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SYMPLECTIC BALL PACKINGS IN HIGHER DIMENSIONS
a reader's guide
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A friendly tour for a high-school reader who knows some
calculus and has met complex numbers, but has never seen
symplectic geometry. We don't reprove the technical lemmas
here. Our only goal is to explain what the paper proves, why
it is hard, and how the pieces fit together, so a curious
young reader can follow the actual paper afterwards.
The technical paper has the same title (September 2026); this
file is the guide to it.
1. THE ONE-PARAGRAPH ANSWER
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Imagine a big "symplectic ball" of capacity R (we will define
capacity in a moment) and k small closed symplectic balls of
capacities R_1, ..., R_k. Can the small balls be fit disjointly
into the interior of the big one by a symplectic map?
Dimension is important. We work in n complex dimensions,
which is 2n real dimensions. The theorem covers every n >= 3,
every k >= 1, and every choice of positive capacities.
Main theorem (this paper). The packing is possible
IF AND ONLY IF both of the following hold:
R_1^n + R_2^n + ... + R_k^n < R^n (volume)
R_i + R_j < R for every pair i != j (two-ball)
There are no other conditions. Not one extra constraint.
Not in dimension 6, not in dimension 8, not ever.
That is exactly Siegel and Yao's Conjecture A, stated in 2025 and
now proven positively in every n >= 3.
The strict inequalities reflect the "closed sources, open target"
convention: the closed balls must live inside the open interior
of the big ball, so no volume can leak onto the boundary.
2. WHAT IS A SYMPLECTIC BALL
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We work in C^n, which as a real vector space is just R^{2n}.
Write a point as z = (z_1, ..., z_n) with z_j = x_j + i y_j.
The closed symplectic ball of capacity R is
B^{2n}(R) = { z in C^n : pi * ( |z_1|^2 + ... + |z_n|^2 ) <= R }
where |z_j|^2 = x_j^2 + y_j^2. This is a round ball of
Euclidean radius sqrt(R/pi). Its volume is R^n / n!.
We will always measure balls by capacity R, never by radius.
Why? Because symplectic geometry sees "symplectic area," and
the symplectic area of a complex 2-plane through the origin is
exactly pi * r^2 = R. Capacity is the natural unit.
The symplectic form on C^n is
omega_0 = dx_1 ^ dy_1 + dx_2 ^ dy_2 + ... + dx_n ^ dy_n.
Each term dx_j ^ dy_j is the signed area element of the
j-th complex coordinate plane C_j = { other coords = 0 }.
So omega_0 is a sum of n independent signed area measurements.
A symplectic embedding f : U -> U' (both open subsets of C^n)
is a smooth injective map with f^* omega_0 = omega_0.
That is: the pullback of the area form equals the original.
Volume of a region is the integral of the top power
omega_0^n / n!. So preserving omega_0 automatically preserves
volume. But preserving omega_0 is much stricter than preserving
volume, because omega_0 records signed 2-dimensional area on
every coordinate plane, not only the total 2n-dimensional
volume.
For n = 1 (ordinary plane), preserving omega_0 = dx ^ dy is
the same as preserving ordinary area. There is no rigidity.
For n >= 2 a gap opens between "symplectic" and
"volume-preserving," and that gap is the whole story.
Gromov's fundamental observation: symplectic geometry is
rigid in a way that volume-preserving geometry is not, in
every dimension n >= 2.
3. THE TWO OBSTRUCTIONS
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Both parts of the main theorem's necessity are short to state.
(3a) VOLUME
Symplectic maps preserve volume. The volume of B^{2n}(R) is
R^n / n!, so the total volume of the packed balls must not
exceed the target volume:
(R_1^n + ... + R_k^n) / n! < R^n / n!
which is exactly the first inequality.
(3b) TWO-BALL (Gromov, 1985)
If you have two balls of capacities R_1, R_2 that you want
to pack disjointly into capacity R, then
R_1 + R_2 < R.
This is the surprising one. Two half-capacity balls in
dimension n = 3 have total volume just 1/4 of the target,
yet they cannot fit: their capacities add to 1, so any
symplectic map that tries to place both is forbidden.
The proof idea, in one paragraph:
A symplectic form like omega_0 admits a compatible
almost-complex structure J (a rule for rotating tangent
vectors by 90 degrees, like multiplication by i in
coordinates). Once you have J, you can ask for
J-holomorphic spheres: maps from the standard round
two-sphere S^2 into the symplectic manifold whose
derivative is complex-linear.
Gromov proved that through every point of C^n there
passes such a sphere, and its symplectic area is
pinned to a specific value tied to the "size" of the
ball it is drawn through. Those spheres behave like
impossible-to-shrink rulers.
Now, if you try to place a second ball B' disjointly
from a first ball B, the Gromov sphere through B
cannot touch B'. That forces B' to be below a
calculable capacity. Doing this carefully gives
R_1 + R_2 < R.
A full proof uses the moduli space of J-holomorphic spheres
and their intersection theory. That is far beyond high
school. The takeaway that matters: these rigid curves
capture information about size that ordinary volume does
not see.
(3c) NECESSITY OF THE SECOND OBSTRUCTION FOR ALL PAIRS
If you have k >= 3 balls and any pair, say i and j, has
R_i + R_j >= R, then those two alone cannot fit. Since
the whole packing contains that pair, the whole packing
cannot fit either. Applying this over all pairs gives
the full set of R_i + R_j < R conditions.
So the necessity direction of the theorem is short.
4. THE SUFFICIENCY STRATEGY IN ONE PAGE
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Normalize the target capacity to 1. A single scaling factor of
sqrt(R_0/R) applied to both source and target gives this for
free; the two factors cancel inside the pullback.
If there is only one ball, place it inside the larger target by
inclusion: done.
Otherwise, the theorem splits into three geometric cases:
CASE 1: EVERY REQUESTED CAPACITY IS < 1/2.
Build a "positive-genus curve" model inside PP^n and pack
everything there. The curve is the complete intersection of
n-1 smooth quadric hypersurfaces. Its genus is at least 1
for every n >= 3, so we get the "handle" required by the
surface packing lemma.
CASE 2: EXACTLY ONE CAPACITY, SAY R_0, IS >= 1/2, AND n >= 4.
Reserve a ball of capacity a > R_0 at the center. Pack the
remaining balls into the shell { a < sum |z_j|^2 < 1 }.
For n >= 4, a two-stage model works: first a hyperplane
inside PP^n, then inside that hyperplane a complete
intersection of quadrics. The second-stage curve has
genus 1 + (n-4) 2^{n-3}, which is >= 1 for n >= 4.
CASE 3: EXACTLY ONE CAPACITY, SAY R_0, IS >= 1/2, AND n = 3.
