K\"ahler packings of projective complex manifolds

TL;DR

This paper establishes a relationship between the multipoint Seshadri constant and the maximum radii of K"ahler ball embeddings in projective complex manifolds, linking algebraic and symplectic geometry concepts.

Contribution

It demonstrates a bidirectional correspondence between Seshadri constants and K"ahler ball embedding radii, providing new insights into geometric embedding constraints.

Findings

Seshadri constants determine K"ahler ball embedding sizes

Maximum radii of embeddings are characterized by algebraic invariants

The work bridges algebraic and symplectic geometry concepts

Abstract

In this note we show that the multipoint Seshadri constant determines the maximum possible radii of embeddings of K\"ahler balls and vice versa.

Equations

$$d\ge \frac{1}{\sqrt{k}}\sum _{i=1}^k{m}_i.$$

$$ϵ({\mathbb{P}}^2,{\mathcal{O}}_{{\mathbb{P}}^2}(1),P_1,\dots ,P_k)=\frac{1}{\sqrt{k}}.$$

$$\varphi =\coprod _{q=1}^k{\varphi }_q:\coprod _{q=1}^k{B}_0(r_q)\to X$$

$${\gamma }_k(X,\omega ;P_1,\dots ,P_n)=\sqrt{ϵ(X,L;P_1,\dots ,P_n)}.$$

$$\tilde{L}:={\pi }^{\ast }L-\sum _{i=1}^k{ϵ}_i{E}_i\text{ for }{ϵ}_i\in {\mathbb{Q}}_{>0}.$$

$$\max \{ϵ\ge 0|{\pi }^{\ast }L-\sum {ϵ}_i{E}_i\text{ is nef }\}.$$

$$\frac{({\pi }^{\ast }L-{ϵ}^{\prime }E)\cdot \tilde{C}}{{\mathrm{mult}}_{\tilde{Q}}\tilde{C}}=\frac{L\cdot C}{{\mathrm{mult}}_{Q}C}-{ϵ}^{\prime }\frac{{\mathrm{mult}}_{P}C}{{\mathrm{mult}}_{Q}C}\ge \frac{{ϵ}_L}{2}\text{ if }\frac{{\mathrm{mult}}_{P}C}{{\mathrm{mult}}_{Q}C}<\frac{{ϵ}_L}{2{ϵ}^{\prime }}.$$

$$\frac{({\pi }^{\ast }L-{ϵ}^{\prime }E)\cdot \tilde{C}}{{\mathrm{mult}}_{\tilde{Q}}\tilde{C}}=\frac{({\pi }^{\ast }L-{ϵ}_{P}E)\cdot \tilde{C}}{{\mathrm{mult}}_{\tilde{Q}}\tilde{C}}+({ϵ}_P-{ϵ}^{\prime })\frac{E\cdot \tilde{C}}{{\mathrm{mult}}_{\tilde{Q}}\tilde{C}}\ge ({ϵ}_P-{ϵ}^{\prime })\frac{{\mathrm{mult}}_{P}C}{{\mathrm{mult}}_{Q}C}\ge ({ϵ}_P-{ϵ}^{\prime })\frac{{ϵ}_L}{2{ϵ}^{\prime }}.$$

$$\frac{({\pi }^{\ast }L-{ϵ}^{\prime }E)\cdot \tilde{C}}{{\mathrm{mult}}_{\tilde{Q}}\tilde{C}}\ge \frac{({\pi }^{\ast }L-{ϵ}^{\prime }E)\cdot \tilde{C}}{{\mathrm{mult}}_P\pi (\tilde{C})}\ge {ϵ}_P-{ϵ}^{\prime }>0.$$

$$\frac{({\pi }^{\ast }L-{ϵ}^{\prime }E)\cdot \tilde{C}}{{\mathrm{mult}}_{\tilde{Q}}\tilde{C}}\ge -\frac{{ϵ}^{\prime }E\cdot \tilde{C}}{{\mathrm{mult}}_{\tilde{Q}}\tilde{C}}\ge 1.$$

$$\frac{({\pi }^{\ast }L-{\sum }_{i=1}^{k-1}{ϵ}_i{E}_1)\cdot C}{{\mathrm{mult}}_{Q}C}$$

$$=\frac{({\pi }^{\ast }L-{\sum }_{i=1}^k{ϵ}_i{E}_1)}{{\mathrm{mult}}_{Q}C}+\frac{{ϵ}_i{E}_i\bar{C}}{{\mathrm{mult}}_{Q}C}$$

$$=\frac{({\pi }^{\ast }L-{\sum }_{i=1}^k{ϵ}_i{E}_1)}{{\mathrm{mult}}_{Q}C}+{ϵ}_i\cdot \frac{{\mathrm{mult}}_{P_k}C}{{\mathrm{mult}}_{Q}C}$$

$$\ge {ϵ}_{{\pi }^{\ast }L-{\sum }_{i=1}^k{ϵ}_i{E}_i}.$$

$$ϵ(X,L;P_1,\dots ,P_k)=\text{ sup}\{ϵ\in {\mathbb{Q}}_{>0}:{\pi }^{\ast }L-\sum _{i=1}^k{ϵ}_i{E}_i\text{ is }\mathbb{Q}\text{-ample}\}.$$

$$l_i:=\{(0,\dots ,0,t\cdot {\delta }_i,{\delta }_{i+1},\dots ,{\delta }_k):t\in \mathbb{R}\}$$

$$(q\cdot \Pi {)}^{-1}(t^{(i)})=B{l}_{P_1,\dots ,P_{i-1}}(X)\cup \bigcup _{j=1}^{i-1}{\mathcal{E}}_j\cap (q\cdot \Pi {)}^{-1}(t^{(j)}).$$

$$(q\cdot\Pi)^{-1}(l_{i})=\Big{(}\bigcup_{j=1}^{i-1}\mathcal{E}_{j}\cap(q\cdot\Pi)^{-1}(l_{i})\Big{)}\cup\tilde{\mathcal{X}}_{i}.$$

$${\tilde{\mathcal{L}}}_{d,m_1,\dots ,m_i}^{(i)}:=d{\Pi }^{(i)\ast }{p}_{i-1}^{\ast }{\tilde{L}}_{d;m_1,\dots ,m_i}^{i-1}-m_i{\mathcal{E}}_i^{(i)}$$

$${\sigma }_{0,0_i}^{(i)}|,\dots ,{\sigma }_{N_i,0_i}^{(i)}|$$

$$0\to \bigcap _{i=1}^k{\mathfrak{m}}_{X,P_i}^{m_i+1}\otimes {\mathcal{O}}_X(dL)\to {\mathcal{O}}_X(dL)\to \underset{i=1}{\overset{k}{⨁}}\text{nicefrac}{\mathcal{O}}_X{\mathfrak{m}}_{X_i,P_i}^{m_i+1}\to 0.$$

