Post on 23-Jul-2020
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic Curves
Matthew Baker’s Project Group 3:The Poincare-Lelong Formula
Xander Faber, Andrew Obus, Jorge Pineiro, Jerome Poineau,Christian Wahle
March 20, 2007
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
Throughout, K will denote an algebraically closed field completewith respect to a nontrivial nonarchimedean norm.
I P1Berk is a tree, infinitely branched at a dense set of points.
The points of P1Berk can be classified into 4 types, of which we
will primarily focus on 3:I Type I points are the classical points of P1(K ).I The set of type II and III points will be denoted by HR
Berk.
I There is a canonical metric on HRBerk. For example, if
ζ0,a, ζ0,b ∈ HRBerk correspond to balls in K centered at zero
with radii a < b, respectively, then
ρ(ζ0,a, ζ0,b) = logv b − logv a.
I An extended real-valued function on P1Berk is continuous
piecewise affine (CPA) roughly if, on small paths [x , y ], it is ofthe form t 7→ Aρ(x , t) + B for some A,B ∈ R.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
Throughout, K will denote an algebraically closed field completewith respect to a nontrivial nonarchimedean norm.
I P1Berk is a tree, infinitely branched at a dense set of points.
The points of P1Berk can be classified into 4 types, of which we
will primarily focus on 3:I Type I points are the classical points of P1(K ).I The set of type II and III points will be denoted by HR
Berk.
I There is a canonical metric on HRBerk. For example, if
ζ0,a, ζ0,b ∈ HRBerk correspond to balls in K centered at zero
with radii a < b, respectively, then
ρ(ζ0,a, ζ0,b) = logv b − logv a.
I An extended real-valued function on P1Berk is continuous
piecewise affine (CPA) roughly if, on small paths [x , y ], it is ofthe form t 7→ Aρ(x , t) + B for some A,B ∈ R.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
Throughout, K will denote an algebraically closed field completewith respect to a nontrivial nonarchimedean norm.
I P1Berk is a tree, infinitely branched at a dense set of points.
The points of P1Berk can be classified into 4 types, of which we
will primarily focus on 3:
I Type I points are the classical points of P1(K ).I The set of type II and III points will be denoted by HR
Berk.
I There is a canonical metric on HRBerk. For example, if
ζ0,a, ζ0,b ∈ HRBerk correspond to balls in K centered at zero
with radii a < b, respectively, then
ρ(ζ0,a, ζ0,b) = logv b − logv a.
I An extended real-valued function on P1Berk is continuous
piecewise affine (CPA) roughly if, on small paths [x , y ], it is ofthe form t 7→ Aρ(x , t) + B for some A,B ∈ R.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
Throughout, K will denote an algebraically closed field completewith respect to a nontrivial nonarchimedean norm.
I P1Berk is a tree, infinitely branched at a dense set of points.
The points of P1Berk can be classified into 4 types, of which we
will primarily focus on 3:I Type I points are the classical points of P1(K ).
I The set of type II and III points will be denoted by HRBerk.
I There is a canonical metric on HRBerk. For example, if
ζ0,a, ζ0,b ∈ HRBerk correspond to balls in K centered at zero
with radii a < b, respectively, then
ρ(ζ0,a, ζ0,b) = logv b − logv a.
I An extended real-valued function on P1Berk is continuous
piecewise affine (CPA) roughly if, on small paths [x , y ], it is ofthe form t 7→ Aρ(x , t) + B for some A,B ∈ R.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
Throughout, K will denote an algebraically closed field completewith respect to a nontrivial nonarchimedean norm.
I P1Berk is a tree, infinitely branched at a dense set of points.
The points of P1Berk can be classified into 4 types, of which we
will primarily focus on 3:I Type I points are the classical points of P1(K ).I The set of type II and III points will be denoted by HR
Berk.
I There is a canonical metric on HRBerk. For example, if
ζ0,a, ζ0,b ∈ HRBerk correspond to balls in K centered at zero
with radii a < b, respectively, then
ρ(ζ0,a, ζ0,b) = logv b − logv a.
I An extended real-valued function on P1Berk is continuous
piecewise affine (CPA) roughly if, on small paths [x , y ], it is ofthe form t 7→ Aρ(x , t) + B for some A,B ∈ R.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
Throughout, K will denote an algebraically closed field completewith respect to a nontrivial nonarchimedean norm.