Same reservation idea, but for n = 3 the plane inside PP^3
is PP^2, and a conic in PP^2 has genus 0 (no handle). We
replace the conic by a plane cubic through two marked
points. Its genus is 1 and removing the marks keeps a
puncture. A further "Kähler cut" makes the shell available.
In Case 2 or 3, "one capacity >= 1/2" is automatic uniqueness:
since every pair satisfies R_i + R_j < 1, another ball with
capacity >= 1/2 would force R_i + R_j >= 1, contradiction.
The engine behind every case is the following three-part
machine, in order:
STEP A: turn volume into pointwise room
(Sections 5 of this guide, corresponding to comparison.tex
of the paper). On a polytope Delta, a continuous concave
function P with positive mean is upgraded to a strict
pointwise inequality via a Hamiltonian isotopy.
STEP B: turn pointwise room into layers
(Section 6, corresponding to surface.tex). A handle in the
base surface provides a closed 1-form with period 1, which
creates a stack of disjoint "layers" along a coordinate.
One closed ball goes into each layer.
STEP C: transport into the real target
(Section 7, corresponding to geometry.tex). Normal-cone
degenerations and Moser isotopies move the packing from the
abstract model into the actual ball, while preserving
disjointness and keeping away from forbidden divisors.
Case-specific details (Sections 8 of this guide) show how to
produce an appropriate "surface plus toric fiber" model whose
volume is large enough for each of the three cases.
5. STEP A: HAMILTONIAN COMPARISON
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This is the beating heart of the proof and lives in
comparison.tex. We take a mean inequality and upgrade it to a
strict pointwise inequality by reshuffling the fiber coordinates.
SETTING.
Fix m >= 1 and a compact polytope
Delta = { p in R^m_{>=0} : b_nu . p <= c_nu, nu = 1..N }
with b_nu in R^m_{>=0} (nonnegative coefficients) and
positive bounds c_nu. If b_nu . p = c_nu we call it an
"outer face" of Delta. The coordinate faces p_j = 0 are
NOT outer faces.
In C^m, take the toric domain
V = { z in C^m : p(z) in Delta }, p_j(z) = pi |z_j|^2,
with the toric form
omega_V = dp_1 ^ d theta_1 + ... + dp_m ^ d theta_m
where theta_j = arg(z_j) / (2 pi). This is just the standard
symplectic form on C^m, restricted to V.
Think of Delta as the "moment polytope" of V: it records how much
area is available along each complex coordinate direction.
Lowering any moment p_j keeps you in Delta (because b_nu has
nonnegative coefficients), so Delta "opens downward."
THE LEMMA.
Let P : Delta -> R be continuous and concave. Suppose
(i) the average of P over Delta is strictly positive;
(ii) there exists 0 < tau < 1 such that
P is bounded below by a positive constant on
Delta minus tau Delta (i.e. outside a shrunken
copy of Delta near its center).
Then there is a smooth toric function K on Delta and a
Hamiltonian diffeomorphism q : V -> V with
( K - P )( q u ) < K( u ) for every u in V.
The Morse-theoretic way to say this: the isotopy from the
identity to q is generated by Hamiltonians that Poisson
commute with the defining moment b_nu . p near each outer
face b_nu . p = c_nu, uniformly in time.
WHY THIS HELPS.
Later, when we build the "layers," we need a strict
pointwise gap between two functions. The lemma produces
exactly that gap from a mean-only hypothesis. Hypothesis
(ii) prevents the leftover area from vanishing inside the
polytope in a way that defeats the argument.
PROOF, IN THREE MOVES.
MOVE 1: increasing rearrangement.
Given a continuous convex f : [0,1] -> R, define its
increasing rearrangement g by
g(h) = min_{0 <= a <= 1-h} max{ f(a), f(a+h) }.
The name is literal. The rearranged function is the "sorted"
version of f, made nondecreasing while preserving:
- the integral of f over [0,1];
- order (f <= g implies R_f <= R_g);
- translation by a constant;
- uniform norm (|R_f - R_g| <= |f - g|).
For a function F(y, p_j) on Delta, define R_j F by taking
the slice at fixed other moments y, viewing F as a convex
function of p_j on [0, L(y)] with L(y) the largest p_j allowed
by Delta, and applying the 1D rearrangement scaled to the
slice length.
Key property: R_j preserves convexity. Reason: on two
slices of lengths h_0 and h_1, if we take the convex
combination (1-t) h_0 + t h_1 and the corresponding convex
combination of minimizing intervals, F evaluated at the two
endpoints of that interval is bounded by the same convex
combo of F at the two original endpoints, so the max-formula
inherited by the rearrangement stays convex.
Apply R_1, R_2, ..., R_m in order; call the composite R.
Since each R_j is nondecreasing in p_j and preserves
monotonicity in other coordinates, R F is coordinatewise
nondecreasing.
MOVE 2: contraction fixed point.
For 0 < lambda < 1, construct
F |--> R( lambda F - P ).
The map is a contraction of the complete metric space of
continuous convex functions on Delta, in uniform norm, with
constant lambda. So it has a unique fixed point
K_lambda.
Because R preserves integrals (Move 1) and P is fixed:
mean( K_lambda ) = -mean(P) / (1 - lambda).
In particular, mean(K_lambda) -> -infinity as lambda -> 1.
Now choose a small positive d < min( mean(P),
inf over
Delta minus tau Delta
of P ),
strictly less than both the mean of P and the boundary
lower bound of P.
Claim: for lambda close enough to 1,
M_lambda := max_over_Delta K_lambda < - d / (1 - lambda).
Prove by contradiction. If M_lambda is not that negative,
then on the outer region Delta minus tau Delta we can use
lambda K_lambda - P <= M_lambda + d (small),
while on the inner core tau Delta we can use convexity to
bound K_lambda far below M_lambda, giving
lambda K_lambda - P << M_lambda - something big.
The "something big" blows up as lambda -> 1, so on both
regions we end up with
max( lambda K_lambda - P ) < M_lambda,
contradicting the fixed-point equation
K_lambda = R( lambda K_lambda - P )
and the fact that max( R F ) <= max( F ).
So the claim holds for lambda close enough to 1.
Fix such a lambda. The pointwise inequality
(1 - lambda) K_lambda <= - d
(which is exactly M_lambda < -d/(1-lambda) rearranged) plus
R's order-preservation and translation-invariance gives
R( K_lambda - P ) <= K_lambda - d.
So: if we apply the full rearrangement R to F_0 := K-P,
the output is at most K - d. Strict gap of size d.
MOVE 3: realize rearrangement by Hamiltonian maps.
The rearrangement R = R_m ... R_1 is abstract. The paper
proves it can be implemented, up to arbitrarily small error,
by an actual Hamiltonian diffeomorphism of V that permutes
only the fiber variables.