$$H^1(X,\bigcap _{i=1}^k{\mathfrak{m}}_{X,P_i}^{m_i+1}\otimes {\mathcal{O}}_X(dL))=H^1(\tilde{X},{\mathcal{O}}_X({\tilde{L}}_{d,m_i})).$$

$$B^{(i)}:=B_0\cup \bigcup _{j=1}^i{\tilde{B}}_j\cup \bigcup _{j=i+1}^k{B}_j.$$

$$m_i\cdot {\omega }_{FS}$$

$$=\frac{i}{2\pi }\partial \bar{\partial }\log (Y_1{\bar{Y}}_1+\dots +Y_n{\bar{Y}}_n+T\bar{T}{)}^{m_i}$$

$$=\frac{i}{2\pi }\partial \bar{\partial }\log (Y_1^{m_i}{\overline{Y_1}}^{m_i}+m_i{Y}_1^{m_i-1}{\bar{Y}}_1^{m_i-1}{Y}_2{\bar{Y}}_2\dots )$$

$$=m_i\cdot {\omega }_{FS}=\frac{i}{2\pi }\partial \bar{\partial }\log (\sum _{\alpha +\beta =m_i}{c}_{\alpha ,\beta }{Y}^{\alpha }{\bar{Y}}^{\alpha }{T}^{\beta }{\bar{T}}^{\beta }),\text{ where }{c}_{a,b}\text{ are positive integers. }$$

Full text

1. Introduction.

Seshadri constants were first introduced by Demailly in [Dem92] as a way of studying local positivity of ample line bundles at a given point of a variety. Since then, Seshadri constants have attracted substantial attention in the field of algebraic geometry and can be used to reformulate many classical ideas. A nice example of this is the famous conjecture of Nagata on plane algebraic curves, which in its original form is as follows:

Conjecture 1**.**

Let $P_{1},\ldots,P_{k}$ be points of $\mathbb{P}^{2}$ in general position and $m_{1},\ldots,m_{k}$ be positive integers. Then for $k\geq 9$ any curve $C\in\mathbb{P}^{2}$ of degree $d$ passing each point $P_{i}$ with multiplicity $m_{i}$ must satisfy

[TABLE]

Using multipoint Seshadri constants this can be rewritten as:

Conjecture 2**.**

Let $P_{1},\ldots,P_{k}$ be points of $\mathbb{P}^{2}$ in general position then for $k\geq 9$ the multipoint Seshadri constant

[TABLE]

Motivated by work of Biran [Bir01], who proved a symplectic analagon between Nagata’s conjecture formulated as a packing problem, Eckl proved in [Eck17] that Nagata’s conjecture is in fact equivalent to a more restricted packing problem, namely a Kähler packing problem. More generally in dimension 2 there is a direct correspondence between sizes of multi-ball Kähler packings and multipoint Seshadri constants. A similar result was obtained by David Witt-Nystrom in [WN] but this time for a variety of any dimension blown up at a single point. The aim of this note is to generalise these results to projective complex manifolds of arbitrary dimension blown up at any number of points (in general position or not). During the writing of this paper a similar result was achived by Trusiani in [Tru18] however there are some differences in the formulation of the statement and its proof. In more details:

Definition 3**.**

Let $(X,\omega)$ be an $n$ dimensional Kähler manifold with Kähler form $\omega$. Then a holomorphic embedding

[TABLE]

is called a Kähler embedding of $k$ disjoint complex flat balls in $\mathbb{C}^{n}$ centered in [math], of radius $r_{q}$, if there exists a Kähler form $\omega^{\prime}$ such that $[\omega^{\prime}]=c_{1}(L)$ and $\phi_{q}^{*}(\omega^{\prime})=\omega_{std}$ is the standard Kähler form on $\mathbb{C}^{n}$ restricted to $B_{0}(r_{q}).$ Let $\gamma_{k}(X,\omega;P_{1},\ldots,P_{n})=\sup\{r>0\colon$ That there exists a Kähler packing with $\phi_{q}(0)=P_{q}\}$. We call $\gamma_{k}$ the $k-$ball packing constant.

Since curvature is an invariant under open holomorphic embeddings we cannot define the packing condition $\phi^{*}_{q}(\omega)=\omega_{std}$ using the original Kähler form $\omega$ on $X$ as long as that form is not flat enough around $q$. However the following theorem shows that under suitable conditions it is possible to find a Kähler form flat enough around $q$ in the cohomology class of $\omega$ as requested in the definition.

Theorem 4**.**

With the notation as above we have that the square of the $k-$ball packing constant is equal to the multipoint Seshadri constant:

[TABLE]

2. Multipoint Seshadri constants.

In this section we start by recalling some facts and definitions related to Seshadri constants. Single point Seshadri constants have been quite well studied and a good introduction to them can be found in Chapter 5 of [Laz04] and [BRH*+*08]. As we are interested in multipoint Seshadri constants we will introduce the notation and some ideas needed later.

Let $X$ be a complex projective manifold of dimension $n$, with $P_{1},\ldots,P_{k}$ distinct points of $X$. Let $L$ be an ample divisor and $\tilde{X}\coloneqq Bl_{P_{1},\dots,P_{k}}(X)\xrightarrow{\pi}X$ be the blow-up of $X$ at the points $P_{1},\ldots,P_{k}$. Let $\pi^{-1}(P_{i})=E_{i}$ denote the exceptional divisor corresponding to the point $P_{i}$ for all $i=1,\ldots,k$ and for $\epsilon_{i}\in\mathbb{Q}_{>0}$ set

[TABLE]

If $L$ is a nef $\mathbb{Q}$-Cartier divisor then [Laz04] defines the multipoint Seshadri constant to be

[TABLE]

Since we require $L$ ample to achieve the desired Kähler packing we show that there is an equivalent definition where we replace $\max$ with $\sup$ and $\geq 0$ with $>0$. The following lemma’s prove that the two definitions are equivalent filling a gap in the literature.

Lemma 5**.**

$\pi^{*}L-\epsilon^{\prime}E$* is ample on $\tilde{X}\xrightarrow{\pi}X$ if $0<\epsilon^{\prime}<\epsilon(X,L;P)$.*

Proof.

