The moduli space of stable coherent sheaves via non-archimedean geometry

Abstract

We provide a construction of the moduli space of stable coherent sheaves in the world of non-archimedean geometry, where we use the notion of Berkovich non-archimedean analytic spaces. The motivation for our construction is Tony Yue Yu's non-archimedean enumerative geometry in Gromov-Witten theory. The construction of the moduli space of stable sheaves using Berkovich analytic spaces will give rise to the non-archimedean version of Donaldson-Thomas invariants. In this paper we give the moduli construction over a non-archimedean field $\kk$. We use the machinery of formal schemes, that is, we define and construct the formal moduli stack of (semi)-stable coherent sheaves over a discrete valuation ring $R$, and taking generic fiber we get the non-archimedean analytic moduli of semistable coherent sheaves over the fractional non-archimedean field $\kk$. For a moduli space of stable sheaves of an algebraic variety $X$ over an algebraically closed field $\kappa$, the analytification of such a moduli space gives an example of the non-archimedean moduli space. We generalize Joyce's $d$-critical scheme structure in \cite{Joyce} or Kiem-Li's virtual critical manifolds in \cite{KL} to the world of formal schemes, and Berkovich non-archimedean analytic spaces. As an application, we provide a proof for the motivic localization formula for a $d$-critical non-archimedean $\kk$-analytic space using global motive of vanishing cycles and motivic integration on oriented formal $d$-critical schemes. This generalizes Maulik's motivic localization formula for motivic Donaldson-Thomas invariants.