nabla^{0,1}=\bar{\partial}$ is its $(0,1)$ -part. Thus the estimates above give

\|\Lambda\|^{2}_{\mathrm{HS}(H_{k})}\leq 2C_{3}k^{n}\left(\|\nabla f_{k}(\Lambda;H_{k})\|^{2}_{L^{2}(\omega_{h})}+\|\nabla\nabla f_{k}(\Lambda;H_{k})\|^{2}_{L^{2}(\omega_{h})}\right),

when $\mathrm{tr}(\Lambda)=0$ .

In general, we write $\Lambda=\Lambda_{0}+cI$ where $c:=\mathrm{tr}(\Lambda)/N_{k}$ and $\Lambda_{0}$ is the trace-free part of $\Lambda$ . By diagonalising $\Lambda_{0}$ and recalling the definition of the Hilbert–Schmidt norm, we find

\|\Lambda\|^{2}_{\mathrm{HS}(H_{k})}=\|\Lambda_{0}\|^{2}_{\mathrm{HS}(H_{k})}+c^{2}N_{k}.

We likewise decompose

f_{k}(\Lambda;H_{k})=f_{k}(\Lambda_{0};H_{k})+c\sum_{i=1}^{N_{k}}|s_{i}|^{2}_{\mathrm{FS}(H_{k})}=f_{k}(\Lambda_{0};H_{k})+c

by recalling the definition (1) of $\mathrm{FS}(H_{k})$ . Writing $\bar{\rho}_{k}(\omega_{h})=\frac{V}{N_{k}}\rho_{k}(\omega_{h})$ and recalling $\mathrm{FS}_{k}(H_{k})=\bar{\rho}_{k}(\omega_{h})^{-1}h^{k}$ as in [rawnsley], the above equation gives

\bar{\rho}_{k}(\omega_{h})f_{k}(\Lambda;H_{k})=\sum_{i=1}^{N_{k}}\lambda_{i}|s_{i}|^{2}_{h^{k}}+c\bar{\rho}_{k}(\omega_{h}).

Integrating this over $X$ , noting that $\{s_{i}\}_{i=1}^{N_{k}}$ is a $\mathrm{Hilb}(h^{k})$ -orthonormal basis and that we have $\int_{X}\bar{\rho}_{k}(\omega_{h})\omega^{n}_{h}/n!=V$ , we get

cV=\int_{X}\bar{\rho}_{k}(\omega_{h})f_{k}(\Lambda;H_{k})\frac{\omega_{h}^{n}}{n!}

since $\mathrm{tr}(\Lambda_{0})=0$ , and hence

|c|=\frac{1}{V}\left|\int_{X}\bar{\rho}_{k}(\omega_{h})f_{k}(\Lambda;H_{k})\frac{\omega_{h}^{n}}{n!}\right|\leq\frac{1}{V}\|\bar{\rho}_{k}(\omega_{h})\|_{L^{2}(\omega_{h})}\|f_{k}(\Lambda;H_{k})\|_{L^{2}(\omega_{h})}\leq 2\|f_{k}(\Lambda;H_{k})\|_{L^{2}(\omega_{h})},

by Cauchy–Schwarz and the Bergman kernel expansion (2).

Collecting all the estimates and setting $C_{4}:=2\max\{C_{3},2V\}>0$ , which depends only on $h$ and not on $k$ , we get

	$\displaystyle\\|\Lambda\\|^{2}_{\mathrm{HS}(H_{k})}$	$\displaystyle\leq 2N_{k}\\|f_{k}(\Lambda;H_{k})\\|^{2}_{L^{2}(\omega_{h})}+2C_{3}k^{n}\left(\\|\nabla f_{k}(\Lambda;H_{k})\\|^{2}_{L^{2}(\omega_{h})}+\\|\nabla\nabla f_{k}(\Lambda;H_{k})\\|^{2}_{L^{2}(\omega_{h})}\right)$
		$\displaystyle\leq C_{4}k^{n}\\|f_{k}(\Lambda;H_{k})\\|^{2}_{W^{2,2}(\omega_{h})}$

for all large enough $k$ , since we have $N_{k}=Vk^{n}+O(k^{n-1})$ by the asymptotic Riemann–Roch theorem. ∎

Remark 2.3.