By Seshadri’s Criterion [Laz04, Thm 1.4.13] if $L$ is ample then there exists $\epsilon_{L}>0$ such that for all points $Q\in X$ and all irreducible curves $C$ containg $Q$ we have that $\frac{L\cdot C}{\operatorname{mult}_{Q}C}>\epsilon_{L}$. Let $\tilde{Q}\in\tilde{X}$ such that $\pi(\tilde{Q})=Q$ and let $\tilde{C}\subset\tilde{X}$ be an irreducible curve with $\tilde{Q}\in\tilde{C}$. Then there are the following cases. If $Q\not\in E$ then

[TABLE]

On the other hand if $\frac{\operatorname{mult}_{P}C}{\operatorname{mult}_{Q}C}\geq\frac{\epsilon_{L}}{2\epsilon^{\prime}}$ then

[TABLE]

If $\tilde{Q}\in E$ with $\tilde{C}\not\in E$ then $\operatorname{mult}_{\tilde{Q}}\tilde{C}\leq\operatorname{mult}_{P}\pi(\tilde{C})$ and

[TABLE]

Finally if $\tilde{Q}\in E$ and $\tilde{C}\subset E$ then

[TABLE]

The final inequality appears since $E\cdot\tilde{C}=\deg\tilde{C}$ on $E\cong\mathbb{P}^{n-1}$ and $\deg\tilde{C}\geq\operatorname{mult}_{\tilde{Q}}\tilde{C}$. Hence we have shown that if $0<\epsilon^{\prime}<\epsilon_{P}$ then $\pi^{*}L-\epsilon^{\prime}E$ is $\mathbb{Q}$-ample.

∎

Corollary 6**.**

$\max\{\epsilon\geq 0:\pi^{*}L-\epsilon E$* is nef $\}=\sup\{\epsilon>0:\pi^{*}L-\epsilon E$ is $\mathbb{Q}$-ample$\}$.∎*

Lemma 7**.**

There exists $\epsilon_{1},\ldots,\epsilon_{k}>0$ such that $\tilde{L}$ is $\mathbb{Q}$-ample for all $i=1,\ldots,k$.

Proof.

By Seshadri’s criterion and Lemma 5 we know there exist some $\epsilon_{1}>0$ such that $\pi^{*}L-\epsilon_{1}E$ is $\mathbb{Q}$-ample on $\tilde{X}_{1}=Bl_{P_{1}}(X)$. Choosing a second point $P_{2}\in X$ Seshadri’s criterion again ensures the existence of some $\epsilon_{2}>0$ such that $\pi^{*}L-\epsilon_{1}E_{1}-\epsilon_{2}E_{2}$ is $\mathbb{Q}$-ample on $\tilde{X}_{2}=Bl_{P_{1},P_{2}}(X)$. Arguing iteratively for $k$ points of $X$ we obtain the claim of the lemma. ∎

From the statement of the above lemma it is not obvious that we can chose all the $\epsilon_{i}$ to be equal. This follows from the next lemma.

Lemma 8**.**

If $\pi^{*}L-\sum_{i=1}^{k}\epsilon_{i}E_{i}$ is $\mathbb{Q}$-ample then $\pi^{*}L-\sum_{i=1}^{k}\epsilon^{\prime}_{i}E_{i}$ is ample if $0<\epsilon^{\prime}_{i}\leq\epsilon_{i}$, for all $i=1,\ldots,k$.

Proof.

Lemma 5 proves the case when $k=1$. If we show that $\pi^{*}L-\sum_{i=1}^{k}\epsilon_{i}E_{1}$ ample implies $\pi_{k-1}^{*}L-\sum_{i=1}^{k-1}\epsilon_{i}E_{1}$ is ample, where $\pi_{k-1}\colon\tilde{X}\to X$ denotes the blow up of $X$ at the first $k-1$ points, then we can simply apply Lemma 5 on the last blow up. Let $C\subset X$ be an irreducible curve and $\bar{C}\subset\tilde{X}$ be the strict transform of $C$. Then

[TABLE]

Thus Seshadri’s Criterion implies the ampleness of $\pi_{k-1}^{*}L-\sum_{i=1}^{k-1}\epsilon_{i}E_{i}$.

∎

Definition 9**.**

For $X$ and $L$ defined as above the multipoint Seshadri constant associated to points $P_{1},\ldots P_{k}$ of X is

[TABLE]

The multipoint Seshadri constant associated to an ample divisor is always $>0$ by Lemma 8.

3. Degeneration of complex projective manifolds to multipoint blow up.

First we fix some general notation and introduce some constructions that will be used later. The notation fixed in this section will be used for the remainder of the report unless otherwise stated.

Let $X$ and $L$ be defined as above and consider the product manifold

Then if $t_{1},\ldots,t_{k}$ represent coordinates of $\mathbb{A}_{\mathbb{C}}^{k}$ and set $Z_{i}\coloneqq\{P_{i}\}\times\{t_{i}=0\}$. These are exactly the coordinate hyperplanes cut out by $t_{i}=0$ over the points $P_{i}$.

Remark 10**.**

For all $1\leq i<j\leq k$ we have $Z_{i}\cap Z_{j}=\emptyset$ since the points $P_{i}\in X$ are distinct.

By blowing up the family $\mathcal{X}$ over the union of all the centres $Z_{i}$ we obtain a new algebraic family over $\mathbb{A}^{k}_{\mathbb{C}}$.

such that the exceptional divisor corresponding to the centre $Z_{i}$ is $\mathcal{E}_{i}\coloneqq\Pi^{-1}(Z_{i})$. Set $\tilde{\mathcal{L}}_{dm_{i},\ldots,m_{k}}=d\Pi^{*}p^{*}_{1}L-\sum_{i=1}^{k}m_{i}\mathcal{E}_{i}$ a divisor on $\tilde{\mathcal{X}}$.

Remark 11**.**

1. (1)

$\tilde{\mathcal{X}}$ is a well defined complex manifold projective over $\mathbb{A}_{\mathbb{C}}^{k}$. 2. (2)

$\mathcal{E}_{i}\cong Z_{i}\times\mathbb{P}_{\mathbb{C}}^{n}$ 3. (3)

(q\circ\Pi)^{-1}\big{(}(t_{1},\ldots,t_{k})\big{)} is the blowup of $X$ in the points $P_{i}$ where $t_{i}=0$.

