Related papers: Klt varieties of general type with small volume

The purpose of this note is twofold. First, we give a quick proof of Ballico-Chiantini's theorem stating that a Fano or Calabi-Yau variety of dimension at least 4 in codimension two is a complete intersection. Second, we improve Barth-Van…

We prove two rigidity theorems for open (complete and noncompact) $n$-manifolds $M$ with nonnegative Ricci curvature and the infimum of volume growth order $<2$. The first theorem asserts that the Riemannian universal cover of $M$ has…

For n >= 2 a construction is given for a large family of compact convex sets K and L in n-dimensional Euclidean space such that the orthogonal projection L_u onto the subspace u^\perp contains a translate of the corresponding projection K_u…

We prove new boundedness results across different areas of algebraic geometry, stemming from a unifying technical starting point: bounding the integer $q > 0$ such that the $q$-th Hodge bundle becomes (semi-)positive for families of stable…

We generalize the Guth--Katz joints theorem from lines to varieties. A special case says that $N$ planes (2-flats) in 6 dimensions (over any field) have $O(N^{3/2})$ joints, where a joint is a point contained in a triple of these planes not…

We prove the existence of good minimal models for any klt algebraically integrable adjoint foliated structure of general type, and that Fano algebraically integrable adjoint foliated structures with total minimal log discrepancies and…

Let $(X,B)$ be an $\epsilon$-lc pair of dimension $d$ with a closed point $x\in X$. Birkar and Shokurov conjectured that there is an effective Cartier divisor $H$ passing through $x$ such that $(X,B+tH)$ is lc near $x$, where $t$ is a…

Given a knot in 3-space, one can associate a sequence of Laurrent polynomials, whose $n$th term is the $n$th colored Jones polynomial. The Generalized Volume Conjecture states that the value of the $n$-th colored Jones polynomial at $\exp(2…

We prove that the abundance conjecture holds on a variety $X$ with mild singularities if $X$ has many reflexive differential forms with coefficients in pluricanonical bundles, assuming the Minimal Model Program in lower dimensions. This…

Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V)>0$ and $P_{24}(V)>1$ (which answers an open problem of J. Kollar and S. Mori). We also prove that the canonical volume has an universal lower bound…

In this paper we give lower and upper bounds for the volume growth of a regular hyperbolic simplex, namely for the ratio of the $n$-dimensional volume of a regular simplex and the $(n-1)$-dimensional volume of its facets. In addition to the…

Smoktunowicz, Lenagan, and the second-named author recently gave an example of a nil algebra of Gelfand-Kirillov dimension at most three. Their construction requires a countable base field, however. We show that for any field $k$ and any…

We construct a family of examples of complete $(2+n)-$dimensional ($n\ge 2$) open manifolds with positive Ricci curvature, sectional curvature bounded from below and infinite Betti numbers $b_2,b_n$, moreover its volume growth can be…

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…

Let $n\geq 2$ be any integer. We study the optimal lower bound $v_{n, n-i}$ of the canonical volume and the optimal upper bound $r_{n,n-i}$ of the canonical stability index for minimal projective $n$-folds of general type, which are…

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…

In this paper, we study the explicit geometry of threefolds, in particular, Fano varieties. We find an explicitly computable positive integer $N$, such that all but a bounded family of Fano threefolds have $N$-complements. This result has…

We prove that in any fixed dimension, K-semistable log Fano cone singularities whose volumes are bounded from below by a fixed positive number form a bounded set. As a consequence, we show that the set of local volumes of klt singularities…

We construct log canonical pairs $(X,B)$ with $B$ a nonzero reduced divisor and $K_X+B$ ample that have the smallest known volume. We conjecture that our examples have the smallest volume in each dimension. The conjecture is true in…

For all nonsingular projective $n$-folds $V$ of general type, we prove the existence of Noether type inequalities in the following form: $$\text{vol}(V)\geq a_{n,k}h^0(\Omega_V^k)-b_{n,k}$$ where $0< k\leq n$, $a_{n,k}$ and $b_{n,k}$ are…