Related papers: Boundedness and volume of generalised pairs
We show that minimal models of $\mathbb{Q}$-factorial NQC log canonical generalised pairs exist, assuming the existence of minimal models of smooth varieties. More generally, we prove that on a $\mathbb{Q}$-factorial NQC log canonical…
In this paper we study regularity and topological properties of volume constrained minimizers of quasi-perimeters in $\sf RCD$ spaces where the reference measure is the Hausdorff measure. A quasi-perimeter is a functional given by the sum…
We develop the moduli theory of boundary polarized CY pairs, which are slc Calabi-Yau pairs $(X,D)$ such that $D$ is ample. The motivation for studying this moduli problem is to construct a moduli space at the Calabi-Yau wall interpolating…
In this article, we study how the absolute coregularity of a projective log pair reflects on its fundamental group. More precisely, we conjecture that for a projective klt log pair $(X,D)$ of absolute coregularity $c$ (and arbitrary…
We study lc pairs polarized by a nef and log big divisor. After proving the minimal model theory for projective lc pairs polarized by a nef and log big divisor, we prove the effectivity of the Iitaka fibrations and some boundedness results…
We provide bounds linear in the rank for the generalized conjugacy diameters, introduced by Kedra, Libman and Martin, for the special linear and symplectic groups defined over the rings of integers of global fields by way of using certain…
In this paper, we construct counterexamples to the boundedness of generalised log canonical models of surfaces with fixed appropriate invariants, where the underlying varieties can have arbitrary Kodaira dimension. This answers a question…
Let $(X, \Delta)/U$ be klt pairs and $Q$ be a convex set of divisors. Assuming that the relative Kodaira dimensions are non-negative, then there are only finitely many log canonical models when the boundary divisors varying in a relatively…
We show that volumes and related limits in terms of the Kodaira-Iitaka dimension exist for line bundles and graded linear series on compact reduced complex analytic spaces.
In this article we study conjectures regarding normalized volume and boundedness of singularities. We focus on singularities with a torus action of complexity 1, threefold singularities, and hypersurface singularities. Given a real value…
We study the behavior of generalized lc pairs with $\mathrm{\textbf b}$-log abundant nef part, a meticulously designed structure on algebraic varieties. We show that this structure is preserved under the canonical bundle formula and…
Stable surfaces and their log analogues are the type of varieties naturally occuring as boundary points in moduli spaces. We extend classical results of Kodaira and Bombieri to this more general setting: if $(X,\Delta)$ is a stable log…
Take a set of balls in $\mathbb R^d$. We find a subset of pairwise disjoint balls whose combined perimeter controls the perimeter of the union of the original balls. This can be seen as a boundary version of the Vitali covering lemma. We…
It is well known that a pair of compact sets in $\mathbb{R}^d$ ($d \in \mathbb{N}$) can be separated by small deformations if the sum of their upper box dimensions is less than $d$. In this paper, we demonstrate that this dimension…
We study deformations of pairs (X,D), with X smooth projective variety and D a smooth or a normal crossing divisor, defined over an algebraically closed field of characteristic 0. Using the differential graded Lie algebras theory and the…
In this paper, we continue to develop the theories on functional pairs and uniform rational polytopes. We show that there is a uniform perturbation for Iitaka dimensions of pseudo-effective lc pairs of fixed dimension with DCC coefficients…
We use the canonical bundle formula for parabolic fibrations to give an inductive approach to the generalized abundance conjecture using nef reduction. In particular, we observe that generalized abundance holds for a klt pair $(X,B)$ if the…
Let $\kk$ be an algebraically closed field of characteristic zero and $\KK$ a finitely generated field over $\kk$. Let $\Sigma$ be a central simple $\KK$-algebra, $X$ a normal projective model of $\KK$ and $\Lambda$ a sheaf of maximal…
In this paper we study boundedness of conjugation invariant norms on diffeomorphism groups of manifold pairs. For the diffeomorphism group ${\mathcal D} \equiv {\rm Diff}(M,N)_0$ of a closed manifold pair $(M, N)$ with $\dim N \geq 1$,…
We consider the moduli space of log smooth pairs formed by a cubic surface and an anticanonical divisor. We describe all compactifications of this moduli space which are constructed using Geometric Invariant Theory and the anticanonical…