Running through an appropriate path through the parameter space $\mathbb{A}_{\mathbb{C}}^{k}$ these blow ups accumulate iteratively. Let $\delta_{1},\ldots,\delta_{k}$ be positive real numbers and define lines in $\mathbb{A}_{\mathbb{C}}^{k}$,

[TABLE]

consisting of points whose first $(i-1)$ coordinates are zero, the $i$-th coordinate is of the form $t\cdot\delta_{i}$ and the remaining $(i+1)$ coordinates are constant. For the point $t^{(i)}=(0,\ldots,0,\delta_{i},\delta_{i+1},\ldots,\delta_{k})\in l_{i}$ for $i=1,\ldots,k$, the preimage is

[TABLE]

Following the line from $l_{i}$ from $t^{(i)}$ to $t^{(i+1)}$ we trace a path through the parameter space such that the preimage of $t^{(1)}$ is simply $X$, the preimage of $t^{(1)}$ is simply the blow up of $X$ at $P_{1}$ with some contributions from the exceptional divsior and so on until we get that the preimage of $t^{(k)}=(0,\ldots,0)$ is the blow up of $X$ at $P_{k}$ with contributions from all exceptional divisors. See Figure 1 for a visualisation.

Similarly for $l_{i}\cong\mathbb{A}_{\mathbb{C}}^{1}$ we have

[TABLE]

Where $\mathcal{E}^{(i)}=\mathbb{P}^{n}_{\mathbb{C}}$ and $\tilde{\mathcal{X}}_{i}$ is the blow up of $\tilde{X}_{i-1}\times l_{i}$ in the point $(P_{i},0)$.

On $\tilde{\mathcal{X}}_{i}$ there exists a family of divisors

[TABLE]

where $\mathcal{E}_{i}^{(i)}\cong\mathbb{P}^{n}_{\mathbb{C}}$ is the exceptional divisor of the blow up $\Pi_{i}$. Specific divisors associated to a particular fiber are denoted $\mathcal{L}^{(i)}_{d,\underline{m},t}$ , where $t$ denotes the parameter on $\mathbb{A}^{1}_{\mathbb{C}}$ and $d,\underline{m}_{i}=(m_{1},\ldots,m_{i})$ record the degree and multiplicity (we will often just write $\tilde{\mathcal{L}}^{(i)}_{t}$ for short if the multiplicity and degree are fixed).

Remark 12**.**

$\tilde{\mathcal{X}}_{i,0}=\tilde{X}_{i}\cup\mathcal{E}_{i}^{(i)}$, where $\mathcal{E}_{i}^{(i)}\cong\mathbb{P}_{\mathbb{C}}^{n}$ and $\tilde{\mathcal{X}}_{i,t}=\Pi^{(i)\ -1}(t)$ is the fiber over $l_{i}\cong\mathbb{A}^{1}_{\mathbb{C}}$. Hence there exists positive integers $d,m_{j}$ such that $\tilde{L}^{(i)}=\tilde{L}^{(i)}_{d;\underline{m}_{i}}\coloneqq d\pi^{*}L-\sum_{j=1}^{i}m_{j}E_{j}$ is an ample divisor on $\tilde{X}_{i}$ (and very ample for $d,m_{j}\gg 0$), if $\frac{m_{j}}{d}=\epsilon_{j}$ for all $\epsilon_{j}$ as in Lemma 7.

Now that we have defined a algebraic family $\tilde{\mathcal{X}}_{i}$ and a family of divisors $\tilde{\mathcal{L}}^{(i)}=\tilde{\mathcal{L}}^{(i)}_{d;\underline{m}_{i}}$ we would like to investigate the global sections of these divisors restricted to the fibers of $\Pi^{(i)}$, and how these global sections are constructed for different $i$. This is shown by the following Theorem:

Theorem 13**.**

For all $i=1,\ldots,k$ there exists global sections $\sigma_{0}^{(i)},\ldots,\sigma_{N_{i}}^{(i)}$ of $\Pi^{(i)}_{*}\tilde{\mathcal{L}}^{(i)}$ such that:

1. (1)

$\sigma_{0}^{(i-1)},\ldots,\sigma_{N_{i}}^{(i-1)}$* trivialise $\Pi^{(i)}_{*}\tilde{\mathcal{L}}^{(i)}$.* 2. (2)

*If $0_{i}$ denotes the zero of the line $l_{i}$ then the sections *

[TABLE]

generate the Kähler form $m_{i}\cdot\omega_{FS}$ on $\mathcal{E}_{i}^{(i)}\cong\mathbb{P}_{\mathbb{C}}^{n}$. 3. (3)

The restricted sections $\sigma_{0,\delta_{i}}^{(i)},\ldots,\sigma_{N_{i},\delta_{i}}^{(i)}$ generate the same Kähler form on $\tilde{\mathcal{X}}_{i,\delta_{i}}\cong\tilde{X}_{i-1}$ as;

\sigma^{(i-1)}_{0,0_{i-1}}\raisebox{-2.15277pt}{|}_{\tilde{X}_{i-1}},\ldots,\sigma^{(i-1)}_{N_{i-1},0_{i-1}}\raisebox{-2.15277pt}{|}_{\tilde{X}_{i-1}}* from the previous family $\tilde{\mathcal{X}}_{i-1}$.*

Proof.

If we show that the dimension of the space of global sections of $\tilde{\mathcal{L}}^{(i)}$ restricted to each fiber of $\Pi^{(i)}$ is the same, Grauert’s semi-continuity theorem [Har77, III.12] implies immediately that $\Pi_{*}^{(i)}\tilde{\mathcal{L}}^{(i)}$ is locally free, hence it is free by the Quillen-Suslin Theorem. Thus there exists trivialising global sections of $\Pi^{*}\tilde{\mathcal{L}}^{(i)}$. But instead of using these powerful theorems we describe the trivialising sections directly because they allow us to prove the properties of the theorem more easily. For the sections to be trivialsing it is enough to show that they form a basis of $H^{0}(\tilde{\mathcal{X}}_{i,t},\tilde{\mathcal{L}}^{(i)}_{t})$ then restricted to $\tilde{\mathcal{X}}_{i,t}$ for all $t\in\mathbb{A}^{1}_{\mathbb{C}}$.

We start by calculating a basis of sections of $H^{0}(X,\mathcal{O}_{X}(dL))$ characterised by their vanishing behaviour at the points $P_{1},\ldots,P_{k}$ so that subsets of this basis can be interpreted as generating sections of $H^{0}(\tilde{X},\tilde{L}^{(i)})$. Let $\mathfrak{m}_{X,P_{i}}$ denote the maximal ideal of $X$ at $P_{i}$ and consider the short exact sequence

[TABLE]

Taking the long exact sequence of cohomology w.r.t to the above sequence gives

By Serre Vanishing $H^{1}(X,\bigcap^{k}_{i=1}\mathfrak{m}_{X,P_{i}}^{m_{i}+1}\otimes\mathcal{O}_{X}(dL))=0$, since applying the projection formula yields

[TABLE]

and we assume $\tilde{L}_{d,m_{i}}$ is ample and $d,\underline{m}_{k}\gg 0$. Hence $\phi\colon H^{0}(X,\mathcal{O}_{X}(dL))\to\bigoplus_{i=1}^{k}\nicefrac{{\mathcal{O}_{X}}}{{\mathfrak{m}_{X,P_{i}}^{m_{i}+1}}}$ is surjective. This gives a basis of sections of $H^{0}(X,\mathcal{O}_{X}(dL))$ which is the union of:

• •

A basis $B_{0}$ of $H^{0}(X,\bigcap_{i=1}^{k}\mathfrak{m}^{m_{i}+1}\otimes\mathcal{O}_{X}(dL))$, which are sections of $\mathcal{O}_{X}(dL)$ vanishing to multiplicity at least $m_{i}+1$ in $P_{i}$ for $i=1,\ldots,k$.

