Let $K$ be a non-archimedean valued field (with any further adjectives attached as necessary). I'm looking for references or information about symplectic structures on rigid $K$-spaces.

For example, if $X$ is an affinoid $K$-variety, then there is a tangent sheaf $\mathcal{T}_X$ on $X_{\text{rig}}$, with sections $$\mathcal{T}_X(U) = \text{Der}_K(U)$$ over affinoid subdomains $U$. It gives rise to a sheaf of $\mathcal{O}_X$-algebras $\mathcal{A} = \text{Sym}_{\mathcal{O}_X} \mathcal{T}_X$. There is then a "relative analytification'' space $Y = \text{Spec}^{\text{an}} \mathcal{A}$, as described by Conrad in Section 2.2 here.

As defined, this $Y$ should be the correct candidate for the cotangent space $T^*X$, and so one might try to make sense of a symplectic structure on $Y$. However, I haven't read anything about tangent spaces or symplectic forms in the rigid setting. Is that because these notions are problematic or because they are obvious? Should tangent spaces to points on $X$ be defined in the same way as if $X$ were a scheme? Thanks for any insight or pointers to the literature.


1 Answer 1


This is only a partial answer but it might give you a place to start. One alternative approach to nonarchimedean geometry is the theory of Berkovich analytic spaces. There's a fully faithful functor from separated strictly $k$-analytic Berkovich spaces to rigid spaces over $k$ (see Section 3.3 of Berkovich's "Spectral theory and analytic geometry over nonarchimedeanfields).

In Chapter 5 of Ducros' "Families of Berkovich spaces" he reviews the notion of a sheaf of relative differentials, smooth morphisms, and discusses tangent spaces. In particular for good $k$-analytic spaces the stalk at of the tangent space at a $k$-point will be $m_x/m_x^2$ where $m_x$ is the maximal ideal of the stalk of the structure sheaf at $x$.


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