Effective local differential topology of algebraic varieties over local fields of positive characteristics

The goal of this paper is to provide a framework for formulating quantitative statements on distances and measures when studying algebraic varieties and maps between them over local fields.

We will concentrate on local fields of the type F ℓ ((t)) and work uniformly with respect to finite extensions of F ℓ .

We introduce a notion of a rectification of an algebraic variety. This notion allows us to define the concept of a ball on a variety and to fix a family of measures on it.

1.1. The framework. For a variety X defined over a finite field F ℓ we introduce the notion of rectification (see Definition 3.1). This notion allows us to define balls in the set X := X(F ) of F -points of the variety, where F is a local field containing F ℓ (see Definition 3.3). Note that the notion of a ball is defined simultaneously for all local fields of the type F ℓ k ((t)). This allows us to formulate uniform statements for all such fields. Notation 1.1. For a variety X and an integer k ∈ N, we will consider two kinds of balls in X(F ℓ k ((t))).

(1) Non-centered balls, denoted by B X,k m , see Definition 3.3(1)(a) for the formal definition. These could be thought of as balls around the origin (though the origin is not necessarily a point in X). Here m ∈ Z is the valuative radius of the ball, i.e. the actual radius is ℓ km . Usually, m will be positive when considering such a ball.

(2) Centered balls, denoted by B X,k m (x), see Definition 3.3(1)(c) for the formal definition. These are balls of valuative radius m ∈ Z around x ∈ X(F ℓ k ((t))). Here the integer m is usually negative.

Although the balls themselves will depend on the rectification, all the statements that we will prove will not. This is due to the fact that for any two rectifications, one can compare between the corresponding balls. See Corollary 3.6(i).

Similarly, we will define the notion of a µ-rectification of smooth algebraic varieties (see Definition 3.1). This notion allows us to fix measures on balls in X (see Definition 3.3). Again, although the measures themselves will depend on the µ-rectification, the results that we will prove will not. This is established in Corollary 3.6(ii) and Lemma 5.5.

Remark 1.2. For the sake of simplicity, we work only with algebraic varieties defined over F ℓ . This is enough for our purposes. However, we believe that with minor modifications, all the statements would be valid also for varieties defined over F ℓ [[t]] and, with slightly more modifications, also for varieties defined over F ℓ ((t)).

1.2. Main results. We prove the following results:

(1) Effective uniform continuity and boundedness of algebraic morphisms on balls. See §1.2.1.

(2) Effective version of the implicit function theorem. See §1.2.2.

(3) The compliment of a small neighborhood around a closed subvariety Z ⊂ X is controlled by large balls in the complement of Z. See §1.2.3. (4) Effective surjectivity of Nisnevich covers. See §1.2.4.

(5) Effective smoothness of push forward of smooth measures with respect to smooth maps. See §1.2.5. (6) Effective bounds on pushforward of smooth measures with respect to smooth maps. See §1.2.6.

1.2.1. Effective uniform continuity and boundedness. Let γ : X → Y be a map of algebraic varieties defined over a finite field F ℓ . This gives maps γ k : X(F ℓ k ((t))) → Y(F ℓ k ((t))). Note that each map γ k is uniformly continuous and bounded on any ball in X(F ℓ k ((t))). We prove that the modulus of continuity and the bound on γ k in a ball B X,k m of a fixed (valuative) radius m are bounded when we vary k.

More formally, we prove:

Proposition A (Proposition 3.5). Let γ : X → Y be a map of rectified algebraic varieties defined over a finite field F ℓ . Then for any m ∈ N there is m ′ > m such that for any k ∈ N we have:

Effective versions of the inverse and the implicit function theorems. We prove an effective versions of the inverse and the implicit function theorems. Informally it means the following:

Let γ : X → Y be an etale (respectively smooth) map of smooth algebraic varieties defined over a finite field F ℓ . We again consider the maps γ k :

Then, in a ball B X,k m the map γ k admits a local inverse (respectively section), with bounded modulus of continuity, when m ∈ N is fixed and k varies.

More formally, we prove the following theorems.

Theorem B (Theorem 4.1). Let γ : X → Y be an étale map of smooth rectified algebraic varieties defined over a finite field F ℓ . Then for any m there is m ′ such that γ| B X,k m is a monomorphism on balls of valuative radius -m ′ .

Theorem C (Theorem 4.2). Let γ : X → Y be a smooth map of smooth rectifie