The purpose of this article is to show that under a part of the generalized Arthur’s $A$-packet conjecture, locally generic cuspidal automorphic representations of a quasi-split group over a number field are of Ramanujan type, that is, are tempered at almost all places. The $A$-packet conjecture allows one to reduce the problem to a special case of a general local question about the components of the corresponding Langlands $L$-packet which is then answered here in its generality.

In this article we study the asymptotic behavior of the regularity of symbolic powers of ideals of points in a weighted projective plane. By a result of Cutkosky, Ein, and Lazarsfeld, regularity of such powers behaves asymptotically like a linear function, which is deeply related to the Seshadri constant of a blowup. We study the difference between regularity of such powers and this linear function. Under some conditions, we prove that this difference is bounded or eventually periodic.

As a corollary, we show that if there exists a negative curve, then the regularity of symbolic powers of a monomial space curve is eventually a periodic linear function. We give a criterion for the validity of Nagata’s conjecture in terms of the lack of existence of negative curves.

The main purpose of this article is to define the notion of Du Bois singularities for pairs and prove a vanishing theorem by using this new notion. The main vanishing theorem specializes to a new vanishing theorem for resolutions of log canonial singularities.

Motivated by a recent conjecture by Hernandez and Leclerc, we embed a Fomin-Zelevinsky cluster algebra into the Grothendieck ring $\mathbf{R}$ of the category of representations of quantum loop algebras ${\mathbf{U}}_q\left(\mathbf{L}\mathfrak{g}\right)$ of a symmetric Kac-Moody Lie algebra, studied earlier by the author via perverse sheaves on graded quiver varieties. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in $\mathbf{R}$, or Lusztig’s dual canonical base. The conjectures that cluster monomials are positive and linearly independent (and probably many other conjectures) of Fomin and Zelevinsky follow as consequences when there is a seed with a bipartite quiver. Simple modules corresponding to cluster monomials factorize into tensor products of “prime” simple ones according to the cluster expansion.

The $F$-threshold $c^J\left(\mathfrak{a}\right)$ of an ideal $\mathfrak{a}$ with respect to an ideal $J$ is a positive characteristic invariant obtained by comparing the powers of $\mathfrak{a}$ with the Frobenius powers of $J$. We study a conjecture formulated in an earlier article that we authored with M. Mustaţă, which bounds $c^J\left(\mathfrak{a}\right)$ in terms of the multiplicities $e\left(\mathfrak{a}\right)$ and $e\left(J\right)$ when $\mathfrak{a}$ and $J$ are zero-dimensional ideals and $J$ is generated by a system of parameters. We prove the conjecture when $\mathfrak{a}$ and $J$ are generated by homogeneous systems of parameters in a Noetherian graded $k$-algebra. We also prove a similar inequality involving, instead of the $F$-threshold, the jumping number for the generalized parameter test submodules.

We construct an enlargement of the classifying space of mixed Hodge structures with polarized graded quotients by adding mixed Hodge theoretic version of $SL\left(2\right)$-orbits. This space has a real analytic structure and a log structure with sign. The $SL\left(2\right)$-orbit theorem in several variables for mixed Hodge structures can be understood naturally with this space.