• •

A set $B_{i}$ of sections of $\mathcal{O}_{X}(dL)$ mapped to [math] in $\nicefrac{{\mathcal{O}_{X}}}{{\mathfrak{m}_{X,P_{j}}^{m_{j}+1}}}$ for $j\neq i$ and to a basis of homogenous polynomials of degree $\leq m_{i}$ in $\nicefrac{{\mathcal{O}_{X}}}{{\mathfrak{m}_{X,P_{i}}^{m_{i}+1}}}$, in variables given by local coordinates around $P_{i}$.

• •

A set of sections $\tilde{B}_{i}$ of $\mathcal{O}_{X}(dL)$ as above, but with homogenous polynomials of degree $=m_{i}$, and that $\tilde{B}_{i}\subset B_{i}$.

Hence a basis of $H^{0}(\tilde{X}_{i},\mathcal{O}_{\tilde{X}_{i}}(\tilde{L}^{(i)}))$ is given by

[TABLE]

To change between a basis of sections on $X_{i-1}$ and on $X_{i}$ we simply skip those sections which vanish to order less than $m_{i}$ at $P_{i}$.

Since $\tilde{\mathcal{X}}_{i}=\tilde{X}_{i-1}\times\mathbb{A}_{\mathbb{C}}^{1}$ we can use these sections to construct the sections of $H^{0}(\tilde{\mathcal{X}}_{i},\tilde{\mathcal{L}}^{(i)})$ using the following procedure:

1. Step 1:

Pull back a section $s\in H^{0}(\tilde{X}_{i-1},\mathcal{O}_{X_{i-1}}(\tilde{L}^{(i-1)}))$ along $p_{i-1}$ to get a section of $p_{i-1}^{*}\tilde{L}^{(i-1)}=\mathcal{L}^{(i-1)}$ on $\mathcal{X}_{i}=\tilde{X}_{i-1}\times\mathbb{A}_{\mathbb{C}}^{1}$. 2. Step 2:

If mult${}_{P_{i}}s<m_{i}$ then the section $t^{m_{i}-\mbox{ mult }_{P_{i}}s}p_{i-1}^{*}s\in H^{0}(\tilde{\mathcal{X}}_{i},\tilde{\mathcal{L}}^{(i-1)})$ has multiplicity $m_{i}$ in $(P_{i},0)$. If mult${}_{P_{i}}s\geq m_{i}$ then mult${}_{(P_{i},0)}p_{i-1}^{*}s\geq m_{i}$. 3. Step 3:

In both cases, we can subtract $m_{i}$ copies of the exceptional divisor $\mathcal{E}_{i}$ from the pullback of this section along $\Pi_{i}$, thus obtaining sections of $H^{0}(\tilde{\mathcal{X}}_{i},\tilde{\mathcal{L}}^{(i)})$.

When starting with the basis sections in $B^{(i-1)}$ , restricting to the general fibers $\tilde{\mathcal{X}}_{i,t}\cong\tilde{X}_{i-1}$ of $\Pi^{(i)}$ (that is $t\neq 0$) we get back the sections in $B^{(i-1)}$ possibly multiplied with some power of $t$. Thus all the sections $\sigma_{1}^{(i-1)},\ldots\sigma_{N_{i}}^{(i-1)}$ obtained from $B^{(i-1)}$ in the way described above are trivializing $\Pi^{(i)}_{*}\tilde{\mathcal{L}}^{(i)}$ outside the central fiber. To understand the restriction of these sections to the central fiber we need to analyse \tilde{\mathcal{L}}^{(i)}\raisebox{-2.15277pt}{|}_{\tilde{\mathcal{X}}_{i,0}} and its global sections. Note, $\tilde{\mathcal{X}}_{i,0}=\tilde{X}_{i}\cup\mathcal{E}_{i}^{(i)}$ and $\tilde{X}_{i}\cap\mathcal{E}_{i}^{(i)}=E_{i}$. Furthermore \tilde{\mathcal{L}}^{(i)}\raisebox{-2.15277pt}{|}_{\tilde{X}_{i}}=\tilde{L}^{(i)}, and \tilde{\mathcal{L}}^{(i)}\raisebox{-2.15277pt}{|}_{\mathcal{E}_{i}^{(i)}}=\mathcal{O}_{\mathcal{E}_{i}^{(i)}}(-m_{i}\mathcal{E}_{i}^{(i)})\cong\mathcal{O}_{\mathbb{P}^{n}}(m_{i}) via the isomorphism $\mathcal{E}_{i}^{(i)}\cong\mathbb{P}^{n}_{\mathbb{C}}$. This means that global sections of \tilde{\mathcal{L}}^{(i)}\raisebox{-2.15277pt}{|}_{\tilde{\mathcal{X}}_{i,0}} consist of a section of \mathcal{L}^{(i)}\raisebox{-2.15277pt}{|}_{\tilde{X}_{i}} and a section of \mathcal{L}^{(i)}\raisebox{-2.15277pt}{|}_{\mathcal{E}_{i}^{(i)}} coinciding when restricted to $E_{i}$.

• •

Global sections of \tilde{\mathcal{L}}^{(i)}\raisebox{-2.15277pt}{|}_{\tilde{X}_{i}} and their restriction to $E_{i}$:

These are all sections of $dL$ on $X$ which vanish to multiplicity $\geq m_{j}$ in $P_{j}$ for $j=1,\ldots,i.$ Equivalently these are global sections of $\tilde{L}^{(i-1)}$ on $\tilde{X}_{i-1}$ vanishing in multiplicity $\geq m_{i}$ in $P_{i}$. To obtain a section of $\tilde{L}^{(i)}$ from a section of $\tilde{L}^{(i-1)}$ we pull back the section along the map $\pi^{(i-1)}\colon\tilde{X}_{i}\to\tilde{X}_{i-1}$ then subtract $m_{i}$ copies of the exceptional divisor. If a section of \tilde{\mathcal{L}}^{(i)}\raisebox{-2.15277pt}{|}_{\tilde{X}_{i}}=\tilde{L}^{(i)} corresponds to a section of $\tilde{L}^{(i-1)}$ with multiplicity greater than $m_{i}$ at $P_{i}$ the restriction of this section to $E_{i}$ is zero. If the multiplicty is exactly $m_{i}$ at $P_{i}$ then \tilde{\mathcal{L}}^{(i)}\raisebox{-2.15277pt}{|}_{E_{i}}\cong\mathcal{O}_{\mathbb{P}^{n-1}}(m_{i}) hence the restriction of such a section of $\tilde{\mathcal{L}}^{(i)}$ to $E_{i}\cong\mathbb{P}^{n}_{\mathbb{C}}$ will be described by a non-zero homogeneous polynomials of degree $m_{i}$ in $Y_{1},\ldots,Y_{n}$.

