Related papers: Log canonical pairs with conjecturally minimal vol…

We study the volumes of divisors in Calabi--Yau type varieties. We show that given a klt Calabi--Yau pair $(X,B)$ and an integral divisor $A$ on $X$, the volume of $A$ is in a fixed discrete set depending only on the dimension and…

We discuss the minimal model program for b-log varieties, which is a pair of a variety and a b-divisor, as a natural generalization of the minimal model program for ordinary log varieties. We show that the main theorems of the log MMP work…

We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in \Bbb R^3.

We prove that the existence of log minimal models in dimension $d$ essentially implies the LMMP with scaling in dimension $d$. As a consequence we prove that a weak nonvanishing conjecture in dimension $d$ implies the minimal model…

In this paper we investigate boundedness and volumes of generalised pairs, and give applications to usual pairs especially to a class of pairs that we call stable log minimal models. Fixing the dimension and a DCC set controlling…

We show that if $(X,Y)$ is a simple normal crossings log Calabi--Yau pair, then there is a real torus of dimension equal to the codimension of the smallest stratum of $Y$ which can be used to construct $W_{2k-1}H^k(X \setminus…

We prove the macroscopic cousins of three conjectures: 1) a conjectural bound of the simplicial volume of a Riemannian manifold in the presence of a lower scalar curvature bound, 2) the conjecture that rationally essential manifolds do not…

Following Shokurov's ideas, we give a short proof of the following klt version of his result: termination of terminal log flips in dimension d implies that any klt pair of dimension d has a log minimal model or a Mori fibre space. Thus, in…

We construct a moduli space of stable projective pairs with a nontrivial action of a connected reductive group. These stable reductive pairs are higher-dimensional analogs of stable n-pointed curves and generalize to the non-commutative…

We study log Calabi--Yau pairs of the form $(\mathbb{P}^3,\Delta)$, where $\Delta$ is a quartic surface, and classify all such pairs of coregularity less than or equal to one, up to volume preserving equivalence. In particular, if…

We treat two different topics on the log minimal model program, especially for four-dimensional log canonical pairs. (a) Finite generation of the log canonical ring in dimension four. (b) Abundance theorem for irregular fourfolds. We obtain…

It is shown that $m$ disjoint sets with fixed Gaussian volumes that partition $\mathbb{R}^{n}$ with minimum Gaussian surface area must be $(m-1)$-dimensional. This follows from a second variation argument using infinitesimal translations.…

We formulate an effective cone conjecture for klt Calabi--Yau pairs $(X,\Delta)$, pertaining to the structure of the cone of effective divisors $\mathrm{Eff}(X)$ modulo the action of the subgroup of pseudo-automorphisms…

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…

Let $(X,B)$ be a log canonical pair over a normal variety $Z$ with maximal Albanese dimension. If $K_X+B$ is relatively abundant over $Z$ (for example, $K_X+B$ is relatively big over $Z$), then we prove that $K_X+B$ is abundant. In…

In this paper, we prove that the log minimal model program in dimension $d-1$ implies the existence of log minimal models for effective lc pairs (eg of nonnegative Kodaira dimension) in dimension $d$. In fact, we prove that the same…

For certain quasismooth Calabi-Yau hypersurfaces in weighted projective space, the Berglund-H\"{u}bsch-Krawitz (BHK) mirror symmetry construction gives a concrete description of the mirror. We prove that the minimal log discrepancy of the…

We show that a weak version of the canonical bundle formula holds for fibrations of relative dimension one. We provide various applications thereof, for instance, using the recent result of Xu and Zhang, we prove the log non-vanishing…

We prove that the first gap of $\mathbb R$-complementary thresholds of surfaces is $\frac{1}{13}$. More precisely, the largest $\mathbb R$-complementary threshold for surfaces that is strictly less than $1$ is $\frac{12}{13}$. This result…

We study the convergence of volume forms on a degenerating holomorphic family of log-Calabi-Yau varieties to a non-Archimedean measure, extending a result of Boucksom and Jonsson. More precisely, let $(X,B)$ be a holomorphic family of sub…