We introduce a diagrammatic monoidal category $\mathcal{\mathscr{H}}ei{s}_k\left(z,t\right)$ which we call the quantum Heisenberg category; here, $k\in \mathbb{Z}$ is “central charge” and $z$ and $t$ are invertible parameters. Special cases were known before: for central charge $k=-1$ and parameters $z=q-q^{-1}$ and $t=-z^{-1}$ our quantum Heisenberg category may be obtained from the deformed version of Khovanov’s Heisenberg category introduced by Licata and Savage by inverting its polynomial generator, while $\mathcal{\mathscr{H}}ei{s}_0\left(z,t\right)$ is the affinization of the HOMFLY-PT skein category. We also prove a basis theorem for the morphism spaces in $\mathcal{\mathscr{H}}ei{s}_k\left(z,t\right)$.

We compute the $L$-characteristic cycle of an $A$-hypergeometric system and higher Euler–Koszul homology modules of the toric ring. We also prove upper semicontinuity results about the multiplicities in these cycles and apply our results to analyze the behavior of Gevrey solution spaces of the system.

Let $G$ be a finite subgroup of $SL\left(2,k\right)$ and let $R=k{\left[x,y\right]}^G$ be the coordinate ring of the corresponding Kleinian singularity. In 1998, Crawley-Boevey and Holland defined deformations ${\mathcal{𝒪}}^{\lambda }$ of $R$ parametrised by weights $\lambda$. In this paper, we determine the singularity categories ${\mathcal{𝒟}}_{sg}\left({\mathcal{𝒪}}^{\lambda }\right)$ of these deformations, and show that they correspond to subgraphs of the Dynkin graph associated to $R$. This generalises known results on the structure of ${\mathcal{𝒟}}_{sg}\left(R\right)$. We also provide a generalisation of the intersection theory appearing in the geometric McKay correspondence to a noncommutative setting.

In this paper we prove that the $p$-adic $L$-function that interpolates the Rankin–Selberg product of a general modular form and a CM form of higher weight divides the characteristic ideal of the corresponding Selmer group. This is one divisibility of the Iwasawa main conjecture for this $p$-adic $L$-function. We prove this conjecture using congruences between Klingen–Eisenstein series and cusp forms on the group $GU\left(3,1\right)$, following the strategy of recent work by C. Skinner and E. Urban. The actual argument is, however, more complicated due to the need to work with general Fourier–Jacobi expansions. This theorem is used to deduce a converse of the Gross–Zagier–Kolyvagin theorem and the $p$-adic part of the precise BSD formula in the rank one case.

Let $X$ be a very general hypersurface of degree $d$ in $P^n$. We investigate positivity properties of the spaces $R_e\left(X\right)$ of degree $e$ rational curves in $X$. We show that for small $e$, $R_e\left(X\right)$ has no rational curves meeting the locus of smooth embedded curves. We show that for $n\le d$, there are no rational curves other than lines in the locus $Y\subset X$ swept out by lines. We exhibit differential forms on a smooth compactification of $R_e\left(X\right)$ for every $e$ and $n-2\ge d\ge \frac{1}{2}\left(n+1\right)$.

We define and study a new class of bialgebras, which generalize certain Turner double algebras related to generic blocks of symmetric groups. Bases and generators of these algebras are given. We investigate when the algebras are symmetric which is relevant to block theory of finite groups. We then establish a double centralizer property related to blocks of Schur algebras.