A heuristic interpretation of Theorem 1.2 is as follows. The image of the Kodaira embedding $\iota_{k}:X\hookrightarrow\mathbb{P}(H^{0}(X,L)^{\vee})$ is not contained in any proper linear subspace, and hence we expect that an appropriately defined norm of $\pi_{T}\xi_{\Lambda}$ should be equivalent to $\|\Lambda\|_{\mathrm{HS}(H_{k})}$ . The estimate depends on how “degenerate” $\iota_{k}(X)$ is in the projective space, i.e. how close it is to being contained in a proper linear subspace. The argument above proves this intuition when we take the reference metric to be $\mathrm{FS}_{k}(H_{k})$ , for which the centre of mass of the embedding is in fact close to the identity matrix by the Bergman kernel expansion (2). Since $\iota_{k}$ is an algebraic morphism, we can also expect that the constant depends at most polynomially in $k$ . This argument seems intuitively plausible, but no written proof seems to be available in the literature.

3. Correction to [yhextremal]

There are several places in [yhextremal] which requires corrections, since they depend on the results in [yhhilb] that are reproduced in [yhextremal, Lemmas 5 and 6]. [yhextremal, Lemma 5] is not directly used in the rest of the argument in [yhextremal], and only affects the paper through [yhextremal, Lemma 6]. The full quantitative statement of [yhextremal, Lemma 6] is used only in [yhextremal, page 3004 in §4.2, after (35)], and all the other places [yhextremal, pages 2982 and 3002] that need [yhextremal, Lemma 6] only need the injectivity of the Fubini–Study map which was also proved by Lempert [Lem21].

Thus it suffices to correct the argument in [yhextremal, page 3004], which deals with the equation

\sum_{i=1}^{N_{k}}\left(d_{i}-\left(1-\frac{F_{m,k}}{4\pi k^{m+2}}\right)\right)|s_{i}|^{2}_{h^{k}_{(m)}}=0,

(5)

where $d_{1},\dots,d_{N_{k}}\in\mathbb{R}$ , $m\in\mathbb{N}$ , $h^{k}_{(m)}=\mathrm{FS}_{k}(H_{k,m})$ , $H_{k,m}\in\mathcal{B}_{k}$ , is a hermitian metric on $L^{k}$ as constructed in [yhextremal, Proposition 1 and Proof of Corollary 1], and $\{s_{i}\}_{i=1}^{N_{k}}$ is an $H_{k,m}$ -orthonormal basis. We need to conclude that there exists $C_{5}>0$ such that

|d_{i}-1|\leq C_{5}k^{c(n)-m}

for all $i=1,\dots,N_{k}$ , and for all large enough $k$ , where $c(n)$ is some constant which depends only on $n$ .

We apply the argument above. We first define two hermitian matrices $A:=\mathrm{diag}(d^{-1}_{1},\dots,d^{-1}_{N_{k}})$ and $B:=I$ with respect to the $H_{k,m}$ -orthonormal basis $\{s_{i}\}_{i=1}^{N_{k}}$ . Note that they take a different form when they are represented with respect to an $H_{k}$ -orthonormal basis, where $H_{k}=\mathrm{Hilb}(h^{k})$ and $\omega_{h}$ is the extremal metric that is used as a reference metric in [yhextremal]. We find

f_{k}(A,B;H_{k,m})=\sum_{i=1}^{N_{k}}\left(d_{i}-1\right)|s_{i}|^{2}_{h^{k}_{(m)}},

and hence (5) can be re-written as

f_{k}(A,B;H_{k,m})=-\frac{F_{m,k}}{4\pi k^{m+2}}.

Taking $H_{k}=\mathrm{Hilb}(h^{k})$ as above, we observe

f_{k}(A,B;H_{k})=\frac{\mathrm{FS}_{k}(H_{k})}{\mathrm{FS}_{k}(H_{k,m})}\left(\frac{\mathrm{FS}_{k}(H_{k,m})}{\mathrm{FS}_{k}(A)}-\frac{\mathrm{FS}_{k}(H_{k,m})}{\mathrm{FS}_{k}(B)}\right),

which implies that

f_{k}(A,B;H_{k})=\frac{\mathrm{FS}_{k}(H_{k})}{\mathrm{FS}_{k}(H_{k,m})}f_{k}(A,B;H_{k,m})=-\frac{\mathrm{FS}_{k}(H_{k})}{\mathrm{FS}_{k}(H_{k,m})}\frac{F_{m,k}}{4\pi k^{m+2}}.

