Higher Solovay Models

In 1970, Solovay wrote a fundamental article where from a model of ZFC with an inaccessible cardinal he built, by Lévy collapsing the inaccessible to $\omega_1$ and considering the inner model $L(\mathbb R)$ of this extension, a model where all sets of reals are Lebesgue measurable and have the Baire and perfect set properties. Later, Bagaria and Woodin introduced an abstract axiomatisation of this model, and they proved that every model satisfying this axiomatisation is the same as a model built with the original Solovay construction.
In recent years, a prolific subject is generalised descriptive set theory, that tries to generalise results about $L(\mathbb R)$ to any model of the form $L(V_{\kappa+1})$. This was done also by Dimonte, Poveda and Thei, that mimicked the Solovay construction as follows. Instead of Lévy collapse, they used some kind of $\Sigma$-Příkrý forcing, and considered the inner model $L(V_{\kappa+1})$ of this forcing extension, for $\kappa$ a singular cardinal. Here, all sets of "$\kappa$-reals" satisfy adapted versions of the Baire and perfect set properties.
The goal of my talk is to introduce an axiomatisation of these higher Solovay models $L(V_{\kappa+1})^M$, and then show the corresponding version of Bagaria-Woodin's theorem and prove some generic absoluteness results about these kinds of models.
Joint work with Sebastiano Thei (Udine).