K\"ahler packings of projective complex manifolds

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


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 , . . . , P k be points of P 2 in general position and m 1 , . . . , m k be positive integers. Then for k ≥ 9 any curve C ∈ P 2 of degree d passing each point P i with multiplicity m i must satisfy Using multipoint Seshadri constants this can be rewritten as: Conjecture 2. Let P 1 , . . . , P k be points of P 2 in general position then for k ≥ 9 the multipoint Seshadri constant ǫ(P 2 , O P 2 (1), P 1 , . . . , P k ) = 1 √ k .
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, ω) be an n dimensional Kähler manifold with Kähler form ω. Then a holomorphic embedding is called a Kähler embedding of k disjoint complex flat balls in C n centered in 0, of radius r q , if there exists a Kähler form ω ′ such that [ω ′ ] = c 1 (L) and φ * q (ω ′ ) = ω std is the standard Kähler form on C n restricted to B 0 (r q ). Let γ k (X, ω; P 1 , . . . , P n ) = sup{r > 0 : That there exists a Kähler packing with φ q (0) = P q }. We call γ k the k−ball packing constant.
Since curvature is an invariant under open holomorphic embeddings we cannot define the packing condition φ * q (ω) = ω std using the original Kähler form ω 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 ω 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: γ k (X, ω; P 1 , . . . , P n ) = ǫ(X, L; P 1 , . . . , P n ).

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 , . . . , P k distinct points of X. Let L be an ample divisor andX := Bl P1,...,P k (X) π − → X be the blow-up of X at the points P 1 , . . . , P k . Let π −1 (P i ) = E i denote the exceptional divisor corresponding to the point P i for all i = 1, . . . , k and for ǫ i ∈ Q >0 set If L is a nef Q-Cartier divisor then [Laz04] defines the multipoint Seshadri constant to be 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 ≥ 0 with > 0. The following lemma's prove that the two definitions are equivalent filling a gap in the literature.
Proof. By Seshadri's Criterion [Laz04,Thm 1.4.13] if L is ample then there exists ǫ L > 0 such that for all points Q ∈ X and all irreducible curves C containg Q we have that L·C multQ C > ǫ L . LetQ ∈X such that π(Q) = Q and letC ⊂X be an irreducible curve withQ ∈C. Then there are the following cases.
On the other hand if multP C multQ C ≥ ǫL 2ǫ ′ then IfQ ∈ E withC ∈ E then multQC ≤ mult P π(C) and The final inequality appears since E ·C = degC on E ∼ = P n−1 and degC ≥ multQC. Hence we have shown that if 0 < ǫ ′ < ǫ P then π * L − ǫ ′ E is Q-ample.
Proof. By Seshadri's criterion and Lemma 5 we know there exist some ǫ 1 > 0 such that π * L−ǫ 1 E is Q-ample onX 1 = Bl P1 (X). Choosing a second point P 2 ∈ X Seshadri's criterion again ensures the existence of some ǫ 2 > 0 such that π * L − ǫ 1 E 1 − ǫ 2 E 2 is Q-ample onX 2 = Bl P1,P2 (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 ǫ i to be equal. This follows from the next lemma.
where π k−1 :X → 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 ⊂ X be an irreducible curve andC ⊂X be the strict transform of C. Then Definition 9. For X and L defined as above the multipoint Seshadri constant associated to points P 1 , . . . P k of X is 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 These are exactly the coordinate hyperplanes cut out by t i = 0 over the points P i .
By blowing up the family X over the union of all the centres Z i we obtain a new algebraic family over A k C .X (1)X is a well defined complex manifold projective over Running through an appropriate path through the parameter space A k C these blow ups accumulate iteratively. Let δ 1 , . . . , δ k be positive real numbers and define lines in A k C , consisting of points whose first (i − 1) coordinates are zero, the i-th coordinate is of the form t · δ i and the remaining (i + 1) coordinates are constant. For the point 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, . . . , 0) is the blow up of X at P k with contributions from all exceptional divisors. See Figure 1 for a visualisation.

