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”GSI’15”
École Polytechnique, October 28, 2015
Heights of toric varieties, entropy
and integration over polytopes
José Ignacio Burgos Gil, Patrice Philippon & Martı́n Sombra
Patrice Philippon, IMJ-PRG
UMR 7586 - CNRS
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Toric varieties
Toric varieties form a remarkable class of algebraic varieties,
endowed with an action of a torus having one Zariski dense open
orbit. Toric divisors are those invariant by the action of the torus.
Together with their toric divisors, they can be described
in terms of combinatorial objects such as lattice fans, support
functions or lattice polytopes
(u1 ,u2 )7→0
(u1 ,u2 )7→−u1
(u1 ,u2 )7→−u2
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Each cone corresponds to an affine toric variety and the fan
encodes how they glue together. If the fan is complete then the
toric variety is proper.
The support function determines a toric divisor D on each
affine toric chart. By duality, the stability set of the support
function is a polytope ∆, which may be empty but which is of
dimension n as soon as D is nef, which is equivalent to the
support function being concave.
One fundamental result is: if D is a toric nef divisor then
degD (X) = n!voln(∆).
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Heights
A height measures the complexity of objects over the field of
rational numbers, say. For a/b ∈ Q× and d = gcd(a, b):
h(a/b) = log max(|a/d|, |b/d|) =
X
log max(|a|v , |b|v ),
v
thanks to the product formula:
Y
|d|v = 1
v
for any d ∈ Q× and where v runs over all the (normalised)
absolute values on Q (usual and p-adic).
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Heights
A height measures the complexity of objects over the field of
rational numbers, say. For a/b ∈ Q× and d = gcd(a, b):
X
h(a/b) = log max(|a/d|, |b/d|) =
log max(|a|v , |b|v ),
v
thanks to the product formula:
Q
×
|d|
=
1,
d
∈
Q
.
v
v
For points of a projective space x = (x0 : . . . : xN ) ∈ PN (Q):
X
X
h(x) =
log kxkv = −
log k`(x)kv ,
v
v
where k·kv is a norm on QN +1 compatible with the absolute value
v
|·|v on Q (usual or p-adic). Metrics on OPN (1): k`(x)kv = |`(x)|
kxkv .
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On an abstract variety equipped with a divisor (X, D),
defined over Q, the suitable arithmetic setting amounts to a
collection of metrics on the space of rational sections of the
divisor, compatible with the absolute values on Q (the collection
is in bijection with the set of absolute values on Q). We denote
D the resulting metrised divisor.
Arithmetic intersection theory allows to define the height of
X relative to D analogously to the degree degD (X):
X
hD (X) =
hv (X)
v
where the local heights hv are defined through an arithmetic
analogue of Bézout formula. Local heights depend on the choice
of auxiliary sections but the global height does not.
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Metrics on toric varieties
On toric divisors, a metric is said toric if it is invariant by
the action of the compact sub-torus of the principal orbit.
There exists a bijection between toric metrics and continuous
functions on the fan, whose difference with the support function
is bounded. The metric is semipositive iff the corresponding
function is concave.
By Legendre duality, the semipositive toric metrics are also in
bijection with the continuous, concave functions on the polytope
associated to the toric divisor, dubbed roof function.
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The roof function is the concave enveloppe of the graph
of the function s 7→ − log kskv,sup, for s running over the toric
sections of the divisor and its multiples.
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Roof function of the pull-back of the canonical metric of P2 on P1 by t7→( 1
t : 2 :t)
v=2
v=∞
v=other
The support function itself corresponds to the so-called
canonical metric. Its roof function is the zero function on the
polytope.
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Heights on toric varieties
Let (X, D) be a toric varieties with a toric divisor (over
Q), equipped with a collection of toric metrics (a toric metrised
divisor).
The (local) roof functions attached to the toric metrised
divisor sum up in the so-called global roof function:
X
ϑ :=
ϑv .
v
We have the analogue of the formula seen for the degree:
Z
hD (X) = (n + 1)!
