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.

We study generalized Kummer surfaces ${\mathrm{Km}}_3(A)$, by which we mean the K3 surfaces obtained by desingularization of the quotient of an abelian surface A by an order-3 symplectic automorphism group. Such a surface carries nine disjoint configurations of two smooth rational curves C, $C^{\prime }$ with $C{C}^{\prime }=1$. This $9{\mathbf{A}}_2$-configuration plays a role similar to the Nikulin configuration of 16 disjoint smooth rational curves on (classical) Kummer surfaces. We study the (generalized) question of Shioda: Suppose that ${\mathrm{Km}}_3(A)$ is isomorphic to ${\mathrm{Km}}_3(B)$. Does that imply that A and B are isomorphic? We answer by the negative in general with two methods: a link between that problem and Fourier–Mukai partners of A, and construction of $9{\mathbf{A}}_2$-configurations on ${\mathrm{Km}}_3(A)$ that cannot be exchanged under the automorphism group.

We study the least doubling constant $C_G$ among all possible doubling measures defined on a path graph G. We consider both finite and infinite cases and show that if $G=\mathbb{Z}$, $C_{\mathbb{Z}}=3$, while for $G=L_n$ the path graph with n vertices, one has $1+2cos(\frac{\mathrm{\pi }}{n+1})\le C_{L_n}<3$, with equality on the lower bound if and only if $n\le 8$. Moreover, we analyze the structure of doubling minimizers on $L_n$ and $\mathbb{Z}$—those measures whose doubling constant is the smallest possible.

Tropical toric varieties are partial compactifications of finite dimensional real vector spaces associated with rational polyhedral fans. We introduce plurisubharmonic functions and a Bedford–Taylor product for Lagerberg currents on open subsets of a tropical toric variety. The resulting tropical toric pluripotential theory provides the link to give a canonical correspondence between complex and non-Archimedean pluripotential theories of invariant plurisubharmonic functions on toric varieties. We will apply this correspondence to solve invariant non-Archimedean Monge–Ampère equations on toric and abelian varieties over arbitrary non-Archimedean fields.

We study the existence of ample uniruled divisors on irreducible holomorphic symplectic manifolds that are deformations of the 10-dimensional example introduced by O’Grady. In particular, we show that for any polarized O’Grady’s 10-dimensional ($OG10$) manifold lying in four specific connected components of the moduli space of polarized $OG10$ manifolds, there exists a multiple of the polarization that is the class of a uniruled divisor.

In this paper, we prove an explicit matching theorem for some Hecke elements in the case of (possibly ramified) cyclic base change for general linear groups over local fields of characteristic zero with odd residual characteristic under a mild assumption. A key observation, based on the works of Waldspurger and Ganapathy–Varma, is to regard the base change lifts with twisted endoscopic lifts and replace the condition for the matching orbital integrals with one for semisimple descent in the twisted space according to Waldspurger’s fundamental work, “L’endoscopie tordue n’est pas si tordue.”

Questions that seek to determine whether a hyperplane arrangement property, be it geometric, arithmetic, or topological, is of a combinatorial nature (i.e., determined by the intersection lattice) are abundant in the literature. To tackle such questions and provide a negative answer, one of the most effective methods is to produce a counterexample. To this end, it is essential to know how to construct arrangements that are lattice equivalent. The more different they are, the more efficient it will be. In this paper, we present a method to construct arrangements of complex projective lines that are lattice equivalent but lie in distinct connected components of their moduli space. To illustrate the efficiency of the method, we apply it to reconstruct all the classical examples of arrangements with disconnected moduli spaces: MacLane, Falk–Sturmfels, Nasir–Yoshinaga, and Rybnikov. Moreover, we employ this method to produce novel examples of arrangements of eleven lines whose moduli spaces are formed by four connected components.