I P1Berk is a tree, infinitely branched at a dense set of points.
The points of P1Berk can be classified into 4 types, of which we
will primarily focus on 3:I Type I points are the classical points of P1(K ).I The set of type II and III points will be denoted by HR
Berk.
I There is a canonical metric on HRBerk. For example, if
ζ0,a, ζ0,b ∈ HRBerk correspond to balls in K centered at zero
with radii a < b, respectively, then
ρ(ζ0,a, ζ0,b) = logv b − logv a.
I An extended real-valued function on P1Berk is continuous
piecewise affine (CPA) roughly if, on small paths [x , y ], it is ofthe form t 7→ Aρ(x , t) + B for some A,B ∈ R.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
Throughout, K will denote an algebraically closed field completewith respect to a nontrivial nonarchimedean norm.
I P1Berk is a tree, infinitely branched at a dense set of points.
The points of P1Berk can be classified into 4 types, of which we
will primarily focus on 3:I Type I points are the classical points of P1(K ).I The set of type II and III points will be denoted by HR
Berk.
I There is a canonical metric on HRBerk. For example, if
ζ0,a, ζ0,b ∈ HRBerk correspond to balls in K centered at zero
with radii a < b, respectively, then
ρ(ζ0,a, ζ0,b) = logv b − logv a.
I An extended real-valued function on P1Berk is continuous
piecewise affine (CPA) roughly if, on small paths [x , y ], it is ofthe form t 7→ Aρ(x , t) + B for some A,B ∈ R.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
Throughout, K will denote an algebraically closed field completewith respect to a nontrivial nonarchimedean norm.
I P1Berk is a tree, infinitely branched at a dense set of points.
The points of P1Berk can be classified into 4 types, of which we
will primarily focus on 3:I Type I points are the classical points of P1(K ).I The set of type II and III points will be denoted by HR
Berk.
I There is a canonical metric on HRBerk. For example, if
ζ0,a, ζ0,b ∈ HRBerk correspond to balls in K centered at zero
with radii a < b, respectively, then
ρ(ζ0,a, ζ0,b) = logv b − logv a.
I An extended real-valued function on P1Berk is continuous
piecewise affine (CPA) roughly if, on small paths [x , y ], it is ofthe form t 7→ Aρ(x , t) + B for some A,B ∈ R.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
The Laplacian
View HRBerk as the direct limit of a direct system of metrized
graphs Γ (with metric induced by ρ).
We build a Laplacian of a CPA function f by restricting to finitemetrized subgraphs Γ of P1
Berk:
I For p ∈ Γ, ∆(f |Γ)(p) = −∑
~v∈TΓ,pd~v (f |Γ)(p).
I ∆(f |Γ) =∑
p∈Γ ∆(f |Γ)(p) δp
I Define ∆(f ) on P1Berk to be the weak limit of the Laplacians
of f |Γ on finite subgraphs Γ (pushed forward to P1Berk).
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
The Laplacian
View HRBerk as the direct limit of a direct system of metrized
graphs Γ (with metric induced by ρ).
We build a Laplacian of a CPA function f by restricting to finitemetrized subgraphs Γ of P1
Berk:
I For p ∈ Γ, ∆(f |Γ)(p) = −∑
~v∈TΓ,pd~v (f |Γ)(p).
I ∆(f |Γ) =∑
p∈Γ ∆(f |Γ)(p) δp
I Define ∆(f ) on P1Berk to be the weak limit of the Laplacians
of f |Γ on finite subgraphs Γ (pushed forward to P1Berk).
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
The Laplacian
View HRBerk as the direct limit of a direct system of metrized
graphs Γ (with metric induced by ρ).
We build a Laplacian of a CPA function f by restricting to finitemetrized subgraphs Γ of P1
Berk:
I For p ∈ Γ, ∆(f |Γ)(p) = −∑
~v∈TΓ,pd~v (f |Γ)(p).
I ∆(f |Γ) =∑
p∈Γ ∆(f |Γ)(p) δp
I Define ∆(f ) on P1Berk to be the weak limit of the Laplacians
of f |Γ on finite subgraphs Γ (pushed forward to P1Berk).
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
The Laplacian
View HRBerk as the direct limit of a direct system of metrized
graphs Γ (with metric induced by ρ).
