Integrability of pushforward measures by analytic maps

Given a map ϕ: X → Y between F-analytic manifolds over a local field F of characteristic 0, we introduce an invariant ϵ_✶(ϕ) measuring the integrability of pushforwards of smooth compactly supported measures by ϕ, together with a local version ϵ_✶(ϕ; x) near x ∈ X. These invariants are closedly tied to the singularities of ϕ. When Y is one-dimensional, we provide an explicit formula for ϵ_✶(ϕ; x) and show that it is asymptotically equivalent to classical singularity invariants, in particular the F-log-canonical threshold lct_F(ϕ− ϕ(x); x). In general, we prove that ϵ_✶(ϕ; x) is bounded below by the F-log-canonical threshold λ = lct_F(J_ϕ; x) of the Jacobian ideal J_ϕ, with equality when dim Y = dim X. For polynomial maps φ: X → Y between smooth algebraic Q-varieties, we show that the condition “ϵ_✶(φ_F) = ∞ over a large family of local fields” is equivalent to φ being flat with semi-log-canonical fibers.