We continue the study of the closures of $GL\left(V\right)$-orbits in the enhanced nilpotent cone $V\times \mathcal{N}$ begun by the first two authors. We prove that each closure is an invariant-theoretic quotient of a suitably defined enhanced quiver variety. We conjecture, and prove in special cases, that these enhanced quiver varieties are normal complete intersections, implying that the enhanced nilpotent orbit closures are also normal.

Let $l$ be a prime number. In this paper, we prove that the isomorphism class of an $l$-monodromically full hyperbolic curve of genus zero over a finitely generated extension of the field of rational numbers is completely determined by the kernel of the natural pro-$l$ outer Galois representation associated to the hyperbolic curve. This result can be regarded as a genus zero analogue of a result due to Mochizuki which asserts that the isomorphism class of an elliptic curve which does not admit complex multiplication over a number field is completely determined by the kernels of the natural Galois representations on the various finite quotients of its Tate module.

Let $A$ be a self-injective algebra over an algebraically closed field. We study the stable dimension of $A$, which is the dimension of the stable module category of $A$ in the sense of Rouquier. Then we prove that $A$ is representation-finite if the stable dimension of $A$ is zero.

Let $L$ be a nonnegative self-adjoint operator on $L^2\left(X\right)$, where $X$ is a space of homogeneous type. Assume that $L$ generates an analytic semigroup $e^{-tL}$ whose kernel satisfies the standard Gaussian upper bounds. We prove that the spectral multiplier $F\left(L\right)$ is bounded on $H_L^p\left(X\right)$ for $0<p\le 1$, the Hardy space associated to operator $L$, when $F$ is a suitable function.

In this paper, we construct (many) Kolyvagin systems out of Stickelberger elements utilizing ideas borrowed from our previous work on Kolyvagin systems of Rubin-Stark elements. The applications of our approach are twofold. First, assuming Brumer’s conjecture, we prove results on the odd parts of the ideal class groups of CM fields which are abelian over a totally real field, and we deduce Iwasawa’s main conjecture for totally real fields (for totally odd characters). Although this portion of our results has already been established by Wiles unconditionally (and refined by Kurihara using an Euler system argument, when Wiles’s work is assumed), the approach here fits well in the general framework the author has developed elsewhere to understand Euler/Kolyvagin system machinery when the core Selmer rank is $r>1$ (in the sense of Mazur and Rubin). As our second application, we establish a rather curious link between the Stickelberger elements and Rubin-Stark elements by using the main constructions of this article hand in hand with the “rigidity” of the collection of Kolyvagin systems proved by Mazur, Rubin, and the author.

We characterize the adjoint $G_2$ orbits in the Lie algebra $\mathfrak{g}$ of $G_2$ as fibered spaces over $S^6$ with fibers given by the complex Cartan hypersurfaces. This combines the isoparametric hypersurfaces of case $(g,m)=(6,2)$ with case $(3,2)$. The fibrations on two singular orbits turn out to be diffeomorphic to the twistor fibrations of $S^6$ and $G_2/SO\left(4\right)$. From the symplectic point of view, we show that there exists a 2-parameter family of Lagrangian submanifolds on every orbit.