Numerical Kodaira Dimension

2014 ◽  
pp. 91-105
Author(s):  
Marco Brunella
Keyword(s):  
Kodaira Dimension

scholarly journals Elliptic Fibrations and Kodaira Dimensions of Schreieder’s Varieties

2021 ◽  
Author(s):  
Alice Garbagnati
Keyword(s):  
K3 Surfaces ◽  
Base Change ◽  
Kodaira Dimension ◽  
Elliptic Surface ◽  
Minimal Resolution ◽  
Elliptic Fibrations ◽  
Smooth Model ◽  
Birational Model ◽  
Hodge Numbers ◽  
Rational Elliptic Surface

Abstract We discuss the birational geometry and the Kodaira dimension of certain varieties previously constructed by Schreieder, proving that in any dimension they admit an elliptic fibration and they are not of general type. The $l$-dimensional variety $Y_{(n)}^{(l)}$, which is the quotient of the product of a certain curve $C_{(n)}$ by itself $l$ times by a group $G\simeq \left ({\mathbb{Z}}/n{\mathbb{Z}}\right )^{l-1}$ of automorphisms, was constructed by Schreieder to obtain varieties with prescribed Hodge numbers. If $n=3^c$ Schreieder constructed an explicit smooth birational model of it, and Flapan proved that the Kodaira dimension of this smooth model is 1, if $c>1$; if $l=2$ it is a modular elliptic surface; if $l=3$ it admits a fibration in K3 surfaces. In this paper we generalize these results: without any assumption on $n$ and $l$ we prove that $Y_{(n)}^{(l)}$ admits many elliptic fibrations and its Kodaira dimension is at most 1. Moreover, if $l=2$, its minimal resolution is a modular elliptic surface, obtained by a base change of order $n$ on a specific extremal rational elliptic surface; if $l\geq 3$ it has a birational model that admits a fibration in K3 surfaces and a fibration in $(l-1)$-dimensional varieties of Kodaira dimension at most 0.

scholarly journals Variation of singular Kähler–Einstein metrics: Positive Kodaira dimension

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Junyan Cao ◽  
Henri Guenancia ◽  
Mihai Păun
Keyword(s):  
General Type ◽  
Einstein Metrics ◽  
Kodaira Dimension ◽  
Einstein Metric ◽  
Canonical Bundle ◽  
Fiber Space ◽  
Generic Fiber ◽  
Lelong Numbers ◽  
Kähler Einstein Metrics

Abstract Given a Kähler fiber space p : X → Y {p:X\to Y} whose generic fiber is of general type, we prove that the fiberwise singular Kähler–Einstein metric induces a semipositively curved metric on the relative canonical bundle K X / Y {K_{X/Y}} of p. We also propose a conjectural generalization of this result for relative twisted Kähler–Einstein metrics. Then we show that our conjecture holds true if the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiberwise Song–Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension).

scholarly journals KODAIRA DIMENSION OF SUBVARIETIES II

2006 ◽  
Vol 17 (05) ◽  
pp. 619-631 ◽  
Author(s):  
THOMAS PETERNELL
Keyword(s):  
Normal Bundle ◽  
General Type ◽  
Kodaira Dimension ◽  
Projective Manifold ◽  
Fundamental Groups ◽  
Weak Version ◽  
Joint Paper ◽  
Rationally Connected

This paper continues the study of non-general type subvarieties begun in a joint paper with Schneider and Sommese [14]. We prove uniruledness of a projective manifold containing a submanifold not of general type whose normal bundle has positivity properties and study moreover the rational quotient. We also relate the fundamental groups and a prove a cohomological criterion for a manifold to be rationally connected (weak version of a conjecture of Mumford).

scholarly journals Towards a classification of rigid product quotient varieties of Kodaira dimension 0

2021 ◽  
Author(s):  
Ingrid Bauer ◽  
Christian Gleissner
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Structure Theorem ◽  
Complete Classification ◽  
Kodaira Dimension ◽  
The Other ◽  
Diagonal Action ◽  
Quotient Varieties ◽  
A Finite Group

AbstractIn this paper the authors study quotients of the product of elliptic curves by a rigid diagonal action of a finite group G. It is shown that only for $$G = {{\,\mathrm{He}\,}}(3), {\mathbb {Z}}_3^2$$ G = He ( 3 ) , Z 3 2 , and only for dimension $$\ge 4$$ ≥ 4 such an action can be free. A complete classification of the singular quotients in dimension 3 and the smooth quotients in dimension 4 is given. For the other finite groups a strong structure theorem for rigid quotients is proven.

scholarly journals Kodaira dimension of moduli of special cubic fourfolds

2019 ◽  
Vol 2019 (752) ◽  
pp. 265-300 ◽  
Author(s):  
Sho Tanimoto ◽  
Anthony Várilly-Alvarado
Keyword(s):  
Complete Intersection ◽  
Cusp Form ◽  
Moduli Space ◽  
General Type ◽  
Kodaira Dimension ◽  
Countable Family ◽  
Prior Work ◽  
Low Weight ◽  
Modular Varieties ◽  
Cubic Fourfold

Abstract A special cubic fourfold is a smooth hypersurface of degree 3 and dimension 4 that contains a surface not homologous to a complete intersection. Special cubic fourfolds give rise to a countable family of Noether–Lefschetz divisors {{\mathcal{C}}_{d}} in the moduli space {{\mathcal{C}}} of smooth cubic fourfolds. These divisors are irreducible 19-dimensional varieties birational to certain orthogonal modular varieties. We use the “low-weight cusp form trick” of Gritsenko, Hulek, and Sankaran to obtain information about the Kodaira dimension of {{\mathcal{C}}_{d}} . For example, if {d=6n+2} , then we show that {{\mathcal{C}}_{d}} is of general type for {n>18} , {n\notin\{20,21,25\}} ; it has nonnegative Kodaira dimension if {n>13} and {n\neq 15} . In combination with prior work of Hassett, Lai, and Nuer, our investigation leaves only twenty values of d for which no information on the Kodaira dimension of {{\mathcal{C}}_{d}} is known. We discuss some questions pertaining to the arithmetic of K3 surfaces raised by our results.

scholarly journals Threefolds with negative Kodaira dimension and positive irregularity

1983 ◽  
Vol 91 ◽  
pp. 163-172 ◽  
Author(s):  
Mauro Beltrametti ◽  
Paolo Francia
Keyword(s):  
Kodaira Dimension ◽  
Complex Field ◽  
Negative Kodaira Dimension

The purpose of this paper is to study threefolds X, with negative Kodaira dimension k(X) and positive irregularity q(X), defined over the complex field C.

Sign in / Sign up

Export Citation Format

Share Document