The key sub-lemma is 2D:
Planar rearrangement lemma. Let f_y : [0,1] -> R be a
jointly continuous family of convex functions,
g_y the increasing rearrangements. For every eps > 0,
there is a smooth family of compactly supported
Hamiltonian isotopies of the disk D(1), with endpoints
psi_y, such that for every z in D(1),
f_y( pi |psi_y(z)|^2 ) <= g_y( pi |z|^2 ) + eps.
The idea: on D(1), the "moment" pi |z|^2 is the area of
the concentric disk through z. So we want a Hamiltonian
map that "sorts" the values of f into increasing order.
The trick is:
- choose a fine grid 0 < h_1 < ... < h_l < 1;
- for each h_j, look at the sublevel set
{ v : f_y(v) < g_y(h_j) + eps' }, an interval;
- inside each interval, choose nested rectangles of
prescribed areas h_1 < h_2 < ... < h_l;
- smooth the rectangle corners into nested smooth
disks of the same area;
- Schlenk's "nested disk lemma" (a standard trick)
gives a Hamiltonian isotopy carrying the standard
concentric disks of areas h_j to the chosen disks;
- now any point of radial area v sits below the "next
grid point up," and evaluating f there is at most
g(v) + 2 eps'.
Lifting to C^m: for the j-th factor, apply the planar
construction to f_y(v) = F(y, L(y) v), and lift the
Hamiltonian by l(y) * B_t(y, z_j / sqrt(l(y))) where
l(y) is a smooth approximation to the slice length L(y)
from below. On the outer strip L(y) < p_j, the maps can
be taken to be the identity (there is nothing to sort
there), which keeps the construction smooth over Delta.
Because the planar Hamiltonian vanishes near the boundary
of the disk, the lift vanishes near the outer face
b_nu . p = c_nu. This is the "outer face preservation"
property used in the Moser step later.
Summing the errors over j = 1, ..., m gives
(K - P) o q <= R(K - P) + sum_j eps_j
<= K - d + sum_j eps_j
< K
provided sum eps_j < d.
Finally approximate K by a smooth function; the error is
smaller than d/4, so the strict gap survives.
QED (comparison lemma).
WHAT WE GET.
From a mean inequality, a strict pointwise inequality. From
a convex function P, we get a smooth toric K and a Hamiltonian
q that turn the mean surplus into a pointwise surplus.
The next section uses that surplus to cut disjoint layers.
6. STEP B: PACKING OVER A HANDLE
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Now we actually place the balls. The setup is a base surface
C^o of genus >= 1 (with finitely many points removed, at least
one) times the toric fiber V from Section 5. A positive-genus
surface has a "handle" -- an essential circle that cannot be
shrunk to a point. That handle is what allows us to make
disjoint layers.
THE MODEL.
We have a family of two-forms on Sigma x V, one for each
compact bordered surface Sigma inside C^o. Each has the
form
Omega = omega_V + d Gamma,
with Gamma a toric horizontal one-form (annihilates vectors
tangent to V, coefficients depending smoothly on both base
point and moment p). The base curvature
d_C Gamma = kappa(p) dp_1 ^ dp_2
is positive, and its integral over Sigma,
A(p) = int over Sigma of kappa(p),
is assumed to converge uniformly to a continuous positive
limit H(p) as Sigma grows.
This "surface form" is exactly what the geometric models of
Section 7 provide.
THE LEMMA (surface packing).
Let r_1, ..., r_k > 0 be positive numbers, each simplex
{ p >= 0 : sum_j p_j <= r_i } strictly inside Delta and
strictly away from every outer face. Define the leftover
function
P(p) = H(p) - sum_i (r_i - sum_j p_j)_+ .
Suppose:
(i) P is concave;
(ii) P has positive average over Delta;
(iii) P is positive on Delta minus tau Delta.
Then for any capacities r_i' < r_i, some model (i.e. some
sufficiently large Sigma) contains a compact packing of the
closed balls of capacities r_i' in the interior of
Sigma x V, with neighborhood embeddings.
That is: the three hypotheses turn into an actual packing
into the model.
PROOF, IN FIVE STEPS.
STEP 1: shrink to a finite model.
Hamiltonian comparison (Section 5) provides K and q
with (K-P)(qu) < K(u). The uniform gap is
g = min_v [ K(q^{-1}v) - K(v) + H(p(v)) - C_r(p(v)) ]
where C_r(p) is the cap sum. g > 0 by the lemma's
hypotheses.
Choose 0 < eps < g/8 (this is very small). Take Sigma
so large that
|A(p) - H(p)| < eps
for all p in Delta (this is possible by uniform
convergence of A to H). From now on, "A" means the
finite base area, "H" the limiting area.
We will use the comparison based on the LIMITING area H,
not the finite A. This is why the lemma is stated in
terms of H.
STEP 2: place the base area on an annulus.
Inside the handle choose an annular region A^+ with a
positive coordinate s (going around the annulus) and an
angular coordinate t in R/Z. A closed one-form
beta on the base is supported in a transverse annulus
B, and on A^+ looks like
beta|A^+ = f(t) dt, f >= 0, integral f = 1.
So beta has period exactly 1 around the essential
circle. This is the "handle form"; its integral around
the handle is 1.
Now modify the base area. Pick a smooth smaller area
kappa_sm(p) that (a) agrees with kappa(p) near the base
boundary, (b) is independent of p on A^+, and (c) has
small total integral:
sup_p int_Sigma kappa_sm(p) < eps.
Let
H_*(p) = int_Sigma ( kappa(p) - kappa_sm(p) ),
kappa_1(p) = kappa_sm(p) + H_*(p) chi_A,
where chi_A is a bump on the annulus A with integral 1.
Then kappa_1 > 0, |H_* - H| < 2 eps, kappa_1 - kappa has
zero total integral. By a standard "relative area
primitive" lemma, we can normalize the base one-form
Gamma so that on the annulus
d_C Gamma_1 = kappa_1, Gamma_1|A = gamma + s H_* beta,
where d gamma = kappa_sm on the annulus. The total form
is still symplectic.
STEP 3: build the "layer path."
Choose smooth positive toric functions h_i on Delta with
h_i(p) > (r_i - sum_j p_j)_+,
0 < sum_i h_i - C_r < eps.
(This makes h_i slightly larger than the "cap" function
for the i-th ball.)
Define
J(v) = K(v) - H_*(p(v)),
R_*(v) = K(q^{-1}v) - J(v) - sum_i h_i(p(v)).
Then R_* >= g - 3 eps > 0 by the gap.
The gap says: on each fiber v, the s-coordinate (going
around the handle) can interpolate a certain coefficient
from J(v) at the lower end of the handle to K(q^{-1}v)
at the upper end, in steps of h_i(p(v)) with positive
leftover R_*(v).