We build a Laplacian of a CPA function f by restricting to finitemetrized subgraphs Γ of P1
Berk:
I For p ∈ Γ, ∆(f |Γ)(p) = −∑
~v∈TΓ,pd~v (f |Γ)(p).
I ∆(f |Γ) =∑
p∈Γ ∆(f |Γ)(p) δp
I Define ∆(f ) on P1Berk to be the weak limit of the Laplacians
of f |Γ on finite subgraphs Γ (pushed forward to P1Berk).
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
The Laplacian
View HRBerk as the direct limit of a direct system of metrized
graphs Γ (with metric induced by ρ).
We build a Laplacian of a CPA function f by restricting to finitemetrized subgraphs Γ of P1
Berk:
I For p ∈ Γ, ∆(f |Γ)(p) = −∑
~v∈TΓ,pd~v (f |Γ)(p).
I ∆(f |Γ) =∑
p∈Γ ∆(f |Γ)(p) δp
I Define ∆(f ) on P1Berk to be the weak limit of the Laplacians
of f |Γ on finite subgraphs Γ (pushed forward to P1Berk).
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
The Poincare-Lelong Formula
I If ϕ ∈ K (T ) is a rational function on P1K , then the function
z 7→ − logv |ϕ(z)| extends to a CPA function on P1Berk.
I Theorem[Poincare-Lelong] For any rational functionϕ ∈ K (T ), we have the identity
∆(− logv |ϕ|) = δdiv(ϕ).
I Key Point: The left side depends on the canonical metric,while the right is independent of it.
I Question: If one replaces the canonical metric by anothermetric ρ′ on HR
Berk, can Poincare-Lelong still hold?
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
The Poincare-Lelong Formula
I If ϕ ∈ K (T ) is a rational function on P1K , then the function
z 7→ − logv |ϕ(z)| extends to a CPA function on P1Berk.
I Theorem[Poincare-Lelong] For any rational functionϕ ∈ K (T ), we have the identity
∆(− logv |ϕ|) = δdiv(ϕ).
I Key Point: The left side depends on the canonical metric,while the right is independent of it.
I Question: If one replaces the canonical metric by anothermetric ρ′ on HR
Berk, can Poincare-Lelong still hold?
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
The Poincare-Lelong Formula
I If ϕ ∈ K (T ) is a rational function on P1K , then the function
z 7→ − logv |ϕ(z)| extends to a CPA function on P1Berk.
I Theorem[Poincare-Lelong] For any rational functionϕ ∈ K (T ), we have the identity
∆(− logv |ϕ|) = δdiv(ϕ).
I Key Point: The left side depends on the canonical metric,while the right is independent of it.
I Question: If one replaces the canonical metric by anothermetric ρ′ on HR
Berk, can Poincare-Lelong still hold?
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
The Poincare-Lelong Formula
I If ϕ ∈ K (T ) is a rational function on P1K , then the function
z 7→ − logv |ϕ(z)| extends to a CPA function on P1Berk.
I Theorem[Poincare-Lelong] For any rational functionϕ ∈ K (T ), we have the identity
∆(− logv |ϕ|) = δdiv(ϕ).
I Key Point: The left side depends on the canonical metric,while the right is independent of it.
I Question: If one replaces the canonical metric by anothermetric ρ′ on HR
Berk, can Poincare-Lelong still hold?
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Berkovich Projective Line
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X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line
Conditions on the Metric
I Say ρ′ is another metric on P1Berk for which Poincare-Lelong
holds, and let ∆′ be the associated Laplacian.
Thus,∆′(− logv |ϕ|) = δdiv(ϕ) for all ϕ ∈ K (T ).
I Consider the (unique) path in P1Berk from 0 to ∞, and let
ϕ = T .
Then ∆′(− logv |T |) = δ0 − δ∞.
I Note that − logv |T |ζ0,r = − logv r , and is locally constant offthe path from 0 to ∞.
I Consider two discs of radii a < b, centered around the origin,and the interval Γ connecting the corresponding points ζ0,a
and ζ0,b.
Since Γ is a finite metrized graph, Poincare-Lelongsays that ∆′(− logv |T |) on Γ is δζ0,a − δζ0,b
.
I So − logv |T | is CPA in the interior of Γ, and is thus clearlyaffine here.
Since a and b can be chosen arbitrarily, − logv |T |is affine with respect to ρ′ in (0,∞).
