Abstract We provide a constructive version of the notion sheaf models univalent type theory. start by relativizing existing theory to presheaves over base category. Any Grothendieck topology category then gives rise family left-exact modalities, and we recover model localizing presheaf with respect this modalities. some examples.

We study holomorphic foliations of arbitrary codimension in smooth complete toric varieties. show that split are stable if some good behaviour their singular set is provided. As an application these results, we exhibit irreducible components the space arise as pullbacks special T-invariant subvarieties.

We construct a sheaf of Fock spaces over the moduli space elliptic curves Ey with ?1(3)-level structure, arising from geometric quantization H1(Ey), and global section this sheaf. The coincides, near appropriate limit points, Gromov–Witten potentials local P2 orbifold [C3??3]. This proves that are quasimodular functions for group ?1(3), as predicted by Aganagic, Bouchard, Klemm, it crepant reso...