Concretely, define a family of forms on A x V by
Omega_lambda = d gamma + D_lambda^*(omega_V + d(G_lambda beta))
where D_lambda is a Hamiltonian isotopy of the fiber
(moving from identity to q as the s-coordinate runs
from 0 to 1), and G_lambda is a scalar function on
A x V determined by the interpolation. The suspension
formula (standard in symplectic topology) shows that
Omega_lambda is symplectic whenever the s-derivative
of the scalar part is nonnegative.
At the ends, the form matches the model from Step 2,
so Omega_lambda extends globally.
Refine G from "just interpolating" to "interpolating
with marked layers": choose disjoint sub-intervals
I_1, ..., I_k of the s-axis, and a residual tail I_*.
Along I_i, increment G by h_i(p(v)) alone. Then the
form on layer i is
omega_V + d(G_f beta) with G_f = a_i(p) + rho_i(s) h_i(p).
The "a_i(p)" part contributes a toric Hamiltonian to the
fiber, and the "rho_i(s) h_i(p)" part contributes the
linear growth in s.
STEP 4: restore the original form via Moser.
The modified form is symplectic and has the same
boundary conditions on the base boundary. But we want
it back on the original model. Use a Moser isotopy:
- the family of forms has the same cohomology class
at each stage,
- the family preserves every outer face b_nu . p
(because the construction uses toric functions and
outer-face-commuting Hamiltonians),
- the base boundary collar is fixed.
The "Moser with outer faces" lemma guarantees an
isotopy that undoes the deformation while keeping the
base boundary collar fixed and (crucially) never
pushing any trajectory across an outer face.
So the final picture: a diffeomorphism psi from the
"layer-modified" model back to the original model.
STEP 5: embed one closed ball in each layer.
In the i-th layer, the form is exactly
omega_V(v') + dx ^ dT
after a coordinate change, where:
(s, T) are base coordinates,
x = L(s,T) + rho_i(s) h_i(p) (a base function),
v' = fiber coordinates with the p-parameters.
The region of this product form that is available in
the i-th layer is
0 < T < 1, 0 < x < h_i(p(v')).
Now we want to fit a closed ball of capacity r_i' < r_i.
A planar trick (a corollary of the nested-disk lemma):
for any A > r_i' and delta > 0, a neighborhood of the
closed disk of area A in the (x, T) plane embeds
symplectically so that
point of radial area w sits at
x < w + delta, 0 < T < 1.
Choose delta < r_i - r_i'. Then for every point of the
ball,
w + sum_j p_j(v') <= r_i' (by the ball's
definition),
x < w + delta < r_i - sum_j p_j(v') (subtract),
hence x < h_i(p(v')) (by h_i > cap).
So the ball fits strictly inside the layer with room to
spare. The layer is compactly contained in the strip,
and the strip is away from the outer faces. So the
closed ball is inside the layer with compact closure,
and the embedding extends to a neighborhood.
Different layers have disjoint base intervals I_i, so
their fiber images are automatically disjoint.
Compose with psi^{-1} from Step 4. Done: we have a
packing of the k closed balls inside the original model.
WHAT WE GET.
From the mean inequality of Section 5 and the handle of a
positive-genus surface, we get a concrete packing in an
explicit surface-plus-toric model. The packing stays away
from the outer faces of Delta, so it survives any further
deformation that respects those faces.
Section 7 builds such models inside the symplectic ball.
7. STEP C: GEOMETRIC MODELS AND TRANSFER
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Now we have to build surface-plus-fiber models inside the
target ball, and to move packings between them. The four
main tools live in geometry.tex.
7.1 RELATIVE KÄHLER MOSER
Classical Moser's method: given a smooth path of symplectic
forms omega_t = omega_0 + d alpha_t with all omega_t
symplectic and nondegenerate, integrate the vector fields
X_t defined by
iota_{X_t} omega_t = -alpha_t
into an isotopy psi_t with psi_t^* omega_t = omega_0.
On a compact complex manifold, this shows: two Kähler forms
in the same cohomology class are related by an isotopy
preserving the Kahler property.
The paper's RELATIVE version: if you also want to
preserve a list of disjoint smooth complex submanifolds
S_1, ..., S_e as SETS, you can arrange that psi_t(S_j) = S_j.
The trick: along each S_j, prescribe the first jet of a
corrector function f_t so that the Moser field is tangent to
S_j.
This is the tool that lets us "leak" past forbidden
divisors when we come back to the target ball. When we say
"remove H_infty from PP^n," we mean "after applying a
Moser isotopy that preserves both H_infty and the blowup
divisor E."
7.2 NORMAL-CONE TRANSFER
Setup. Let T be a smooth projective variety, Z a smooth
closed submanifold, L an ample rational polarization. Let
T' = Bl_Z T, Y = PP_Z( 1 + N_{Z/T} ),
where N_{Z/T} is the normal bundle of Z inside T, and
PP_Z(1 + N_{Z/T}) is the projective bundle obtained by
adding a trivial line to N_{Z/T}.
Choose rational d > 0. Suppose two classes are ample:
on T': pi^* L - d E_Z,
on Y: L|_Z + d U.
(Here E_Z is the exceptional divisor and U is the dual
tautological class.)
THE LEMMA.
Let F subset T be a finite union of disjoint smooth
complex submanifolds (a set we want to keep avoiding).
Let omega_T be any Kähler form in class L, and omega_Y
any Kähler form in class L|_Z + d U invariant under
scalar rotation of the normal summand of Y.
Then every compact subset of
Y minus ( D_infty union pi_Y^{-1}(F cap Z) )
has a neighborhood admitting a symplectic embedding
into
( T minus F, omega_T ),
with its given form omega_Y.
Here D_infty = PP_Z(N_{Z/T}) is the "infinity
section" of Y.
Idea of the proof.
Use the CLASSICAL construction of deformation to the
normal cone:
X = Bl_{Z x {0}} ( T x A^1 ) -> A^1.
Its fiber at t != 0 is T; its fiber at t = 0 is the
union of T' and Y, glued along D_infty.
Put a rational polarization L - d [Y] on the total
space X. It restricts to pi^* L - d E_Z on T' and
to L|_Z + d U on Y. Ampleness on each component
plus finiteness of the map
T' u Y (disjoint) -> central fiber
implies ampleness on the central fiber, which is an
open condition; so the total polarization is ample
near zero.
For any compact K in Y away from D_infty and
pi_Y^{-1}(F), the map Y -> T along the deformation
(i.e. flowing from Y into a nearby smooth fiber T_t)
is defined on a neighborhood of K. That flow is
our symplectic embedding.
In short: deformation to the normal cone is a "video"
that starts at Y (at t = 0) and ends at T (at t -> 0 but
nonzero). The construction carries symplectic structures
along the video.