Figure 1
Similarly for OnX i there exists a family of divisors is the exceptional divisor of the blow up Π i . Specific divisors associated to a particular fiber are denoted L (i) d,m,t , where t denotes the parameter on A 1 C and d, m i = (m 1 , . . . , m i ) record the degree and multiplicity (we will often just writeL (i) t for short if the multiplicity and degree are fixed).
(2) If 0 i denotes the zero of the line l i then the sections (3) The restricted sections σ Proof. If we show that the dimension of the space of global sections ofL (i) restricted to each fiber of Π (i) is the same, Grauert's semi-continuity theorem [Har77, III.12] implies immediately that Π (i) * L (i) is locally free, hence it is free by the Quillen-Suslin Theorem. Thus there exists trivialising global sections of Π * 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 (X i,t ,L (i) t ) then restricted toX i,t for all t ∈ A 1 C . We start by calculating a basis of sections of H 0 (X, O X (dL)) characterised by their vanishing behaviour at the points P 1 , . . . , P k so that subsets of this basis can be interpreted as generating sections of H 0 (X,L (i) ). Let m X,Pi denote the maximal ideal of X at P i and consider the short exact sequence Taking the long exact sequence of cohomology w.r.t to the above sequence gives dL)) = 0, since applying the projection formula yields and we assumeL d,mi is ample and d, m k ≫ 0. Hence φ : OX /m m i +1 X,P i is surjective. This gives a basis of sections of H 0 (X, O X (dL)) which is the union of: , which are sections of O X (dL) vanishing to multiplicity at least m i + 1 in P i for i = 1, . . . , k.
• A set B i of sections of O X (dL) mapped to 0 in OX /m m j +1 X,P j for j = i and to a basis of homogenous polynomials of degree ≤ m i in OX /m m i +1 X,P i , in variables given by local coordinates around P i . • A set of sectionsB i of O X (dL) as above, but with homogenous polynomials of degree = m i , and thatB i ⊂ B i .
Hence a basis of H 0 (X i , OX i (L (i) )) is given by 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 . SinceX i =X i−1 ×A 1 C we can use these sections to construct the sections of H 0 (X i ,L (i) ) using the following procedure: Step 1: Pull back a section s ∈ H 0 (X i−1 , O Xi−1 (L (i−1) )) along p i−1 to get a section of p * i−1L Step 2: If mult Pi s < m i then the section t mi− mult P i s p * i−1 s ∈ H 0 (X i ,L (i−1) ) has multiplicity m i in (P i , 0). If mult Pi s ≥ m i then mult (Pi,0) p * i−1 s ≥ m i .
Step 3: In both cases, we can subtract m i copies of the exceptional divisor E i from the pullback of this section along Π i , thus obtaining sections of H 0 (X i ,L (i) ).
When starting with the basis sections in B (i−1) , restricting to the general fibersX i,t ∼ =Xi−1 of Π (i) (that is t = 0) we get back the sections in B (i−1) possibly multiplied with some power of t. Thus all the sections σ  These are all sections of dL on X which vanish to multiplicity ≥ m j in P j for j = 1, . . . , i. Equivalently these are global sections ofL (i−1) onX i−1 vanishing in multiplicity ≥ m i in P i . To obtain a section ofL (i) from a section ofL (i−1) we pull back the section along the map π (i−1) :X i →X i−1 then subtract m i copies of the exceptional divisor. If a section ofL (i) |X i =L (i) corresponds to a section ofL (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 thenL (i) | Ei ∼ = O P n−1 (m i ) hence the restriction of such a section of L (i) to E i ∼ = P n C will be described by a non-zero homogeneous polynomials of degree m i in Y 1 , . . . , Y n . i . The coordinates Y 1 , . . . , Y n come from local coordinates y 1 , . . . , y n around P i , the T coordinate comes from the affine base parameter t, and i . Such a section ofL i−1 corresponds to a pair of sections onX i,0 that restrict to the same non-zero homogeneous monomial on E i . Finally if a sections ofL i−1 onX 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 mi− mult P i s before we can subtract copies of the exceptional divisor. This type of section corresponds to a pair of sections onX i,0 consisting of the zero section onX i and a section on E (i) i that restricts to zero on E i . In terms of the sets B 0 , B j andB j , making up the basis B (i) ofL (i) we find that a basis of global sections ofL (i−1) vanishing to multiplicity > m i at P i can be written as B 0 ∪ i−1 j=1B j ∪ k i+1 B j . A basis of global sections vanishing to multiplicity exactly m i at P i isB i and finally a basis of global sections vanishing to multiplicity less than m i at P i is given by B i −B i . The union of all three basis provides a basis B (i−1) for X i,t ∼ =Xi−1. The union of the basis corresponding to all sections ofL (i−1) vanishing to multiplicity less than or equal to m i at P i is isomorphic to a basis of global sections ofL where the identification is given by the correspondence between homogeneous polynomials in local coordinates around P i and homogeneous coordinates of E where c a,b are positive integers.
If we choose the sections σ (i) j that do not vanish on the exceptional divisor (i.e. coming from sections of L (i−1) vanishing with multiplicity ≤ m i in P i ) such that they restrict to the basis of monomials √ c α,β Y α T β of the homogeneous polynomials of degree m i , then m i · ω F S is the Kähler form on E (i) generated by the restriction of these sections σ (i) j . Note that c α,β is a positive integer, so √ c α,β is just the usual real square root.
To achieve property (3) we have to construct the trivializing sections ofL (i) onX i iteratively, starting with i = k. OnX k we construct sections σ (k) 0 , . . . , σ (k) N k as above, which restricted toX k,δ k provides a basis of sections ofL (k−1) onX k−2 vanishing with multiplicity ≥ m k−1 in P k−1 . We can complete these sections to a basis of all sections ofL (k−2) onX 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 ofL (i) onX k−1 as above, because by its construction it can be split up into the subsets B 0 , B i ,B i and B (i) . The Kähler form on E (i) is also as requested since we can choose the completing basis sections ofL (k−2) as required above. Iterating this process forX k−2 , . . . ,X 1 we deduce property (3) for each i = 1 . . . , k.

