Canonical ASVG State Spaces for Projective Varieties Axiomatic Construction in Fréchet Geometry

A canonical analytic state space for Axiomatic Second–Variation Geometry (ASVG) on smooth projective varieties is constructed through a metric–independent Hodge–Fréchet completion of differential forms.
This space satisfies a universal mapping property ensuring that every symmetric ASVG second variation extends uniquely and continuously.
A complete classification of ASVG kernels is then obtained: admissible kernels are precisely the positive, holomorphically invariant pseudodifferential operators that are polynomial in the Hodge Laplacian up to smoothing terms.
The resulting framework provides the analytic foundation for subsequent geometric and Hodge–theoretic applications of ASVG.