Related papers: Anticanonical minimal models and Zariski decomposi…

We study ample divisors X with only rational singularities on abelian varieties that decompose into a sum of two lower dimensional subvarieties, X=V+W. For instance, we prove an optimal lower bound on the degree of the corresponding…

We are interested in the long time asymptotic behavoir of solutions to the scalar Zakharov system \[ i u_{t} + \Delta u = nu,\] \[n_{tt} - \Delta n= \Delta |u|^2\] and the Klein-Gordon Zakharov system \[ u_{tt} - \Delta u + u = - nu,\] \[…

Let $(G,G^\Gamma)$ be a Klein four symmetric pair. The author wants to classify all the Klein four symmetric pairs $(G,G^\Gamma)$ such that there exists at least one nontrivial unitarizable simple $(\mathfrak{g},K)$-module $\pi_K$ that is…

In this note we show that given a lc pair $(X, \Delta)$, a large enough multiple of the bundle $K_X+ \Delta$ is effective provided that its Chern class contains an effective $\bQ$-divisor.

In this paper, we study the minimal model theory for threefolds in mixed characteristic. As a generalization of a result of Kawamata, we show that the MMP holds for strictly semi-stable schemes over an excellent Dedekind scheme $V$ of…

We prove that the integral points are potentially Zariski dense in the complement of a reduced effective singular anticanonical divisor in a smooth del Pezzo surface, with the exception of $\mathbb{P}^2$ minus three concurrent lines (for…

In this work, the Kaluza-Klein reduction of the Dirac equation on a 6 dimensional spacetime $\mathbb{M}^{1+5} := \mathbb{M}^{1+2} \times \mathbb{S}^3$ is studied. Because of the group structure on $\mathbb{S}^3$, $\mathbb{M}^{1+5}$ can be…

Given a number field $K$, we show that certain $K$-integral representations of closed surface groups can be deformed to being Zariski dense while preserving many useful properties of the original representation. This generalizes a method…

The property of countable metacompactness of a topological space gets its importance from Dowker's 1951 theorem that the product of a normal space X with the unit interval is again normal iff X is countably metacompact. In a recent paper,…

Let X be a compact K\"ahler threefold that is not uniruled. We prove that X has a minimal model.

Let $(X,\Delta)$ be a projective log canonical pair such that $\Delta \geq A$ where $A \geq 0$ is an ample $\mathbb{R}$-divisor. We prove that either $(X,\Delta)$ has a good minimal model or a Mori fibre space. Moreover, if $X$ is…

We prove a differential analogue of Hilbert's irreducibility theorem. Let $\mathcal{L}$ be a linear differential operator with coefficients in $C(\mathbb{X})(x)$ that is irreducible over $\overline{C(\mathbb{X})}(x)$, where $\mathbb{X}$ is…

Semi-log canonical varieties are a higher-dimensional analogue of stable curves. They are the varieties appearing as the boundary $\Delta$ of a log canonical pair $(X,\Delta)$, and also appear as limits of canonically polarized varieties in…

Let $(X,\Delta)$ be a pair. We study how the condition $\kappa(K_X + \Delta)=0$ causes surjectivity or birationality of the Albanese map and the Albanese morphism of $X$ in both characteristic $0$ and characteristic $p > 0$. In particular…

Let $X$ be a compact K\"ahler fourfold with klt singularities and vanishing first Chern class, smooth in codimension two. We show that $X$ admits a Beauville-Bogomolov decomposition: a finite quasi-\'etale cover of $X$ splits as a product…

In this article, we establish the existence of a good minimal model for a compact K\"ahler klt pair $(X, B)$ when the Albanese map of $X$ is a projective morphism and the general fiber of $(X, B)$ has a good minimal model.

Let $\Delta$ be a finite set of nonzero linear forms in several variables with coefficients in a field $\mathbf K$ of characteristic zero. Consider the $\mathbf K$-algebra $C(\Delta)$ of rational functions generated by $\{1/\alpha \mid…

Let $(X,B)$ be a projective log canonical pair such that $B$ is a $\Q$-divisor, and that there is a surjective morphism $f\colon X\to Z$ onto a normal variety $Z$ satisfying: $K_X+B\sim_\Q f^*M$ for some $\Q$-divisor $M$, and the augmented…

Let $G$ be a connected simple non-compact real reductive Lie group with a maximal compact subgroup $K$. This note aims to show that any non-decreasable $K$-type (in the sense of the first named author) is unitarily small (in the sense of…

Let $K$ be a field and $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$. Let $\Delta$ be a simplicial complex on $n$ vertices and $I=I_{\Delta}$ be its Stanley-Reisner ideal. In this paper, we show that if $I$…