Intuition for why this is useful. A projective manifold
has a "zoom into Z" limit where it becomes the projective
bundle Y. If Y is a nicer target (e.g. a toric model over
a curve), we can do the packing there and zoom back. This
is how the paper transfers packings from the "surface plus
toric fiber" model into the actual ball.
7.3 EXPLICIT NORMAL-BUNDLE FORM
Suppose Z is a curve (complex dimension 1) and
N_{Z/T} = A^b + A^b + ... + A^b (l summands),
where A is an ample line bundle on Z with Chern curvature
sigma > 0, and b > 0 is an integer. If bd < 1, the
class sigma + d U on Y = PP_Z(1 + N) has an explicit
Kahler representative of the shape
omega_Y = sigma + d( sum_{j=1}^l p_j alpha_j ),
p_j >= 0, sum_j p_j < d,
where alpha_j is the angular connection on the j-th normal
line (period one, curvature -b sigma), and p_j are the
standard fiber moments.
The (p_1, ..., p_l) coordinates are nonnegative and their
sum is bounded by d. So the fiber over each point of Z is
a polytope of ALLOWED moment values, which is exactly the
"Delta" of Section 5's toric domain V. This is the
precise sense in which Y is a "surface plus toric fiber"
model.
For Z = C a curve and Sigma subset C a bordered subsurface,
the model restricted to Sigma x V has a toric horizontal
primitive Gamma with base curvature
kappa(p) = ( 1 - b sum_j p_j ) sigma.
The integral of kappa over Sigma tends to (deg A) times
( 1 - b sum_j p_j ) as Sigma grows towards C. That is
exactly the limiting base area H(p) used by the surface
packing lemma (Section 6).
So the normal-bundle model Y is precisely a surface-plus-
fiber model, with H(p) = (deg A) (1 - b sum p_j). This is
the reason the constructions work.
7.4 BLOWUP COMPLEMENT IS A SHELL
Let n >= 2, o a point of PP^n, H_infty a hyperplane not
containing o. Let T = Bl_o PP^n, E the exceptional
divisor, H the pullback of the hyperplane class of PP^n.
For 0 < a < 1, the complement T minus (E union H_infty)
with its standard Kähler form in class H - aE is
symplectomorphic to the shell
{ z in C^n : a < pi sum_j |z_j|^2 < 1 }.
Proof sketch. Use the "symplectic cut" construction
(Lerman 1995): act on PP^n x C by a circle, reduce at a
suitable level, and you get exactly the blowup T with its
standard form. Removing E and H_infty leaves the shell.
A direct model-check then matches the affine Fubini-Study
coordinates to standard complex coordinates in C^n.
This is the punchline: "blowup a point, remove two
divisors, get the shell." The shell is precisely what
Cases 2 and 3 need for the reserved-ball reduction.
7.5 PUTTING IT TOGETHER
A model for the target often looks like this:
PP^n -> (blow up at o) T = Bl_o PP^n -> D subset T
(a curve or
a hyperplane)
|
v
Y = PP_D(1 + N_{D/T})
with the class L|_D + d U on Y. That class is ample by
choosing d small.
The surface-packing lemma (Section 6) works inside Y when
we can identify Y's base D as a punctured curve of positive
genus. So:
- pack into Y,
- transfer back to T via normal-cone transfer,
- identify T minus (E u H_infty) with a shell in C^n
via the blowup-complement lemma,
- add the reserved central ball if applicable.
That is the whole pipeline. Sections 8 flesh it out for
the three cases.
8. THE THREE CASES IN DETAIL
----------------------------------------------------------------
Normalize the target capacity to 1. In every case, we produce
an ample model with base a punctured positive-genus curve, whose
limiting base area H(p) has all the required properties.
8.1 CASE 1: ALL CAPACITIES BELOW 1/2
Take PP^n. Choose a hyperplane H_infty (a generic one). Pick
n-1 quadrics Q_1, ..., Q_{n-1} in general position. Their
intersection
C = Q_1 cap Q_2 cap ... cap Q_{n-1} subset PP^n
is a smooth connected curve of degree 2^{n-1} (Bertini's
theorem ensures this for a generic choice of quadrics). Its
genus is
g(C) = 1 + (n-3) * 2^{n-2},
which is 1 for n=3, 5 for n=4, 17 for n=5, and so on. In any
case, POSITIVE for every n >= 3.
Its NORMAL bundle in PP^n is ( O_C(2) )^{oplus (n-1)}: one
degree-2 line bundle per quadric. Each quadric gives a
section of O(1) that vanishes precisely on C, so the normal
direction of C inside Q_1 cap ... cap Q_j has class the
corresponding O(2).
Deform to normal cone. T = PP^n, Z = C, L = O(1). The
resulting model is
Y = PP_C( 1 + O_C(2)^{oplus (n-1)} ),
with class L|_C + d U. Ampleness of the two components
(Bl_C PP^n with pi^*L - d E_C, and Y with L|_C + d U) holds
for 0 < d < 1/2, which is standard (adjunction in
PP^n).
The base area (from the explicit normal-bundle form, Section
7.3) is
H_0(p) = 2^m ( 1 - 2 sum_{j=1}^m p_j ), m = n-1.
(The 2^m is deg O_C(1) = 2^{n-1} = 2^m, the degree of C as
a curve in PP^n. The factor (1 - 2 sum p_j) is 1 - b sum p_j
with b = 2, one per summand of the normal bundle O_C(2).
Also m = n-1 is the number of normal summands.)
The truncation is at sum p_j <= h for small h. Its integral
is
J_m(h) = int_{sum p_j <= h} 2^m ( 1 - 2 t ) dp
= (2^m h^m / m!) ( 1 - 2mh / (m+1) )
which tends to 1 / n! as h -> 1/2.
And 1 / n! is EXACTLY the volume of the unit ball B^{2n}(1).
So as h approaches 1/2, the model captures ALL of the volume.
Fixing parameters. Given requested capacities R_1, ..., R_k
all < 1/2 with sum R_i^n < 1, choose rationals
r_* < h < d < 1/2,
where r_* = max r_i with each r_i chosen in
(R_i, 1/2), and J_m(h) > sum r_i^n / n!.
These exist because J_m(h) -> 1/n! as h -> 1/2, and the
volume inequality is strict.
Now define P(p) = H_0(p) - sum_i (r_i - t)_+, t = sum p_j.
Check:
concavity: H_0 is affine in t, each cap (r_i - t)_+
is convex, so P is concave.
mean positive: int P dp = J_m(h) - sum r_i^n/n! > 0.
boundary: taking tau with r_*/h < tau < 1, we
have that for p outside tau Delta,
all caps vanish, and H_0(p) >= positive.
So all hypotheses of the surface packing lemma hold.
Apply surface packing lemma (Section 6) with parameters
r_i' < r_i. Get a compact packing of the closed balls of
capacities r_i' inside one model Y.