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line
Conditions on the Metric
I Say ρ′ is another metric on P1Berk for which Poincare-Lelong
holds, and let ∆′ be the associated Laplacian. Thus,∆′(− logv |ϕ|) = δdiv(ϕ) for all ϕ ∈ K (T ).
I Consider the (unique) path in P1Berk from 0 to ∞, and let
ϕ = T .
Then ∆′(− logv |T |) = δ0 − δ∞.
I Note that − logv |T |ζ0,r = − logv r , and is locally constant offthe path from 0 to ∞.
I Consider two discs of radii a < b, centered around the origin,and the interval Γ connecting the corresponding points ζ0,a
and ζ0,b.
Since Γ is a finite metrized graph, Poincare-Lelongsays that ∆′(− logv |T |) on Γ is δζ0,a − δζ0,b
.
I So − logv |T | is CPA in the interior of Γ, and is thus clearlyaffine here.
Since a and b can be chosen arbitrarily, − logv |T |is affine with respect to ρ′ in (0,∞).
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line
Conditions on the Metric
I Say ρ′ is another metric on P1Berk for which Poincare-Lelong
holds, and let ∆′ be the associated Laplacian. Thus,∆′(− logv |ϕ|) = δdiv(ϕ) for all ϕ ∈ K (T ).
I Consider the (unique) path in P1Berk from 0 to ∞, and let
ϕ = T .
Then ∆′(− logv |T |) = δ0 − δ∞.I Note that − logv |T |ζ0,r = − logv r , and is locally constant off
the path from 0 to ∞.I Consider two discs of radii a < b, centered around the origin,
and the interval Γ connecting the corresponding points ζ0,a
and ζ0,b.
Since Γ is a finite metrized graph, Poincare-Lelongsays that ∆′(− logv |T |) on Γ is δζ0,a − δζ0,b
.
I So − logv |T | is CPA in the interior of Γ, and is thus clearlyaffine here.
Since a and b can be chosen arbitrarily, − logv |T |is affine with respect to ρ′ in (0,∞).
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line
Conditions on the Metric
I Say ρ′ is another metric on P1Berk for which Poincare-Lelong
holds, and let ∆′ be the associated Laplacian. Thus,∆′(− logv |ϕ|) = δdiv(ϕ) for all ϕ ∈ K (T ).
I Consider the (unique) path in P1Berk from 0 to ∞, and let
ϕ = T . Then ∆′(− logv |T |) = δ0 − δ∞.
I Note that − logv |T |ζ0,r = − logv r , and is locally constant offthe path from 0 to ∞.
I Consider two discs of radii a < b, centered around the origin,and the interval Γ connecting the corresponding points ζ0,a
and ζ0,b.
Since Γ is a finite metrized graph, Poincare-Lelongsays that ∆′(− logv |T |) on Γ is δζ0,a − δζ0,b
.
I So − logv |T | is CPA in the interior of Γ, and is thus clearlyaffine here.
Since a and b can be chosen arbitrarily, − logv |T |is affine with respect to ρ′ in (0,∞).
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line
Conditions on the Metric
I Say ρ′ is another metric on P1Berk for which Poincare-Lelong
holds, and let ∆′ be the associated Laplacian. Thus,∆′(− logv |ϕ|) = δdiv(ϕ) for all ϕ ∈ K (T ).
I Consider the (unique) path in P1Berk from 0 to ∞, and let
ϕ = T . Then ∆′(− logv |T |) = δ0 − δ∞.I Note that − logv |T |ζ0,r = − logv r , and is locally constant off
the path from 0 to ∞.
I Consider two discs of radii a < b, centered around the origin,and the interval Γ connecting the corresponding points ζ0,a
and ζ0,b.
Since Γ is a finite metrized graph, Poincare-Lelongsays that ∆′(− logv |T |) on Γ is δζ0,a − δζ0,b
.
I So − logv |T | is CPA in the interior of Γ, and is thus clearlyaffine here.
Since a and b can be chosen arbitrarily, − logv |T |is affine with respect to ρ′ in (0,∞).
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line
Conditions on the Metric
I Say ρ′ is another metric on P1Berk for which Poincare-Lelong
holds, and let ∆′ be the associated Laplacian. Thus,∆′(− logv |ϕ|) = δdiv(ϕ) for all ϕ ∈ K (T ).