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 , . . . , P k ∈ X. Let ǫ 0 = ǫ(X; L, P 1 , . . . , P k ) denote the multipoint Seshadri constant of L on X in P 1 , . . . , P k . Then, for any radius r < √ ǫ 0 there exists a Kähler packing of k flat Kähler balls of radius r into X.
Proof. First we construct familiesX i as in the previous section. Then we embed Fubini-Study Kähler balls of large enough volume on E i , for i = k, k − 1, . . . , 1. These balls can be deformed to Kähler balls on non-central fibersX i,δi ∼ =Xi−1, and then iteratively to non-central fibersX j,δj ∼ =Xj−1 for j = i − 1, . . . , 1 if the δ j are chosen small enough. Doing this carefully the deformed balls will not intersect onX i,δi ∼ = X, so after gluing in standard Kähler balls into the Fubini-Study Kähler balls we obtain the claim. In more details: (1) Let ∆ δ ⊂ A 1 C denote the open disk of radius δ, with affine parameter t. Choose local coordinates y 1 , . . . , y n of X around P i , so t, y 1 , . . . , y n are local coordinates around (0, P i ) inX i . Then over the open subset U t ⊂X i where these coordinates are defined there is a chart of the blow up of X i in (P i , 0) with coordinates t, z 1 , . . . , z n such that the blow up map to U t is described by (t, z 1 , . . . , z n ) → (t, y 1 , . . . , y n ) := (t, tz 1 , . . . , tz n ). Because the coordinates y i are bounded the central fiber of the induced projection of The non-central fibers are not quite isomorphic to A n C but contain balls B R (0) with R arbitrarily large close to t = 0. Thus for R arbitrarily large we can find δ sufficently small and an embedding ι : ∆ δ × B R (0) ֒→X i . See Figure 2.
(2) Choosing in (1) R large enough and δ small enough implies that there exists embeddings of Fubini-Study Kähler balls B R ′ (0) with R ′ ≤ R, of volume arbitrarily close to the volume of (E i ) in all fibers of ∆ δ × B R (0) over t ∈ ∆ δ with respect to the same Fubini-Study Kähler form ω (i) i .
(3) By continuity, for t small enough these Fubini-Study forms ω (i) i differ to an arbitrarily small amount from the Kähler form ω i,t onX i,t obtained from the trivializing sections ofL (i) pulled back via the embedding ι i . This allows us to glue in the Fubini-Study Kähler balls of step (2) into non-central fibers X i,t provided with the Kähler form ω i,t for t ≪ 1 small enough. Assume that ω i , s i,t are functions constructed in the usual way from the appropriate sections. Then choose a partition of unity (ρ 1 , ρ 2 ) such that ρ 2| B R ′ (0) ≡ 1 and ρ 2| B R ′ (0) ≡ 0. The glued 2-formω i,t is given by i 2π ∂∂ log(ρ 1 s . This form is obviously closed, and it is non-degenerate because s (i) i − s i,t gets arbitrarily small for t ≪ 0, thus ρ 1 (s (i) i − s i,t ) + s i,t is arbitrarily close to s i,t . Consequentlyω i,t is a Kähler form.
(4) The i th Fubini-Study Kähler ball onX i,δi does not intersect the (i + 1) st , . . . , k th Kähler ball constructed before: On the central fiberX i,0 these balls lie on E (i) i − E i andX i − E i ∼ =Xi−1 − P i , so do not intersect. This will not change when we deform the balls toX i,δi if we choose δ i small enough.
(5) Assume ω n F S = 1, and B1(0) ω std = 1. Then there exists a Kähler embedding (B r (0), ω std ) ֒→ (CP n , ω F S ) , for all r < 1 (for more details on this embedding see [Eck17]). Hence for all r < √ m i there exists a Kähler embedding (B r (0), ω std ) ֒→ (CP n , m i ω F S ). Rescaling by d i we obtain a Kähler embedding (B r (0), ω std ) ֒→ (CP n , mi di ω F S ) for all r < mi di . Since ω (i) i = m i ω F S the embeddings constructed above imply that mi di < ǫ 0 , but since mi di can be chosen arbitrarily close to ǫ 0 we can conclude that √ ǫ 0 ≤ γ k , where γ 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 π of the centres whose curvature lies in the first Chern class of π * L − γE i .
Remark 15. The method to prove Theorem 14 also allows us to construct Kähler packings of balls with radius r 1 , . . . , r k arbitrarily close to ǫ 1 , . . . , ǫ k as long as π * L − k i=1 ǫ 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 P 2 blown up at 2 and 3 points with new shadows.