The packing lives away from D_infty (the infinity section
of Y) and away from pi_Y^{-1}(C cap H_infty). Therefore
it transfers back into PP^n minus H_infty via normal-cone
transfer.
PP^n minus H_infty is the open ball B^{2n}(1) via
affine Fubini-Study coordinates (the map
z -> z / sqrt(pi (1 + sum |z|^2))
is a symplectomorphism). So the packing lands in the
desired target.
Neighborhood statement: all constructions are compact and
defined on neighborhoods; shrink at each composition. Since
R_i < r_i' < r_i, we have the required slack.
8.2 CASE 2: ONE LARGE CAPACITY, DIMENSION n >= 4
Let A >= 1/2 be the large capacity. Choose rationals
1/2 < a < 1, A < a, s = 1 - a.
The reserved central ball has capacity A < a. The remaining
balls (R_1, ..., R_k) must fit into the shell
{ a < pi sum |z_j|^2 < 1 }.
For each i, choose r_i > R_i so that r_i < s and
sum r_i^n < 1 - a^n.
Construct a "two-stage" model.
Stage 1: PP^n -> hyperplane D = PP^{n-1} -> normal cone.
Let T = Bl_o PP^n, L = H - aE. Choose a hyperplane D
not containing o. Deform to normal cone of D. Component
class on Bl_D T is L - d_1 D, ample for d_1 < s (this
is standard for blowups at a point of projective space).
Component class on Y_1 = PP_D(1 + O_D(1)) is L|_D +
d_1 U_1, ample for d_1 < 1 - a = s. Pick rationals
r_* < h_1 < d_1 < s.
The base area function is
H_0^(1)(u) = 1 - u, u = first moment on D,
i.e. deg(O_D(1)) = 1.
Stage 2. Inside D = PP^{n-1}, run Case 1 with the
appropriate dimension (n-1 instead of n, so n-1 >= 3
requires n >= 4). The curve
C = intersection of (n-2) generic quadrics in D
is smooth of genus g(C) = 1 + (n-4) * 2^{n-3}, which
is 1 for n=4, 5 for n=5, 17 for n=6, and so on. Positive
for every n >= 4.
The stage-2 model is Y_2 = PP_C(1 + O_C(2)^{oplus
(n-2)}), with base area
H_0^(2)(v) = 2^{n-2} ( 1 - 2 sum_{j=2}^{n-1} v_j ),
v = (v_2, ..., v_{n-1}) the second-stage moments.
The truncation parameters are r_* < h_2 < d_2 < 1/2.
Combining the two stages. The stage-2 model pulls back
into the stage-1 fiber (Stage 1 gives a ball for the first
normal direction over D, then Stage 2 gives another toric
ball inside D). After the "moment change" coordinate
(z_1, z_2, ..., z_{n-1}) with u = pi |z_1|^2,
v_j = pi |z_j|^2 (j >= 2), define
zeta_1 = z_1, zeta_j = sqrt(1 - u) z_j (j >= 2),
with new moments
p_1 = u, p_j = (1 - u) v_j (j >= 2).
This is a smooth diffeomorphism on u < 1 including all axes.
Set w = p_2 + ... + p_{n-1}. The combined polytope is
Delta(h_1, h_2) = { p >= 0 : p_1 <= h_1,
w + h_2 p_1 <= h_2 }.
Combined base area function (verified in Section 5 of
applications.tex):
H_0(p) = 2^{n-2} ( 1 - p_1 - 2w ).
Its coefficient (1 - p_1 - 2 w) stays positive on Delta
because Delta has p_1 <= h_1 < 1 and w + h_2 p_1 <= h_2,
so w <= h_2 (1 - p_1) and 1 - p_1 - 2w
>= (1 - p_1)(1 - 2 h_2) >= (1 - h_1)(1 - 2 h_2) > 0
when h_1 < 1 and h_2 < 1/2.
Volume. The integral of H_0 over Delta(h_1, h_2) is
B_q(h_2) * ( 1 - (1 - h_1)^n ) / n,
where
B_q(h_2) = 2^q ( h_2^q / q! - 2q h_2^{q+1} / (q+1)! )
-> 1 / (q+1)! as h_2 -> 1/2,
q = n - 2. So (B_q vol) -> ( 1 - a^n ) / n! as
h_1 -> s and h_2 -> 1/2.
And (1 - a^n) / n! is EXACTLY the volume of the shell
{ a < pi sum |z_j|^2 < 1 }.
Same concavity / mean / outer-region checks as Case 1. The
cap functions are (r_i - p_1 - w)_+ and the "+" is still
concave. Choose truncations so that int P dp > 0, and
r_*/h_1, r_*/h_2 are both < tau < 1. Then P is positive
on Delta minus tau Delta.
Apply surface packing lemma to get a compact packing inside
Y_2. Then transfer:
Y_2 -> (embed via phi into) D minus (C cap H_infty)
-> (transfer to) Y_1 minus (its infinity + bad set)
-> (transfer to) T minus (E union H_infty)
-> (identify) shell in C^n.
Add the reserved central ball of capacity A < a at the
center. All disjoint, all with compact closure, all with
neighborhood embeddings.
8.3 CASE 3: ONE LARGE CAPACITY, DIMENSION n = 3
Same setup as Case 2, but n = 3 so the plane inside PP^3
is PP^2, and a conic in PP^2 has genus 0. No handle.
Replace conic by a plane CUBIC curve C through two
distinguishable points q_1, q_2. Such C has genus 1.
Moreover, C minus { q_1, q_2 } still has (non-compact)
genus 1. So the surface packing lemma still works, with
C^o = C minus { two points and possibly extra points }.
Extra wrinkle: we need to cut the moment at p_1 = c
for c < s, to allow Stage 1 and Stage 2 to fit together.
This is a KÄHLER CUT.
Setup for the cut.
V_hat = Bl_{(q_1,0), (q_2,0)} PP_{PP^2}(1 + O_{PP^2}(3)).
This is a projective bundle over PP^2 with a line
bundle O(3), blown up at two points.
The standard circle rotating the O(3) summand lifts to
V_hat. Its Hamiltonian is p_1 (the first moment).
Normalize p_1 = 0 on the "bottom" (the strict transform
of the zero section).
The fixed loci of the circle are at levels:
0, s, d_1,
where s = 1 - a (mass of the exceptional curves) and
d_1 the ordinary fiber mass. Both are positive for
0 < c < s.
The cut.
Take the Kähler reduction of V_hat x C by the
diagonal circle at level
p_1 + pi |v|^2 = c.