I Consider the (unique) path in P1Berk from 0 to ∞, and let
ϕ = T . Then ∆′(− logv |T |) = δ0 − δ∞.I Note that − logv |T |ζ0,r = − logv r , and is locally constant off
the path from 0 to ∞.I Consider two discs of radii a < b, centered around the origin,
and the interval Γ connecting the corresponding points ζ0,a
and ζ0,b.
Since Γ is a finite metrized graph, Poincare-Lelongsays that ∆′(− logv |T |) on Γ is δζ0,a − δζ0,b
.I So − logv |T | is CPA in the interior of Γ, and is thus clearly
affine here.
Since a and b can be chosen arbitrarily, − logv |T |is affine with respect to ρ′ in (0,∞).
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line
Conditions on the Metric
I Say ρ′ is another metric on P1Berk for which Poincare-Lelong
holds, and let ∆′ be the associated Laplacian. Thus,∆′(− logv |ϕ|) = δdiv(ϕ) for all ϕ ∈ K (T ).
I Consider the (unique) path in P1Berk from 0 to ∞, and let
ϕ = T . Then ∆′(− logv |T |) = δ0 − δ∞.I Note that − logv |T |ζ0,r = − logv r , and is locally constant off
the path from 0 to ∞.I Consider two discs of radii a < b, centered around the origin,
and the interval Γ connecting the corresponding points ζ0,a
and ζ0,b. Since Γ is a finite metrized graph, Poincare-Lelongsays that ∆′(− logv |T |) on Γ is δζ0,a − δζ0,b
.
I So − logv |T | is CPA in the interior of Γ, and is thus clearlyaffine here.
Since a and b can be chosen arbitrarily, − logv |T |is affine with respect to ρ′ in (0,∞).
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line
Conditions on the Metric
I Say ρ′ is another metric on P1Berk for which Poincare-Lelong
holds, and let ∆′ be the associated Laplacian. Thus,∆′(− logv |ϕ|) = δdiv(ϕ) for all ϕ ∈ K (T ).
I Consider the (unique) path in P1Berk from 0 to ∞, and let
ϕ = T . Then ∆′(− logv |T |) = δ0 − δ∞.I Note that − logv |T |ζ0,r = − logv r , and is locally constant off
the path from 0 to ∞.I Consider two discs of radii a < b, centered around the origin,
and the interval Γ connecting the corresponding points ζ0,a
and ζ0,b. Since Γ is a finite metrized graph, Poincare-Lelongsays that ∆′(− logv |T |) on Γ is δζ0,a − δζ0,b
.I So − logv |T | is CPA in the interior of Γ, and is thus clearly
affine here.
Since a and b can be chosen arbitrarily, − logv |T |is affine with respect to ρ′ in (0,∞).
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line
Conditions on the Metric
I Say ρ′ is another metric on P1Berk for which Poincare-Lelong
holds, and let ∆′ be the associated Laplacian. Thus,∆′(− logv |ϕ|) = δdiv(ϕ) for all ϕ ∈ K (T ).
I Consider the (unique) path in P1Berk from 0 to ∞, and let
ϕ = T . Then ∆′(− logv |T |) = δ0 − δ∞.I Note that − logv |T |ζ0,r = − logv r , and is locally constant off
the path from 0 to ∞.I Consider two discs of radii a < b, centered around the origin,
and the interval Γ connecting the corresponding points ζ0,a
and ζ0,b. Since Γ is a finite metrized graph, Poincare-Lelongsays that ∆′(− logv |T |) on Γ is δζ0,a − δζ0,b
.I So − logv |T | is CPA in the interior of Γ, and is thus clearly
affine here. Since a and b can be chosen arbitrarily, − logv |T |is affine with respect to ρ′ in (0,∞).
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line
Conditions on the Metric (cont’d)
I Since − logv |T | is affine on (0,∞), we have that ∃c ∈ Rsuch that ∀0 < a ≤ b, − logv a− (− logv b) = cρ′(ζ0,a, ζ0,b).
I So ρ′(ζ0,a, ζ0,b) = logv (b/a)c =
ρ(ζ0,a,ζ0,b)c .
I Since multiplying distances by 1/c multiplies slopes by c , wehave ∆′(− logv |T |) = cδ0 − cδ∞.
Thus c = 1, and ρ = ρ′ on(0,∞)!
