Moduli space of irregular singular $λ$-connections and the relative symplectic form

We study the moduli space of $λ$-connections with ramified singularities on a smooth projective curve $X$. We construct a quasi-projective relative moduli scheme $M^α_{\mathrm{Hod}} \to Λ$ of $α$-stable $λ$-connections over a base parameter space $Λ$. This moduli space connects the irregular de Rham moduli space ($λ=1$) and the irregular Dolbeault moduli space of Higgs bundles equipped with non-zero irregular local exponents ($λ=0$). Furthermore, we show that $M^α_{\mathrm{Hod}}$ is smooth over $Λ$ and carries a canonical relative symplectic form $ω_{M^α_{\mathrm{Hod}}/Λ}$, yielding a natural Poisson structure on the total space.