Because c < s, every point of the level lies over the
attracting basin of the bottom section. The complex
quotient is a projective bundle
Y = PP_D( 1 + N_{D/V_hat} ), D = Bl_{q_1,q_2} PP^2,
with class L|_D + c U_c. This Y is a projective bundle
over D (a plane surface blown up at two points) with
normal bundle
N_{D/T} = O_D(3M - e_1 - e_2),
in classes M = pullback of plane hyperplane class,
e_i = exceptional divisors over q_1, q_2.
Ample classes: L = 3b M - s (e_1 + e_2) with
b = (2 - a)/3 = (1 + s)/3, s = 1 - a, and L is the
class on D.
Now Stage 1 is a quadric-surface normal model (different
from Case 2's hyperplane):
D = Bl_{q_1,q_2} PP^2,
Y_1 = PP_D(1 + O_D(3M - e_1 - e_2)).
Stage 2 is a cubic curve model living in D:
C = strict transform in D of a smooth plane cubic
through q_1, q_2.
Y_2 = PP_C(1 + O_C(3)) is a toric model over C.
deg_C O(3) = 9.
Base area function (Section 4 of sixdim.tex).
The full projective system on the cubic has the
curve-vanishing weight computed as follows. Two
marked points, each contributing one logarithmic
mass (s - p_1 - p_2)_+ to the base area. And
deg O_C(3) = 9, so the base area is
H_0(p) = 9 ( b - p_1 - p_2 ) - 2 ( s - p_1 - p_2 )_+.
Here b = (1 + s) / 3 as above.
The 9 b is the constant coefficient at zero moment:
it comes from deg O_C(3) = 9 times the parameter b.
The -9 p_1 - 9 p_2 is the base area loss per unit
moment. The -2(s - t)_+ is the loss due to the two
marked-point atoms, each missing mass (s - t)_+.
Volume.
Integrate over Delta_infty = { 0 <= p_1 <= s,
0 <= p_1 + p_2 <= b }:
int H_0(p) dp = (1 - a^3) / 6,
which is EXACTLY the volume of the shell
{ a < pi sum |z_j|^2 < 1 } in complex dimension 3.
(The computation is direct integration in one variable.)
Concavity of H_0.
As a function of t = p_1 + p_2, H_0 is piecewise affine
with slopes -7 for t < s and -9 for t > s. Concave.
Apply surface packing lemma with truncated polytope
Delta = { p >= 0 : p_1 <= h_1, p_1 + p_2 <= h_2 },
for h_1 near s and h_2 near b. The cap supports are
inside Delta; there is enough volume for all balls.
Transfer chain:
cubic model Y_2 ->
moment-preserving parallel transport into V_hat
(p_1 <= c preserved exactly) ->
lower Kähler cut at p_1 = c ->
Y_1 = PP_D(1 + O_D(3M - e_1 - e_2)) ->
normal-cone transfer into T = Bl_o PP^3 minus
(E union H_infty) ->
identified with shell in C^3 ->
add reserved central ball of capacity A < a.
All disjoint, all compact, all with neighborhood embeddings.
8.4 WHY THESE THREE COVER EVERYTHING
Given R_1, ..., R_k with sum R_i^n < 1 and all pairs
R_i + R_j < 1:
- if k = 1, put it by inclusion.
- if all R_i < 1/2, use Case 1.
- otherwise exactly one is >= 1/2 (uniqueness from
the two-ball inequality). Call it A.
- if n >= 4, use Case 2.
- if n = 3, use Case 3.
In every case, the assumed hypotheses give exactly the
properties needed (concavity, positive mean, boundary gap).
So the sufficiency direction of the theorem is proven.
9. WHY DIMENSION FOUR IS DIFFERENT
----------------------------------------------------------------
The theorem is for n >= 3, i.e. 6 real dimensions and up. What
about n = 1 (2 real dimensions) and n = 2 (4 real dimensions)?
n = 1 (ordinary plane, real dimension 2):
Symplectic = area preserving. Packing is purely a matter
of area; the criterion is just sum R_i < R. No pairwise
condition needed. Nothing interesting.
n = 2 (real dimension 4):
This is the richest case. Gromov proved the two-ball
obstruction in 1985. McDuff and Polterovich (1994) then
showed that volume + pairwise is NOT sufficient: there
are infinitely many additional obstructions, coming from
exceptional classes in blowups of CP^2 and their
J-holomorphic curves. These are families of "rigid rulers"
that further restrict packing configurations.
Biran (1997, 1999) established "packing stability" in this
dimension: on a closed symplectic four-manifold with rational
symplectic class, any sufficiently large number of EQUAL
balls can fill an arbitrarily large fraction of the volume.
Buse-Hind (2011, 2013) extended equal-ball stability to
higher-dimensional balls and to all closed rational
symplectic manifolds. Buse-Hind-Opshtein (2016) proved
strong stability for unequal balls, provided all capacities
are sufficiently small.
But the sharp criterion (exactly which configurations fit)
remains an open, and much harder, question in dimension 4.
n >= 3 (6 real dimensions and up):
The extra dimensions give "floppiness." The paper proves
that volume + two-ball IS sufficient. All other
obstructions disappear. So the answer is exactly:
pack iff (sum R_i^n) < R^n AND (all R_i+R_j < R).
This is a strange but common phenomenon in high-dimensional
geometry: things that are rigid in low dimensions become
flexible once there is enough room.
10. WHAT THE PROOF DOES AND DOES NOT GIVE
----------------------------------------------------------------
The proof is a pure existence proof. Specifically:
It tells you a packing EXISTS.
It does NOT tell you an algorithm for constructing one.
It does NOT tell you the shapes of the packed balls.
It does NOT give numerical effective bounds on how small
the neighborhoods around the closed balls can be.
That last point is why every "embedding" here means "symplectic
embedding defined on an open neighborhood of the closed source
ball." This is the standard convention, but it's worth
being explicit about.
What the proof does open up:
- The surface-plus-toric-fiber machine (Sections 5-6) is
quite general. Similar constructions (symplectic folding,
Moser-type normalizations, positive-genus bases) appear
across symplectic topology.
- The interplay between packing and the algebraic structure
of the Hamiltonian diffeomorphism group (Edtmair 2025)
was a motivation. The paper's argument doesn't use those
results, but the connection is real.
- The normal-cone transfer with a lower Kähler cut
(Section 7.2 / 8.3) is a reusable device.
What the proof does NOT settle:
- Dimension four. As discussed, the criterion here is
different (richer set of obstructions) and is still an
active research area.
- Stabilized packing: packings after crossing with a fixed
symplectic surface. Siegel and Yao's original paper
studied these too; they are not resolved by the present
theorem.
- Complex dimension 1 and 2 - trivially resolved and
richer-respectively, as noted above.
11. GLOSSARY
----------------------------------------------------------------
Ball of capacity R:
B^{2n}(R) = { z in C^n : pi sum |z_j|^2 <= R }.