I We can repeat this argument for the path between any twoK -rational points y and z by using the function T−y
T−z in placeof T (where, by convention, we take T −∞ to be 1).
Thus,ρ = ρ′ on all type II and type III points.
I By continuity, ρ = ρ′ on type IV points, and thus on all ofHBerk.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line
Conditions on the Metric (cont’d)
I Since − logv |T | is affine on (0,∞), we have that ∃c ∈ Rsuch that ∀0 < a ≤ b, − logv a− (− logv b) = cρ′(ζ0,a, ζ0,b).
I So ρ′(ζ0,a, ζ0,b) = logv (b/a)c =
ρ(ζ0,a,ζ0,b)c .
I Since multiplying distances by 1/c multiplies slopes by c , wehave ∆′(− logv |T |) = cδ0 − cδ∞.
Thus c = 1, and ρ = ρ′ on(0,∞)!
I We can repeat this argument for the path between any twoK -rational points y and z by using the function T−y
T−z in placeof T (where, by convention, we take T −∞ to be 1).
Thus,ρ = ρ′ on all type II and type III points.
I By continuity, ρ = ρ′ on type IV points, and thus on all ofHBerk.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line
Conditions on the Metric (cont’d)
I Since − logv |T | is affine on (0,∞), we have that ∃c ∈ Rsuch that ∀0 < a ≤ b, − logv a− (− logv b) = cρ′(ζ0,a, ζ0,b).
I So ρ′(ζ0,a, ζ0,b) = logv (b/a)c =
ρ(ζ0,a,ζ0,b)c .
I Since multiplying distances by 1/c multiplies slopes by c , wehave ∆′(− logv |T |) = cδ0 − cδ∞.
Thus c = 1, and ρ = ρ′ on(0,∞)!
I We can repeat this argument for the path between any twoK -rational points y and z by using the function T−y
T−z in placeof T (where, by convention, we take T −∞ to be 1).
Thus,ρ = ρ′ on all type II and type III points.
I By continuity, ρ = ρ′ on type IV points, and thus on all ofHBerk.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line
Conditions on the Metric (cont’d)
I Since − logv |T | is affine on (0,∞), we have that ∃c ∈ Rsuch that ∀0 < a ≤ b, − logv a− (− logv b) = cρ′(ζ0,a, ζ0,b).
I So ρ′(ζ0,a, ζ0,b) = logv (b/a)c =
ρ(ζ0,a,ζ0,b)c .
I Since multiplying distances by 1/c multiplies slopes by c , wehave ∆′(− logv |T |) = cδ0 − cδ∞. Thus c = 1, and ρ = ρ′ on(0,∞)!
I We can repeat this argument for the path between any twoK -rational points y and z by using the function T−y
T−z in placeof T (where, by convention, we take T −∞ to be 1).
Thus,ρ = ρ′ on all type II and type III points.
I By continuity, ρ = ρ′ on type IV points, and thus on all ofHBerk.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line
Conditions on the Metric (cont’d)
I Since − logv |T | is affine on (0,∞), we have that ∃c ∈ Rsuch that ∀0 < a ≤ b, − logv a− (− logv b) = cρ′(ζ0,a, ζ0,b).
I So ρ′(ζ0,a, ζ0,b) = logv (b/a)c =
ρ(ζ0,a,ζ0,b)c .
I Since multiplying distances by 1/c multiplies slopes by c , wehave ∆′(− logv |T |) = cδ0 − cδ∞. Thus c = 1, and ρ = ρ′ on(0,∞)!
I We can repeat this argument for the path between any twoK -rational points y and z by using the function T−y
T−z in placeof T (where, by convention, we take T −∞ to be 1).
Thus,ρ = ρ′ on all type II and type III points.
I By continuity, ρ = ρ′ on type IV points, and thus on all ofHBerk.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line
Conditions on the Metric (cont’d)
I Since − logv |T | is affine on (0,∞), we have that ∃c ∈ Rsuch that ∀0 < a ≤ b, − logv a− (− logv b) = cρ′(ζ0,a, ζ0,b).
I So ρ′(ζ0,a, ζ0,b) = logv (b/a)c =
ρ(ζ0,a,ζ0,b)c .
I Since multiplying distances by 1/c multiplies slopes by c , wehave ∆′(− logv |T |) = cδ0 − cδ∞. Thus c = 1, and ρ = ρ′ on(0,∞)!
