We discuss the behavior of Landau–Ginzburg models for toric orbifolds near the large volume limit. This enables us to express mirror symmetry as an isomorphism of Frobenius manifolds that aquire logarithmic poles along a boundary divisor. If the toric orbifold admits a crepant resolution, we construct a global moduli space on the B-side and show that the associated $t{t}^{\ast }$-geometry exists globally.

We consider categories of posets with $\mathfrak{C}$-valued structure sheaves for any category $\mathfrak{C}$ and see how they possess poset-indexed lax colimits that are both easy to describe and “weakly equivalent” to their ordinary colimits in a certain sense. We employ this construction to study descent problems on schematic spaces—a particular scheme-like kind of ringed poset—proving a general Seifert–Van Kampen theorem for their étale fundamental group that recovers and generalizes the homonym result for schemes in the topology of flat monomorphisms. The techniques are general enough to consider their applications in many other frameworks.

Let F be a number field. We estimate the kernels and cokernels of the codescent maps of the étale wild kernels over various p-adic Lie extensions. For this, we propose a novel approach of viewing the étale wild kernel as an appropriate fine Selmer group in the sense of Coates–Sujatha. This viewpoint reduces the problem to a control theorem of the said fine Selmer groups, which in turn allows us to employ the strategies developed by Mazur and Greenberg. As applications of our estimates on the kernels and cokernels of the codescent maps, we establish asymptotic growth formulas for the étale wild kernels in the various said p-adic Lie extensions. We then relate these growth formulas to Greenberg’s conjecture (and its noncommutative analogue). Finally, we shall give some examples to illustrate our results.