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Global sections of \tilde{\mathcal{L}}^{(i)}\raisebox{-2.15277pt}{|}_{\mathcal{E}_{i}^{(i)}} and their restriction to $E_{i}$:

To describe these we introduce homogeneous coordinates $[T:Y_{1}:\ldots:Y_{n}]$ on $\mathcal{E}_{i}^{(i)}$. The coordinates $Y_{1},\ldots,Y_{n}$ come from local coordinates $y_{1},\ldots,y_{n}$ around $P_{i}$, the $T$ coordinate comes from the affine base parameter $t$, and $T=0$ describes $E_{i}\subset\mathcal{E}_{i}^{(i)}$. Hence the restriction of a section of \tilde{\mathcal{L}}^{(i)}\raisebox{-2.15277pt}{|}_{\mathcal{E}_{i}^{(i)}} to $E_{i}$ is obtained by setting $T=0$. Since \tilde{\mathcal{L}}^{(i)}\raisebox{-2.15277pt}{|}_{\mathcal{E}_{i}^{(i)}}\cong\mathcal{O}_{\mathbb{P}^{n}}(m_{i}) sections of \tilde{\mathcal{L}}^{(i)}\raisebox{-2.15277pt}{|}_{\mathcal{E}_{i}^{(i)}} are non-zero homogeneous polynomials of degree $m_{i}$ in $T,Y_{1},\ldots,Y_{n}$.

Sections of $\tilde{\mathcal{L}}^{(i)}$ are therefore described by pairs of sections of $\tilde{X}_{i}$ and $\mathcal{E}_{i}^{(i)}$ both vanishing on $E_{i}$ and pairs of sections of $\tilde{X}_{i}$ and $\mathcal{E}_{i}^{(i)}$ restricting to the same non-zero homogeneous polynomial in $Y_{i},\ldots,Y_{n}$ of degree $m_{i}$ on $E_{i}.$ A basis of global sections of \tilde{\mathcal{L}}^{(i)}\raisebox{-2.15277pt}{|}_{\tilde{\mathcal{X}}_{i,0}} can therefore be built from the basis of $H^{0}(\tilde{X}_{i},L^{(i)})$ and H^{0}(\tilde{\mathcal{E}}_{i}^{(i)},\tilde{\mathcal{L}}^{(i)}\raisebox{-2.15277pt}{|}_{\mathcal{E}_{i}^{(i)}} as the set of

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Pairs of basis sections on $\tilde{X}_{i}$ vanishing on $E_{i}$ and the zero section on $\mathcal{E}_{i}^{(i)}$,

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Pairs of the zero section on $\tilde{X}_{i}$ and a basis section on $\mathcal{E}_{i}^{(i)}$ vanishing on $E_{i}$,

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Pairs of basis sections of $\tilde{X}_{i}$ and $\mathcal{E}_{i}^{(i)}$ that restrict to the same non-zero homogeneous polynomial of degree $m_{i}$ on $E_{i}$.

Now we want to show that the restriction of the $\sigma^{(i)}_{0}\ldots\sigma^{(i)}_{N_{i}}$ to $\tilde{\mathcal{X}}_{i,0}$ yields a basis of \tilde{\mathcal{L}}^{(i)}\raisebox{-2.15277pt}{|}_{\tilde{\mathcal{X}}_{i,0}}. To this purpose, we follow steps 1-3 to construct $\sigma_{j}^{(i)}$ from a section of $\tilde{L}_{i-1}$ and then identify the pair of sections describing the restriction to $\tilde{\mathcal{X}}_{i,0}$. If a section of $\tilde{L}_{i-1}$ vanishes to multiplicity strictly greater than $m_{i}$ at $p_{i}$ then it corresponds to a pair of sections on $\tilde{\mathcal{L}}^{(i)}_{0}$ consisting of a section of $\tilde{\mathcal{L}}^{(i)}$ that restricts to zero on $E_{i}$, and the zero section on $\mathcal{E}_{i}^{(i)}$. If the section of $\tilde{L}_{i-1}$ on $\tilde{X}_{i-1}$ vanishes to exactly multiplicity $m_{i}$ at $P_{i}$ then pulling back and subtracting $m_{i}$ copies of the exceptional divisor we obtain non-zero sections on $\tilde{X}_{i}$ and on $\mathcal{E}^{(i)}_{i}$. Such a section of $\tilde{L}_{i-1}$ corresponds to a pair of sections on $\tilde{\mathcal{X}}_{i,0}$ that restrict to the same non-zero homogeneous monomial on $E_{i}$. Finally if a section$s$ of $\tilde{L}_{i-1}$ on $\tilde{X}_{i-1}$ vanishes to multiplicity strictly less than $m_{i}$ at $P_{i}$ then after pulling back along $p_{i-1}$ we must multiply by $t^{m_{i}-\mbox{ mult }_{P_{i}}s}$ before we can subtract copies of the exceptional divisor. This type of section corresponds to a pair of sections on $\tilde{\mathcal{X}}_{i,0}$ consisting of the zero section on $\tilde{X}_{i}$ and a section on $\mathcal{E}_{i}^{(i)}$ that restricts to zero on $E_{i}$.

