Related papers: ACC for local volumes and boundedness of singulari…

We prove the ACC conjecture for local volumes. Moreover, when the local volume is bounded away from zero, we prove Shokurov's ACC conjecture for minimal log discrepancies.

Recent study in K-stability suggests that klt singularities whose local volumes are bounded away from zero should be bounded up to special degeneration. We show that this is true in dimension three, or when the minimal log discrepancies of…

We consider the following conjecture: on a klt germ (X,x), for every finite set I there is a positive integer N with the property that for every R-ideal J on X with exponents in I, there is a divisor E over X that computes the minimal log…

In this article we prove the following version of the Weak-BAB conjecture for $3$-folds in char $p>5$: Fix a DCC set $I\subset [0, 1)$ and an algebraically closed field $k$ of characteristic $p>5$. Let $\mathfrak{D}$ be a collection of klt…

We prove that a set of density one satisfies the local-global conjecture for integral Apollonian gaskets. That is, for a fixed integral, primitive Apollonian gasket, almost every (in the sense of density) admissible (passing local…

The classical double bubble theorem characterizes the minimizing partitions of $\mathbb{R}^n$ into three chambers, two of which have prescribed finite volume. In this paper we prove a variant of the double bubble theorem in which two of the…

Let $\Gamma$ be a finite set, and $X\ni x$ a fixed klt germ. For any lc germ $(X\ni x,B:=\sum_{i} b_iB_i)$ such that $b_i\in \Gamma$, Nakamura's conjecture, which is equivalent to the ACC conjecture for minimal log discrepancies for fixed…

We show that a set of K-semistable log Fano cone singularities is bounded if and only if their local volumes are bounded away from zero, and their minimal log discrepancies of Koll\'ar components are bounded from above. As corollaries, we…

The swampland distance conjecture (SDC) addresses the ability of effective field theory to describe distant points in moduli space. It is natural to ask whether there is a local version of the SDC: is it possible to construct local…

The Abundance conjecture predicts that on a minimal projective klt pair $(X,\Delta)$, the adjoint divisor $K_X+\Delta$ is semiample. When $\chi(X,\mathcal O_X)\neq0$, we give a necessary and sufficient condition for the conjecture to hold…

Given an ambient variety $X$ and a fixed subvariety $Z$ we give sufficient conditions for the existence of a boundary $\Delta$ such that $Z$ is a log canonical center for the pair $(X, \Delta)$. We also show that under some additional…

The strong cosmic censorship conjecture states (SCCC) that one cannot extend spacetime beyond the Cauchy horizon with a square-integrable connection. This conjecture was postulated to save the deterministic nature of the most successful…

We establish an optimal upper bound for local volumes of Gorenstein canonical non-hypersurface threefold singularities. Specifically, we show that a klt threefold singularity with local volume at least $9$ is either a hypersurface…

The Adams conjecture states that the local theta correspondence sends a local Arthur packet to another local Arthur packet. M{\oe}glin confirmed the conjecture when lifting to groups of sufficiently high rank and also showed that it fails…

In this note, we estimate the upper bound of volume of closed positively or nonnegatively curved Alexandrov space $X$ with strictly convex boundary. We also discuss the equality case. In particular, the Boundary Conjecture holds when the…

We prove that the ACC conjecture for minimal log discrepancies holds for threefolds in $[1-\delta,+\infty)$, where $\delta>0$ only depends on the coefficient set. We also study Reid's general elephant for pairs, and show Shokurov's…

We show that local Lorentz covariance arises canonically as the group of transformations between local thermal states in the framework of Local Quantum Physics, given the following three postulates: (i) Local observable algebras are…

We make some improvements to our previous results. First, we prove a version of our volume growth theorem which does not require any assumption on the first Betti number. Second, we show that our local regularity theorem only requires a…

In this article we introduce a generalization of locally conformally Kaehler metrics from complex manifolds to complex analytic spaces with singularities and study which properties of locally conformally Kaehler manifolds still hold in this…

Shokurov's ACC Conjecture says that the set of all log canonical thresholds on varieties of bounded dimension satisfies the Ascending Chain Condition. This conjecture was proved for log canonical thresholds on smooth varieties in [EM1].…