Introduction Poincaré-Lelong for the Berkovich Projective Line

Poincaré-Lelong for General Algebraic Curves

Matthew Baker’s Project Group 3: The Poincaré-Lelong Formula

Xander Faber, Andrew Obus, Jorge Pineiro, Jérôme Poineau, Christian Wahle

March 20, 2007

X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincaré-Lelong Formula

Introduction Poincaré-Lelong for the Berkovich Projective Line

Poincaré-Lelong for General Algebraic Curves The Berkovich Projective Line

Throughout, K will denote an algebraically closed field complete with 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 of the form t 7→ Aρ(x , t) + B for some A,B ∈ R.

X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincaré-Lelong Formula

Introduction Poincaré-Lelong for the Berkovich Projective Line

Poincaré-Lelong for General Algebraic Curves The Berkovich Projective Line

Throughout, K will denote an algebraically closed field complete with 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 of the form t 7→ Aρ(x , t) + B for some A,B ∈ R.

X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincaré-Lelong Formula

Introduction Poincaré-Lelong for the Berkovich Projective Line

Poincaré-Lelong for General Algebraic Curves The Berkovich Projective Line

Throughout, K will denote an algebraically closed field complete with 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 of the form t 7→ Aρ(x , t) + B for some A,B ∈ R.

X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincaré-Lelong Formula

Introduction Poincaré-Lelong for the Berkovich Projective Line

Poincaré-Lelong for General Algebraic Curves The Berkovich Projective Line

Throughout, K will denote an algebraically closed field complete with 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 of the form t 7→ Aρ(x , t) + B for some A,B ∈ R.

X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincaré-Lelong Formula

Introduction Poincaré-Lelong for the Berkovich Projective Line

Poincaré-Lelong for General Algebraic Curves The Berkovich Projective Line

Throughout, K will denote an algebraically closed field complete with 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 of the form t 7→ Aρ(x , t) + B for some A,B ∈ R.

X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincaré-Lelong Formula

Introduction Poincaré-Lelong for the Berkovich Projective Line

Poincaré-Lelong for General Algebraic Curves The Berkovich Projective Line

Throughout, K will denote an algebraically closed field complete with 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 of the form t 7→ Aρ(x , t) + B for some A,B ∈ R.

X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincaré-Lelong Formula

Introduction Poincaré-Lelong for the Berkovich Projective Line

Poincaré-Lelong for General Algebraic Curves The Berkovich Projective Line

Throughout, K will denote an algebraically closed field complete with 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 of the form t 7→ Aρ(x , t) + B for some A,B ∈ R.

X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincaré-Lelong Formula

Introduction Poincaré-Lelong for the Berkovich Projective Line

Poincaré-Lelong for General Algebraic Curves The Berkovich Projective Line

Throughout, K will denote an algebraically closed field complete with 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 of the form t 7→ Aρ(x , t) + B for some A,B ∈ R.

X. Faber, A. Obus, J. Pineiro, J. Poineau, C. Wahle The Poincaré-Lelong Formula

Introduction Poincaré-Lelong for the Berkovich Projective Line

Poincaré-Lelong for General Algebraic Curves The 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 finite metrized subgraphs Γ of P1Berk:

I For p ∈ Γ, ∆(f |Γ)(p) = − ∑

~v∈TΓ,p d~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 Poincaré-Lelong Formula

Introduction Poincaré-Lelong for the Berkovich Projective Line

Poincaré-Lelong for General Algebraic Cur