I We can repeat this argument for the path between any twoK -rational points y and z by using the function T−y
T−z in placeof T (where, by convention, we take T −∞ to be 1). Thus,ρ = ρ′ on all type II and type III points.
I By continuity, ρ = ρ′ on type IV points, and thus on all ofHBerk.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe case of the Berkovich Projective Line
Conditions on the Metric (cont’d)
I Since − logv |T | is affine on (0,∞), we have that ∃c ∈ Rsuch that ∀0 < a ≤ b, − logv a− (− logv b) = cρ′(ζ0,a, ζ0,b).
I So ρ′(ζ0,a, ζ0,b) = logv (b/a)c =
ρ(ζ0,a,ζ0,b)c .
I Since multiplying distances by 1/c multiplies slopes by c , wehave ∆′(− logv |T |) = cδ0 − cδ∞. Thus c = 1, and ρ = ρ′ on(0,∞)!
I We can repeat this argument for the path between any twoK -rational points y and z by using the function T−y
T−z in placeof T (where, by convention, we take T −∞ to be 1). Thus,ρ = ρ′ on all type II and type III points.
I By continuity, ρ = ρ′ on type IV points, and thus on all ofHBerk.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Case of a General Curve
Main Result
Let X denote a connected smooth proper algebraic curve over K .
I There is a canonical metric ρ on the associated analytic spaceXBerk and a notion of Laplacian.
I Theorem.[Thuillier] For any rational function ϕ ∈ K (X ), thefunction z 7→ − logv |ϕ| lies in CPA(XBerk), and
∆(− logv |ϕ|) = δdiv(ϕ).
I Theorem.[F., O., P., P., W.] Under the assumption that theresidue field of K is countable, the canonical metric is theunique metric on XBerk such that Thuillier’s result holds.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Case of a General Curve
Main Result
Let X denote a connected smooth proper algebraic curve over K .
I There is a canonical metric ρ on the associated analytic spaceXBerk and a notion of Laplacian.
I Theorem.[Thuillier] For any rational function ϕ ∈ K (X ), thefunction z 7→ − logv |ϕ| lies in CPA(XBerk), and
∆(− logv |ϕ|) = δdiv(ϕ).
I Theorem.[F., O., P., P., W.] Under the assumption that theresidue field of K is countable, the canonical metric is theunique metric on XBerk such that Thuillier’s result holds.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Case of a General Curve
Main Result
Let X denote a connected smooth proper algebraic curve over K .
I There is a canonical metric ρ on the associated analytic spaceXBerk and a notion of Laplacian.
I Theorem.[Thuillier] For any rational function ϕ ∈ K (X ), thefunction z 7→ − logv |ϕ| lies in CPA(XBerk), and
∆(− logv |ϕ|) = δdiv(ϕ).
I Theorem.[F., O., P., P., W.] Under the assumption that theresidue field of K is countable, the canonical metric is theunique metric on XBerk such that Thuillier’s result holds.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Case of a General Curve
Main Result
Let X denote a connected smooth proper algebraic curve over K .
I There is a canonical metric ρ on the associated analytic spaceXBerk and a notion of Laplacian.
I Theorem.[Thuillier] For any rational function ϕ ∈ K (X ), thefunction z 7→ − logv |ϕ| lies in CPA(XBerk), and
∆(− logv |ϕ|) = δdiv(ϕ).
I Theorem.[F., O., P., P., W.] Under the assumption that theresidue field of K is countable, the canonical metric is theunique metric on XBerk such that Thuillier’s result holds.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Case of a General Curve
Idea of the Proof
The proof has essentially two steps:
I Use the fact that − logv |ϕ| ∈ CPA(XBerk) to show that thenew metric must be a constant scaling of the canonical metric.
I Choose a clever rational function ϕ and compute∆(− logv |ϕ|) locally to see that the two metrics must agree.
The proof is trickier than for P1Berk because we cannot immediately
construct rational functions with prescribed divisors and poles.To get around this, we apply the following result:
Theorem.[Rumely] Suppose K has countable residue field, and letX be a connected smooth proper curve over K . Let U ⊂ XBerk bean open neighborhood and let ζ ∈ X (K ) \ U(K ) be any classicalpoint. Then there exists a rational function ϕ ∈ K (X ) such thatall of the zeros of ϕ lie in U, and the only pole of ϕ is at ζ.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Case of a General Curve
Idea of the Proof
The proof has essentially two steps:
I Use the fact that − logv |ϕ| ∈ CPA(XBerk) to show that thenew metric must be a constant scaling of the canonical metric.