In terms of the sets $B_{0},B_{j}$ and $\tilde{B}_{j}$, making up the basis $B^{(i)}$ of $\tilde{L}^{(i)}$ we find that a basis of global sections of $\tilde{L}^{(i-1)}$ vanishing to multiplicity $>m_{i}$ at $P_{i}$ can be written as $B_{0}\cup\bigcup^{i-1}_{j=1}\tilde{B}_{j}\cup\bigcup_{i+1}^{k}B_{j}$. A basis of global sections vanishing to multiplicity exactly $m_{i}$ at $P_{i}$ is $\tilde{B}_{i}$ and finally a basis of global sections vanishing to multiplicity less than $m_{i}$ at $P_{i}$ is given by $B_{i}-\tilde{B}_{i}$. The union of all three basis provides a basis $B^{(i-1)}$ for $\tilde{\mathcal{X}}_{i,t}\cong\tilde{X}_{i-1}$. The union of the basis corresponding to all sections of $\tilde{L}^{(i-1)}$ vanishing to multiplicity less than or equal to $m_{i}$ at $P_{i}$ is isomorphic to a basis of global sections of \tilde{\mathcal{L}}^{(i)}\raisebox{-2.15277pt}{|}_{\mathcal{E}^{(i)}_{i}} where the identification is given by the correspondence between homogeneous polynomials in local coordinates around $P_{i}$ and homogeneous coordinates of $\mathcal{E}^{(i)}_{i}\cong\mathbb{P}^{n}_{\mathbb{C}}$.

Thus, the sections $\sigma_{0}^{(i)},\ldots,\sigma_{N_{i}}^{(i)}$ also restrict to a basis of \tilde{\mathcal{L}}^{(i)}\raisebox{-2.15277pt}{|}_{\tilde{\mathcal{X}}_{i,0}}, hence trivialize $\Pi^{(i)}_{*}\tilde{\mathcal{L}}^{(i)}$ over all fibers of the family.

Now let us construct a Kähler form using the trivializing sections of $\tilde{\mathcal{L}}^{(i)}$ restricted to $\mathcal{E}_{i}^{(i)}\cong\mathbb{P}^{n}_{\mathbb{C}}$. On this exceptional divisor we can choose homogeneous coordinates $T,Y_{1},\ldots Y_{n}$ as above. Then

[TABLE]

If we choose the sections $\sigma_{j}^{(i)}$ that do not vanish on the exceptional divisor (i.e. coming from sections of $\tilde{L}^{(i-1)}$ vanishing with multiplicity $\leq m_{i}$ in $P_{i}$) such that they restrict to the basis of monomials $\sqrt{c_{\alpha,\beta}}Y^{\alpha}T^{\beta}$ of the homogeneous polynomials of degree $m_{i}$, then $m_{i}\cdot\omega_{FS}$ is the Kähler form on $\mathcal{E}^{(i)}$ generated by the restriction of these sections $\sigma_{j}^{(i)}$. Note that $c_{\alpha,\beta}$ is a positive integer, so $\sqrt{c_{\alpha,\beta}}$ is just the usual real square root.

To achieve property (3) we have to construct the trivializing sections of $\tilde{\mathcal{L}}^{(i)}$ on $\tilde{\mathcal{X}}_{i}$ iteratively, starting with $i=k$. On $\tilde{\mathcal{X}}_{k}$ we construct sections $\sigma_{0}^{(k)},\ldots,\sigma_{N_{k}}^{(k)}$ as above, which restricted to $\tilde{\mathcal{X}}_{k,\delta_{k}}$ provides a basis of sections of $\tilde{L}^{(k-1)}$ on $\tilde{X}_{k-2}$ vanishing with multiplicity $\geq m_{k-1}$ in $P_{k-1}$. We can complete these sections to a basis of all sections of $\tilde{L}^{(k-2)}$ on $\tilde{X}_{k-2}$ by adding sections which vanish with multiplicity $<m_{k-1}$ in $P_{k-1}$. This basis can be used to construct trivializing sections of $\tilde{\mathcal{L}}^{(i)}$ on $\tilde{\mathcal{X}}_{k-1}$ as above, because by its construction it can be split up into the subsets $B_{0},B_{i},\tilde{B}_{i}$ and $B^{(i)}$. The Kähler form on $\mathcal{E}^{(i)}$is also as requested since we can choose the completing basis sections of $\tilde{L}^{(k-2)}$ as required above. Iterating this process for $\tilde{\mathcal{X}}_{k-2},\ldots,\tilde{\mathcal{X}}_{1}$ we deduce property (3) for each $i=1\ldots,k$. ∎

4. Kähler packings of projective, complex, manifolds.

Theorem 14**.**

Let $X$ be a projective complex manifold of dimension $n$. $L$ an ample line bundle on $X$ and $P_{1},\ldots,P_{k}\in X$. Let $\epsilon_{0}=\epsilon(X;L,P_{1},\ldots,P_{k})$ denote the multipoint Seshadri constant of $L$ on $X$ in $P_{1},\ldots,P_{k}$. Then, for any radius $r<\sqrt{\epsilon_{0}}$ there exists a Kähler packing of $k$ flat Kähler balls of radius $r$ into $X$.

Proof.

First we construct families $\tilde{\mathcal{X}}_{i}$ as in the previous section. Then we embed Fubini-Study Kähler balls of large enough volume on $\mathcal{E}^{(i)}_{i}-E_{i}$ provided with the Kähler metric $\omega_{i}^{(i)}$ induced by the global sections of $\tilde{\mathcal{L}}^{(i)}$ constructed in Theorem 13 on $\mathcal{E}_{i}^{(i)}$, for $i=k,k-1,\ldots,1$. These balls can be deformed to Kähler balls on non-central fibers $\tilde{\mathcal{X}}_{i,\delta_{i}}\cong\tilde{X}_{i-1}$, and then iteratively to non-central fibers $\tilde{\mathcal{X}}_{j,\delta_{j}}\cong\tilde{X}_{j-1}$ for $j=i-1,\ldots,1$ if the $\delta_{j}$ are chosen small enough. Doing this carefully the deformed balls will not intersect on $\tilde{\mathcal{X}}_{i,\delta_{i}}\cong X$, so after gluing in standard Kähler balls into the Fubini-Study Kähler balls we obtain the claim. In more details:

1. (1)

Let $\Delta_{\delta}\subset\mathbb{A}^{1}_{\mathbb{C}}$ denote the open disk of radius $\delta$, with affine parameter $t$. Choose local coordinates $y_{1},\ldots,y_{n}$ of $X$ around $P_{i}$, so $t,y_{1},\ldots,y_{n}$ are local coordinates around $(0,P_{i})$ in $\tilde{\mathcal{X}}_{i}$. Then over the open subset $\mathcal{U}_{t}\subset\tilde{\mathcal{X}}_{i}$ where these coordinates are defined there is a chart of the blow up of $\mathcal{X}_{i}$ in $(P_{i},0)$ with coordinates $t,z_{1},\ldots,z_{n}$ such that the blow up map to $\mathcal{U}_{t}$ is described by $(t,z_{1},\ldots,z_{n})\mapsto(t,y_{1},\ldots,y_{n})\coloneqq(t,tz_{1},\ldots,tz_{n})$. Because the coordinates $y_{i}$ are bounded the central fiber of the induced projection of $\mathcal{U}_{t}$ onto $\mathbb{A}^{1}_{\mathbb{C}}$ is $\mathcal{E}-E\cong\mathbb{A}^{n}_{\mathbb{C}}$. The non-central fibers are not quite isomorphic to $\mathbb{A}^{n}_{\mathbb{C}}$ but contain balls $B_{R}(0)$ with $R$ arbitrarily large close to $t=0$. Thus for $R$ arbitrarily large we can find $\delta$ sufficently small and an embedding $\iota\colon\Delta_{\delta}\times B_{R}(0)\hookrightarrow\tilde{\mathcal{X}}_{i}$. See Figure 2. 2. (2)

