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We prove Yun's minimal reduction conjecture for all classical groups. More precisely, for any topologically nilpotent regular semisimple element $\gamma$, we show that the associated minimal reduction set…

The author has been interested in regions surrounded by cylinders of real algebraic hypersurfaces and their shapes and polynomials associated to them. Here, we formulate and investigate natural decompositions into such cylinders of real…

We prove equivalent numerical conditions for a complete spherical variety to admit a toric structure, and for the smoothness of an arbitrary spherical variety along any given G-orbit. The conditions are in terms of spherical skeletons, a…

We study the infinitesimal variation of Hodge structure associated with families of reduced algebraic curves with singularities. The analysis applies to curves beyond the nodal case and is not restricted to plane curves, encompassing curves…

In this article we give corrections and addendum to the article ``Flops and Poisson deformations of symplectic varieties, Publ. Res. Inst. Math. Sci. {\bf 44} (2008) 259 - 314''.

We prove a Cartier duality for gerbes of algebraic and analytic vector bundles as an anti-equivalence of Hopf algebras in the category of kernels of analytic stacks. As an application, we prove that the category of solid quasi-coherent…

We introduce a notion of restricted pyramid configurations for computing the 1-leg Donaldson-Thomas $\mathbb{Z}_2\times\mathbb{Z}_2$-vertex. We study a special type of restricted pyramid configurations with the prescribed 1-leg partitions,…

We study the collection of measures obtained via push-forward along a map between smooth varieties over p-adic fields. We investigate when the stalks of this collection are finite-dimensional. We provide an algebro-geometric criterion…

The $\mathbb{A}^1$-Euler characteristic is a refinement in algebraic geometry of the classical topological Euler characteristic, which can be constructed using motivic homotopy theory. This invariant is a quadratic form rather than an…

In this note, we show that for a smooth algebraic variety $Y$ and a smooth $m$-section $X$ of the $\mathbb{P}^1$-bundle \[ f : \mathbb{P}(\mathcal{O}_Y \oplus \mathcal{O}_Y(E)) \longrightarrow Y, \] where $E$ is an effective divisor on $Y$…

For each $N\geq 2$, Asakura and Otsubo have recently introduced a smooth family of algebraic curves $\{X_{N,\lambda}\}_{\lambda \in \mathbb{P}^1\setminus \{0, 1, \infty\}}$ in characteristic 0 that is closely related to hypergeometric…

We generalize the equivariant intermediate Jacobian torsor obstruction over $\mathbb{C}$ to algebraically closed fields of characteristic zero. It is an obstruction to the (projective) linearizability problem of finite group actions on…

We investigate a slight weakening of the classical property of strong approximation, which we call almost strong approximation, for connected reductive algebraic groups over global fields with respect to special sets of valuations. While…

Let $X$ be a smooth proper rigid analytic space over a complete algebraically closed field extension $K$ of $\mathbb{Q}_p$. We establish a Hodge--Tate decomposition for $X$ with $G$-coefficients, where $G$ is any commutative locally…

We prove a crepant transformation correspondence in genus zero Gromov-Witten theory for toric stack bundles related by crepant wall-crossings of the toric fibers. Specifically, we construct a symplectic transformation that identifies…

Let $C$ be a completely algebraic closed non-archimedean field over $\mathbb{Q}_p$ and $\alpha,r$ be two positive integers. Denote by $B_\alpha$ the ring $\mathbb{B}_{\mathrm{dR}}^+(C)/(\ker\theta)^\alpha$. This paper first constructs a…

In this paper, we first prove that every Mori dream surface $X$ satisfies the bounded cohomology property (BCP for short). Namely, there exists a constant $c_X>0$ such that $h^1(\mathcal O_X(C))\le c_Xh^0(\mathcal O_X(C))$ for every curve…

This semi-expository work covers central aspects of the theory of relative tensor products as developed in Higher Algebra, as well as their application to Koszul duality for algebras in monoidal oo-categories. Part of our goal is to expand…

A primitive multiple scheme is a Cohen-Macaulay scheme $Y$ such that the associated reduced scheme $X=Y_{red}$ is smooth, irreducible, and that $Y$ can be locally embedded in a smooth variety of dimension $\dim(X)+1$. If $n$ is the…

We develop a new method for the computation of $(3,3)$-isogenies between principally polarized abelian surfaces. The idea is to work with models in $\mathbb{P}^8$ induced by a symmetric level-$3$ theta structure. In this setting, the action…