(6)

We recall $\mathrm{FS}_{k}(H_{k})=\bar{\rho}_{k}(\omega_{h})^{-1}h^{k}$ [rawnsley], and

\mathrm{FS}_{k}(H_{k,m})=e^{k\phi_{(m)}}h^{k}

in the terminology of [yhextremal, Proposition 1]; the only important point here is that $k\phi_{(m)}$ converges in $C^{\infty}$ as $k\to\infty$ , and hence its derivatives are all uniformly bounded for all large enough $k$ . Together with the expansion (2) of the Bergman function, we thus find that all derivatives of $\mathrm{FS}_{k}(H_{k})/\mathrm{FS}_{k}(H_{k,m})$ are bounded uniformly for all large enough $k$ , and hence there exists $C_{6}>0$ such that

\left\|\frac{\mathrm{FS}_{k}(H_{k})}{\mathrm{FS}_{k}(H_{k,m})}\frac{F_{m,k}}{4\pi k^{m+2}}\right\|_{W^{2,2}(\omega_{h})}\leq C_{6}k^{-m-2}\|F_{m,k}\|_{W^{2,2}(\omega_{h})}

holds for all large enough $k$ . As pointed out in [yhextremal, page 2995, before the equation (20)], $\|F_{m,k}\|_{W^{2,2}(\omega_{h})}$ is uniformly bounded for all large enough $k$ . Thus there exists a constant $C_{7}>0$ such that

\|f_{k}(A,B;H_{k})\|_{W^{2,2}(\omega_{h})}\leq C_{7}k^{-m-2}.

(7)

We now recall that $A^{-1}=\mathrm{diag}(d_{1},\dots,d_{N_{k}})$ and $B^{-1}=I$ with respect to the $H_{k,m}$ -orthonormal basis $\{s_{i}\}_{i=1}^{N_{k}}$ . When we represent $H_{k}$ as a matrix with respect to an $H_{k,m}$ -orthonormal basis, we find that each entry of $H_{k}$ can be bounded uniformly for all large enough $k$ by the construction of $H_{k,m}$ as given in [yhextremal, Proof of Corollary 1] (see also [donnum, Appendix]), since it is defined as a $\mathrm{Hilb}(\tilde{h}^{k})$ for some perturbation $\tilde{h}$ of $h$ such that $\log(\tilde{h}/h)=O(1/k)$ . Thus there exists a constant $C_{8}>0$ such that

\|A^{-1}-B^{-1}\|_{\mathrm{HS}(H_{m,k})}\leq C_{8}N_{k}\|A^{-1}-B^{-1}\|_{\mathrm{HS}(H_{k})},

holds for all large enough $k$ . By recalling the asymptotic Riemann–Roch theorem $N_{k}=Vk^{n}+O(k^{n-1})$ , we find that there exists a constant $C_{9}>0$ such that

	$\displaystyle\|d_{i}-1\|$	$\displaystyle\leq\\|A^{-1}-B^{-1}\\|_{\mathrm{HS}(H_{m,k})}$
		$\displaystyle\leq C_{9}k^{n}\\|A^{-1}-B^{-1}\\|_{\mathrm{HS}(H_{k})}$
		$\displaystyle\leq C_{9}C_{h}k^{2n}\\|f_{k}(A,B;H_{k})\\|_{W^{2,2}(\omega_{h})}$
		$\displaystyle\leq C_{7}C_{9}C_{h}k^{2n-m-2}$

for all $i=1,\dots,N_{k}$ and for all large enough $k$ , by Theorem 1.2 and (7), as required.

	$\displaystyle\|d_{i}-1\|$	$\displaystyle\leq\\|A^{-1}-B^{-1}\\|_{\mathrm{HS}(H_{m,k})}$
		$\displaystyle\leq C_{9}k^{n}\\|A^{-1}-B^{-1}\\|_{\mathrm{HS}(H_{k})}$
		$\displaystyle\leq C_{9}C_{h}k^{2n}\\|f_{k}(A,B;H_{k})\\|_{W^{2,2}(\omega_{h})}$
		$\displaystyle\leq C_{7}C_{9}C_{h}k^{2n-m-2}$

Quantitative injectivity of the Fubini–Study map

Abstract.

1. Introduction

Problem 1.1.

Theorem 1.2.

2. Proof of Theorem 1.2

Lemma 2.1.

Proof.

Lemma 2.2.

Proof.

Proof of Theorem 1.2.

Remark 2.3.

3. Correction to [yhextremal]

References