ϑ.
∆
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Metrics from polytopes
Let `F (x) = hx, uF i + `F (0) be the linear forms defining a
polytope Γ ⊂ Rn, with F running over its facets and kuF k =
voln−1(F )
nvoln(Γ) . Let ∆ ⊂ Γ be another polytope, the restriction of
1X
ϑ := −
`F log(`F )
c
F
to ∆, is the roof function of some (archimedean) metric on the
toric variety X and divisor D defined by ∆, hence D.
Example: the roof function of the Fubini-Study metric on Pn is
−(1/2)(x0 log(x0) + . . . + xn log(xn))
Pn −2u 1
where x0 = 1 − x1 − . . . − xn (dual to − 2 log 1 + i=1 e i ).
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Height as average entropy
Let x ∈ Γ and βx be the (discrete) random variable that
maps y ∈ Γ to the face F of Γ such that y ∈ Cone(x, F ):
voln−1(F )
P (βx = F ) = dist(x, F )
.
nvoln(Γ)
x
•
∆
Γ
F
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Height as average entropy
Let x ∈ Γ and βx be the (discrete) random variable that
maps y ∈ Γ to the face F of Γ such that y ∈ Cone(x, F ):
voln−1(F )
P (βx = F ) = dist(x, F )
.
nvoln(Γ)
The entropy
E(βx) = −
X
P (βx = F ) log(P (βx = F ))
F
satisfies
1
·
voln(∆)
Z
hD (X)
c
E(βx)dvoln(x) =
·
.
n + 1 degD (X)
∆
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Integration over polytopes
An aggregate of ∆ in a direction u ∈ Rn is the union of
all the faces of ∆ contained in {x ∈ Rn | hx, ui = λ} for some
λ ∈ R.
Definition – Let V be an aggregate in the direction of u ∈ Rn,
we set recursively: If u = 0, then Cn(∆, 0, V ) = voln(V ) and
Ck (∆, 0, V ) = 0 for k 6= n. If u 6= 0, then
Ck (∆, u, V ) = −
X huF , ui
F
kuk2
Ck (F, πF (u), V ∩ F ),
where the sum is over the facets F of ∆. This recursive formula
implies that Ck (∆, u, V ) = 0 for all k > dim(V ).
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Proposition [2, Prop.6.1.4] – Let ∆ ⊂ Rn be a polytope of
dimension n and u ∈ Rn. Then, for any f ∈ C n(R),
dim(V )
Z
f (n)(hx, ui)dvoln(x) =
∆
X
X
V ∈∆(u)
k=0
Ck (∆, u, V )f (k)(hV, ui).
The coefficients Ck (∆, u, V ) are determined by this identity.
Tn
Example: If ∆ = Conv(ν0, . . . , νn) = i=0{x; hx, uii ≥ λi} is a
simplex and u ∈ Rn \ {0}, then C0(∆, u, ν0) equals
ε det(u1, . . . , un)n−1
n!voln(∆)
Qn
= Qn
,
i=1hν0 − νi, ui
i=1 det(u1, . . . , ui−1, u, ui+1, . . . , un)
with ε the sign of (−1)n det(u1, . . . , un).
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References
[1] G.Everest & T.Ward, Heights of Polynomials and entropy in
Algebraic Dynamics, Universitext, Springer Verlag (1999).
[2] J.I.Burgos Gil, P.Philippon & M.Sombra, Arithmetic
geometry of toric varieties. Metrics, measures and heights,
Astérisque 360, Soc. Math. France, 2014.
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[3] J.I.Burgos Gil, A.Moriwaki, P.Philippon & M.Sombra,
Arithmetic positivity on toric varieties, J. Algebraic Geom.,
2016, to appear, e-print arXiv:1210.7692v3.
[4] J.I.Burgos Gil, P.Philippon & M.Sombra, Successive minima
of toric height functions, Ann. Inst. Fourier, Grenoble, 2015,
to appear, e-print arXiv:1403.4048v2.
Ouf!
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