Radius sqrt(R/pi). Volume R^n / n!.
Capacity, symplectic:
The natural "area" unit. A capacity-one ball has a complex
line through the origin of symplectic area exactly one.
Symplectic form omega_0:
The real 2-form sum_j dx_j ^ dy_j on C^n = R^{2n}. It
measures signed area on each complex coordinate plane and
sums those areas.
Symplectic embedding:
A smooth injection f : U -> U' with f^* omega_0 = omega_0.
Volume-preserving embedding:
A smooth injection with f^* vol = vol. Much weaker than
symplectic in dimension >= 2n >= 4. The gap between the
two is what symplectic geometry is about.
Gromov's two-ball obstruction:
In any n >= 2, symplectic balls of capacities R_1, R_2
cannot both fit disjointly into capacity R unless
R_1 + R_2 < R.
J-holomorphic curve:
A map from a Riemann surface (usually S^2) into a
symplectic manifold that is holomorphic with respect to a
compatible J. These are the "rigid rulers" that detect
non-volume obstructions.
Conjecture A (Siegel-Yao 2025):
For n >= 3, volume + two-ball is sufficient. Now a
theorem.
Toric domain:
V = { z in C^m : p(z) in Delta } for a "downward" polytope
Delta in the nonnegative orthant. Standard form:
omega_V = sum dp_j ^ d theta_j.
Outer face:
A face of Delta of the form b_nu . p = c_nu with c_nu > 0.
The coordinate faces p_j = 0 are not outer.
Hamiltonian comparison (Section 5):
The lemma that upgrades a positive-mean inequality into a
strict pointwise inequality via increasing rearrangements
and a contractive fixed-point argument. Output: K (toric)
and q (Hamiltonian) with (K-P) o q < K.
Increasing rearrangement:
Given a convex function f on [0,1], its increasing
rearrangement g is the (unique up to values) function with
g nondecreasing and the level sets of g having the same
length as those of f. Formula:
g(h) = min_{0 <= a <= 1-h} max{ f(a), f(a+h) }.
Surface packing lemma (Section 6):
Packs disjoint closed balls in a surface-plus-toric model,
given a concave P with positive mean and a boundary
positivity property, and given a positive-genus punctured
base with a handle-form beta of period 1.
Handle form beta:
A closed one-form on the surface with int(beta) = 1 around
an essential circle in the handle. This is the mechanism
that creates disjoint "layers" along one coordinate.
Deformation to the normal cone:
A one-parameter family over A^1 whose fiber at t != 0 is
a given T and whose fiber at t = 0 splits into the blowup
Bl_Z T and the projectivized normal bundle
PP_Z(1 + N_{Z/T}).
Used to move packings between these pieces.
Normal-cone transfer (Section 7.2):
The lemma that transports a compact packing from
PP_Z(1 + N_{Z/T}) into T itself, via the deformation to
the normal cone, while avoiding F subset T.
Moser isotopy (Section 7.1):
Given a smooth path of symplectic forms omega_t = omega_0 +
d alpha_t, integrate the vector fields X_t defined by
iota_{X_t} omega_t = -alpha_t into an isotopy psi_t with
psi_t^* omega_t = omega_0. The engine behind "symplectically
equalize two forms in the same cohomology class."
Relative Kähler Moser:
The version of Moser that preserves a list of complex
submanifolds as sets. Uses a normal first-jet correction
along each S_j.
Kähler cut (symplectic cut):
Take V x C, act by the standard diagonal circle, reduce at
a circle-invariant level p_1 + pi |v|^2 = c. The result
is a symplectic quotient. Used in Case 3 to build the
critical projective bundle Y over the two-point blowup of
PP^2.
Complete intersection of quadrics:
The set { Q_1 = ... = Q_{n-1} = 0 } in PP^n for n-1
general homogeneous quadrics. Smooth of degree 2^{n-1},
genus 1 + (n-3) 2^{n-2}, normal bundle
O_C(2)^{oplus (n-1)}. This is the "handle" for Cases 1
and 2.
Shell:
{ z in C^n : a < pi sum |z_j|^2 < 1 }. Symplectically
equal to Bl_o PP^n minus (E union H_infty) for suitable
standard classes. The target of the large-ball reduction.
12. A SUGGESTED READING ORDER FOR THE PAPER
----------------------------------------------------------------
The technical paper (symplectic_ball_packings_original.txt) is
organized in this order, with the main files as follows:
intro.tex statement of Theorem + history + sketch
comparison.tex Hamiltonian comparison; rearrangement; disks
surface.tex surface packing; handle form; layers; Moser
geometry.tex Kähler Moser; normal-cone transfer; explicit
bundle form; blowup-complement-is-shell
applications.tex Case 1 (all small); Case 2 (n >= 4, large)
sixdim.tex Case 3 (n = 3, large); cubic; Kähler cut
conclusion.tex sufficiency assembly
bibliography.tex references
A good first-pass reading order:
1. intro.tex, sections 1-2 (statement + easy necessity).
2. comparison.tex, statement of Lemma cmp:comparison and
its fixed-point proof.
3. surface.tex, statement of Lemma surf:packing and Step 5
(how the walls of a layer hold a closed ball).
4. geometry.tex, statement of Lemma geo:transfer and Lemma
geo:shell ("normal-cone transfer" and
"blowup-complement-is-shell").
5. applications.tex, Proposition app:small (Case 1).
6. applications.tex, Proposition app:large (Case 2).
7. sixdim.tex, Proposition six:large (Case 3).
8. conclusion.tex.
Sections 3-4 of the paper play a supporting role (relative
Kähler Moser, projective transfer). They can be skimmed on a
first reading.
For a truly thorough reading, pay attention to:
- the two "regularity" tricks in the surface packing lemma
(Step 2: relative area primitive; Step 3: layer path);
- why the concavity of P is preserved under the fixed-point
iteration;
- why the cubic model in Case 3 needs deg O_C(3) = 9 and
not 3.
Each of these is where a sloppy proof would break.
13. THE ONE-LINE MORAL
----------------------------------------------------------------
Symplectic geometry is more than volume. It measures
2-dimensional signed area on every coordinate plane, and this
strict structure produces extra obstructions you might not
expect (Gromov's two-ball rules).
But those extra obstructions are tameable in dimension >= 6.
Once n >= 3, only volume and the two-ball rule survive, and
the paper shows they are SUFFICIENT. The construction builds
surface-plus-toric models inside the target, reshuffles fiber
coordinates via a Hamiltonian isotopy to produce disjoint
layers, and transports the resulting packing back to the ball
via normal-cone degenerations and Moser isotopies.
The final answer, in every dimension n >= 3, for every k, and
for every choice of positive capacities:
pack iff (sum R_i^n < R^n) and (all R_i + R_j < R).
Not one condition more. Not one condition less.
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