7.1 Definitions [03VL]

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7.1 Definitions

Let $B^{sm}$ be a manifold with ${\bf Z}$ -affine structure. The sheaf of ${\bf Z}$ -affine functions $Aff_{{\bf Z}}:=Aff_{{{\bf Z}},B^{sm}}$ gives rise to an exact sequence of sheaves of abelian groups

0\to{{\bf R}}\to Aff_{{\bf Z}}\to(T^{\ast})^{{\bf Z}}\to 0\,\,.

Let $K$ be a complete non-archimedean field with a valuation map $val$ . We give two equivalent definitions of a $K$ -affine structure on $B^{sm}$ compatible with a given ${\bf Z}$ -affine structure.

Definition 11

A $K$ -affine structure on $B^{sm}$ compatible with the given ${\bf Z}$ -affine structure is a sheaf $Aff_{K}$ of abelian groups on $B^{sm}$ , an exact sequence of sheaves

0\to K^{\times}\to Aff_{K}\to(T^{\ast})^{{\bf Z}}\to 0\,\,,

together with a homomorphism $\Phi$ of this exact sequence to the exact sequence of sheaves of abelian groups

0\to{{\bf R}}\to Aff_{{\bf Z}}\to(T^{\ast})^{{\bf Z}}\to 0\,\,,

such that $\Phi=id$ on $(T^{\ast})^{{\bf Z}}$ and $\Phi=val$ on $K^{\times}$ .

Since $B^{sm}$ carries a ${\bf Z}$ -affine structure, we have an associated $GL(n,{{\bf Z}})\ltimes{{\bf R}}^{n}$ -torsor on $B^{sm}$ , whose fiber over a point $x$ consists of all ${\bf Z}$ -affine coordinate systems at $x$ .

Definition 12

A $K$ -affine structure on $B^{sm}$ compatible with the given ${\bf Z}$ -affine structure is a $GL(n,{{\bf Z}})\ltimes(K^{\times})^{n}$ -torsor on $B^{sm}$ such that the application of $val^{\times n}$ to $(K^{\times})^{n}$ gives the initial $GL(n,{{\bf Z}})\ltimes{{\bf R}}^{n}$ -torsor.

Equivalence of two definitions from above is obvious in local ${\bf Z}$ -affine coordinates. The reason is that the set of automorphisms of the exact sequence of groups

0\to K^{\times}\to K^{\times}\times{{\bf Z}}^{n}\to{{\bf Z}}^{n}\to 0

identical on $K^{\times}$ coincides with the group $GL(n,{{\bf Z}})\ltimes(K^{\times})^{n}$ .

Finally, we can formulate the Fixed Point Property for $K$ -affine structures (see Section 3.1 for ${\bf Z}$ -affine case):
Fixed Point Property for $K$ -affine structures. In the notation of the end of Section 3.1, for any $b\in B^{sing}$ and sufficiently small neighborhood $U$ of $b$ the lifted monodromy representation $\pi_{1}(U)\to GL(n,{{\bf Z}})\ltimes(K^{\times})^{n}$ has fixed vectors in $K^{\times n}$ , and the ${{\bf R}}$ -affine span of the corresponding (under the valuation map) vectors in ${{\bf R}}^{n}$ coincides with the set of fixed points of the monodromy representation $\pi_{1}(U)\to GL(n,{{\bf Z}})\ltimes{{\bf R}}^{n}$ .