Related papers: On the Q-divisor method and its application

We show that 3-fold terminal flips and divisorial contractions may be factored into a sequence of flops, blow-downs to a smooth curve in a smooth 3-fold or divisorial contractions to points with minimal discrepancies.

Given an NQC log canonical generalized pair $(X,B+M)$ whose underlying variety $X$ is not necessarily $\mathbb{Q}$-factorial, we show that one may run a $(K_X+B+M)$-MMP with scaling of an ample divisor which terminates, provided that…

We prove that for every compact, connected, differentiable 3--manifold $M$ there is a compact complex manifold $X$ which can be obtained from projective 3--space by a sequence of smooth, real blow ups and downs such that $M$ is…

We use a generalised Kummer construction to realise all but one known weight four newforms with complex multiplication and rational Fourier coefficients in smooth Calabi-Yau threefolds defined over the rational numbers. The Calabi-Yau…

We employ three different methods to prove the following result on prescribed scalar curvature plus mean curvature problem: Let $(M^n,g_0)$ be a $n$-dimensional smooth compact manifold with boundary, where $n \geq 3$, assume the conformal…

We construct a surface with irregularity $q=2,$ geometric genus $p_g=3,$ self-intersection of the canonical divisor $K^2=16$ and canonical map of degree $16.$

We give a bound on which singularities may appear on Koll\'ar--Shepherd-Barron--Alexeev stable surfaces for a wide range of topological invariants and use this result to describe all stable numerical quintic surfaces (KSBA-stable surfaces…

We prove that there exists a positive integer $\nu_{n}$ depending only on $n$ such that for every smooth projective $n$-fold of general type $X$ defined over {\bf C}, $\mid mK_{X}\mid$ gives a birational rational map from $X$ into a…

Let $X$ be a smooth irreducible complex projective curve of genus $g\,\geq\, 2$, and let $D\,=\,x_1+\dots+x_r$ be a reduced effective divisor on $X$. Denote by $U_{\alpha}(L)$ the moduli space of stable parabolic vector bundles on $X$ of…

We determine the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong K\"ahler with torsion…

This paper concerns the construction of minimal varieties with small canonical volumes. The first part devotes to establishing an effective nefness criterion for the canonical divisor of a weighted blow-up over a weighted hypersurface, from…

This is the third of a series of papers studying real algebraic threefolds, but the methods are mostly independent from the previous two. Let $f:X\to S$ be a map of a smooth projective real algebraic 3-fold to a surface $S$ whose general…

Let (X,D) be a dlt pair, where X is a normal projective variety. Let S denote the support of the rounddown of D, and K the canonical divisor of X. We show that any smooth family of canonically polarized varieties over X\S is isotrivial if…

The aim of this note is to give an alternative proof of a theorem of Koras and Russell, that is, a characterization of smooth contractible affine varieties endowed with a hyperbolic action of the group…

We prove two results. First, we establish that the normal bundle of any smooth curve of genus 7 having maximal Clifford index is stable. Note that 7 is the smallest genus for which such a result could possibly hold. We then show that rank…

Let $S$ be a rational surface with $\dim|-K_S|\ge 1$ and let $\pi: X\rightarrow S$ be a ramified cyclic covering from a nonruled smooth surface $X$. We show that for any integer $k\ge 3$ and ample divisor $A$ on $S$, the adjoint divisor…

In this paper, we establish a structure theorem for minimal projective klt varieties $X$ that satisfiy Miyaoka's equality $3c_2(X) = c_1(X)^2$. Specifically, we prove that the canonical divisor $K_X$ is semi-ample and that the Kodaira…

We study the conormal sheaves and singular schemes of 1-dimensional foliations on smooth projective varieties $X$ of dimension 3 and Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is…

For a weighted projective line, the stable category of its vector bundles modulo lines bundles has a natural triangulated structure. We prove that, for any positive integers $p, q, r$ and $r'$ with $r'\leq r$, there is an explicit…

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…