We calculate the category of branes in the Landau–Ginzburg $A$-model with background $M={\mathbb{C}}^n$ and superpotential $W=z_1\cdots z_n$ in the form of microlocal sheaves along a natural Lagrangian skeleton. Our arguments employ the framework of perverse schobers, and our results confirm expectations from mirror symmetry.

Let $G$ be a real reductive group, and let $\chi$ be a character of a reductive subgroup $H$ of $G$. We construct $\chi$-invariant linear functionals on certain cohomologically induced representations of $G$, and we show that these linear functionals do not vanish on the bottom layers. Applying this construction, we prove two Archimedean nonvanishing hypotheses which are vital to the arithmetic study of special values of certain $L$-functions via modular symbols.

We propose a precise formula relating the height of certain diagonal cycles on the product of unitary Shimura varieties and the central derivative of some tensor product $L$-functions. This can be viewed as a refinement of the arithmetic Gan–Gross–Prasad conjecture. We use the theory of arithmetic theta lifts to prove some endoscopic cases of it for $U(2)\times U(3)$.