Choosing in (1) $R$ large enough and $\delta$ small enough implies that there exists embeddings of Fubini-Study Kähler balls $B_{R^{\prime}}(0)$ with $R^{\prime}\leq R$, of volume arbitrarily close to the volume of $(\mathcal{E}^{(i)}_{i},\omega_{i}^{(i)})$ in all fibers of $\Delta_{\delta}\times B_{R}(0)$ over $t\in\Delta_{\delta}$ with respect to the same Fubini-Study Kähler form $\omega_{i}^{(i)}$. 3. (3)

By continuity, for $t$ small enough these Fubini-Study forms $\omega_{i}^{(i)}$ differ to an arbitrarily small amount from the Kähler form $\omega_{i,t}$ on $\tilde{\mathcal{X}}_{i,t}$ obtained from the trivializing sections of $\tilde{\mathcal{L}}^{(i)}$ pulled back via the embedding $\iota_{i}$. This allows us to glue in the Fubini-Study Kähler balls of step (2) into non-central fibers $\tilde{\mathcal{X}}_{i,t}$ provided with the Kähler form $\omega_{i,t}$ for $t\ll 1$ small enough. Assume that $\omega_{i}^{(i)}=\frac{i}{2\pi}\partial\bar{\partial}\log(s_{i}^{(i)})$ and $\omega_{i,t}=\frac{i}{2\pi}\partial\bar{\partial}\log s_{i,t}$ on $\{t\}\times B_{R}(0)$, where $s_{i}^{(i)},s_{i,t}$ are functions constructed in the usual way from the appropriate sections. Then choose a partition of unity $(\rho_{1},\rho_{2})$ such that \rho_{2}\raisebox{-2.15277pt}{|}_{B_{R^{\prime}}(0)}\equiv 1 and \rho_{2}\raisebox{-2.15277pt}{|}_{B_{R^{\prime}}(0)}\equiv 0. The glued 2-form $\tilde{\omega}_{i,t}$ is given by $\frac{i}{2\pi}\partial\bar{\partial}\log(\rho_{1}s_{i}^{(i)}+\rho_{2}s_{i,t})=\frac{i}{2\pi}\partial\bar{\partial}\log(\rho_{1}(s_{i}^{(i)}-s_{i,t})+s_{i,t})$. This form is obviously closed, and it is non-degenerate because $s_{i}^{(i)}-s_{i,t}$ gets arbitrarily small for $t\ll 0$, thus $\rho_{1}(s_{i}^{(i)}-s_{i,t})+s_{i,t}$ is arbitrarily close to $s_{i,t}$. Consequently $\tilde{\omega}_{i,t}$ is a Kähler form. 4. (4)

The $i^{th}$ Fubini-Study Kähler ball on $\tilde{\mathcal{X}}_{i,\delta_{i}}$ does not intersect the $(i+1)^{st},\ldots,k^{th}$ Kähler ball constructed before: On the central fiber $\tilde{\mathcal{X}}_{i,0}$ these balls lie on $\mathcal{E}_{i}^{(i)}-E_{i}$ and $\tilde{X}_{i}-E_{i}\cong\tilde{X}_{i-1}-P_{i}$, so do not intersect. This will not change when we deform the balls to $\tilde{\mathcal{X}}_{i,\delta_{i}}$ if we choose $\delta_{i}$ small enough. 5. (5)

Assume $\int\omega^{n}_{FS}=1$, and $\int_{B_{1}(0)}\omega_{std}=1$. Then there exists a Kähler embedding $(B_{r}(0),\omega_{std})\hookrightarrow(\mathbb{C}\mathbb{P}^{n},\omega_{FS})$ , for all $r<1$ (for more details on this embedding see [Eck17]). Hence for all $r<\sqrt{m_{i}}$ there exists a Kähler embedding $(B_{r}(0),\omega_{std})\hookrightarrow(\mathbb{C}\mathbb{P}^{n},m_{i}\omega_{FS})$. Rescaling by $d_{i}$ we obtain a Kähler embedding $(B_{r}(0),\omega_{std})\hookrightarrow(\mathbb{C}\mathbb{P}^{n},\frac{m_{i}}{d_{i}}\omega_{FS})$ for all $r<\sqrt{\frac{m_{i}}{d_{i}}}$. Since $\omega_{i}^{(i)}=m_{i}\omega_{FS}$ the embeddings constructed above imply that $\frac{m_{i}}{d_{i}}<\epsilon_{0}$, but since $\frac{m_{i}}{d_{i}}$ can be chosen arbitrarily close to $\epsilon_{0}$ we can conclude that $\sqrt{\epsilon_{0}}\leq\gamma_{k}$, where $\gamma_{k}$ is the $k-$ball packing constant.

∎

Proof of Theorem 4.

The claim that the k-point Seshadri constant is less or equal to the Kähler packing constant is a direct consequence of Theorem 14. The converse argument is a consequence of the symplectic blow up construction [MP94]. The embedded balls allow us to construct Kähler forms on the blow up $\pi$ of the centres whose curvature lies in the first Chern class of $\pi^{*}L-\sum\gamma E_{i}$. ∎

Remark 15**.**

The method to prove Theorem 14 also allows us to construct Kähler packings of balls with radius $r_{1},\ldots,r_{k}$ arbitrarily close to $\epsilon_{1},\ldots,\epsilon_{k}$ as long as $\pi^{*}L-\sum_{i=1}^{k}\epsilon_{i}E_{i}$ is nef.

As a final remark in [Eck17] it was shown that when $X$ is a toric variety and $L$ a toric invariant divisor, the sections that generate the Kähler form also induce a moment map whose image is the well known toric polytope associated to $X$ and $L$. Furthermore we find that the cut off triangles of the polytope are the shadows under the moment map of the glued in balls. In future work I will show that we can generalise this somewhat by replacing the moment polytopes by Newton-Okounkov bodies and a non-toric moment map. To illustrate these ideas I will construct examples of $\mathbb{P}^{2}$ blown up at 2 and 3 points with new shadows.