I Choose a clever rational function ϕ and compute∆(− logv |ϕ|) locally to see that the two metrics must agree.
The proof is trickier than for P1Berk because we cannot immediately
construct rational functions with prescribed divisors and poles.To get around this, we apply the following result:
Theorem.[Rumely] Suppose K has countable residue field, and letX be a connected smooth proper curve over K . Let U ⊂ XBerk bean open neighborhood and let ζ ∈ X (K ) \ U(K ) be any classicalpoint. Then there exists a rational function ϕ ∈ K (X ) such thatall of the zeros of ϕ lie in U, and the only pole of ϕ is at ζ.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Case of a General Curve
Idea of the Proof
The proof has essentially two steps:
I Use the fact that − logv |ϕ| ∈ CPA(XBerk) to show that thenew metric must be a constant scaling of the canonical metric.
I Choose a clever rational function ϕ and compute∆(− logv |ϕ|) locally to see that the two metrics must agree.
The proof is trickier than for P1Berk because we cannot immediately
construct rational functions with prescribed divisors and poles.To get around this, we apply the following result:
Theorem.[Rumely] Suppose K has countable residue field, and letX be a connected smooth proper curve over K . Let U ⊂ XBerk bean open neighborhood and let ζ ∈ X (K ) \ U(K ) be any classicalpoint. Then there exists a rational function ϕ ∈ K (X ) such thatall of the zeros of ϕ lie in U, and the only pole of ϕ is at ζ.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Case of a General Curve
Idea of the Proof
The proof has essentially two steps:
I Use the fact that − logv |ϕ| ∈ CPA(XBerk) to show that thenew metric must be a constant scaling of the canonical metric.
I Choose a clever rational function ϕ and compute∆(− logv |ϕ|) locally to see that the two metrics must agree.
The proof is trickier than for P1Berk because we cannot immediately
construct rational functions with prescribed divisors and poles.
To get around this, we apply the following result:
Theorem.[Rumely] Suppose K has countable residue field, and letX be a connected smooth proper curve over K . Let U ⊂ XBerk bean open neighborhood and let ζ ∈ X (K ) \ U(K ) be any classicalpoint. Then there exists a rational function ϕ ∈ K (X ) such thatall of the zeros of ϕ lie in U, and the only pole of ϕ is at ζ.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Case of a General Curve
Idea of the Proof
The proof has essentially two steps:
I Use the fact that − logv |ϕ| ∈ CPA(XBerk) to show that thenew metric must be a constant scaling of the canonical metric.
I Choose a clever rational function ϕ and compute∆(− logv |ϕ|) locally to see that the two metrics must agree.
The proof is trickier than for P1Berk because we cannot immediately
construct rational functions with prescribed divisors and poles.To get around this, we apply the following result:
Theorem.[Rumely] Suppose K has countable residue field, and letX be a connected smooth proper curve over K . Let U ⊂ XBerk bean open neighborhood and let ζ ∈ X (K ) \ U(K ) be any classicalpoint. Then there exists a rational function ϕ ∈ K (X ) such thatall of the zeros of ϕ lie in U, and the only pole of ϕ is at ζ.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula
IntroductionPoincare-Lelong for the Berkovich Projective Line
Poincare-Lelong for General Algebraic CurvesThe Case of a General Curve
Idea of the Proof
The proof has essentially two steps:
I Use the fact that − logv |ϕ| ∈ CPA(XBerk) to show that thenew metric must be a constant scaling of the canonical metric.
I Choose a clever rational function ϕ and compute∆(− logv |ϕ|) locally to see that the two metrics must agree.
The proof is trickier than for P1Berk because we cannot immediately
construct rational functions with prescribed divisors and poles.To get around this, we apply the following result:
Theorem.[Rumely] Suppose K has countable residue field, and letX be a connected smooth proper curve over K . Let U ⊂ XBerk bean open neighborhood and let ζ ∈ X (K ) \ U(K ) be any classicalpoint. Then there exists a rational function ϕ ∈ K (X ) such thatall of the zeros of ϕ lie in U, and the only pole of ϕ is at ζ.
X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincare-Lelong Formula