Varieties of general type with doubly exponential asymptotics

Abstract

We construct smooth projective varieties of general type with the smallest known volume and others with the most known vanishing plurigenera in high dimensions. The optimal volume bound is expected to decay doubly exponentially with dimension, and our examples achieve this decay rate. We consider the analogous questions for other types of varieties. For example, in every dimension we conjecture the terminal Fano variety of minimal volume, and the canonical Calabi-Yau variety of minimal volume. In each case, our examples exhibit doubly exponential behavior.

Pluricanonical Maps and the Fujita Conjecture

“…Added in proof: Burt Totaro mentioned to me that similar examples are to be found in Theorem 3 of [BPT13], which contains also other examples where the minimal model does not have a Cartier canonical divisor K X . Under the condition that the minimal models have a canonical divisor K X which is not Cartier, then there are recent more extreme and striking examples by Esser, Wang, and Totaro; these examples are such that there are about 2 2 n/2 vanishing plurigenera in dimension n, see Theorem 1.1 of [ETW21] and ensuing discussion.…”

Remark on complements on surfaces

“…Similar topics and alternative directions include the estimation of the lower bound n (cf. [15,42]), the boundedness of the the anti-canonical volume of Fano varieties (cf. [35,36,11,37,12,38,13,14,23,22,24,4]), estimation of (𝜖, 𝑛)-complement [9], the explicit M c Kernan-Shokurov conjecture [18], precise bounds of mlds [21,33,31], etc.…”

On explicit birational geometry for minimal n-folds of canonical dimension n-1

“…• By Esser-Totaro-Wang [22], for n ≥ 3, r n > 2 2 (n−2)/2 and v n < 1 2 2 n/2 . For a given minimal variety X of general type, the canonical dimension of X is defined as d 1 := dim ϕ 1 (X).…”

“…In each dimension, we will present optimal examples achieving the bounds in ( 1) and ( 2), which will be weighted projective hypersurfaces of general type. For an introduction to weighted projective hypersurfaces, see [22,Section 2].…”