scholarly journals On subadditivity of Kodaira dimension in positive characteristic over a general type base

2017 ◽  
Vol 27 (1) ◽  
pp. 21-53 ◽  
Author(s):  
Zsolt Patakfalvi
Keyword(s):  
General Type ◽  
Positive Characteristic ◽  
Kodaira Dimension

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 Algebraic Surfaces of General Type with Small c21 in Positive Characteristic

2008 ◽  
Vol 191 ◽  
pp. 111-134 ◽  
Author(s):  
Christian Liedtke
Keyword(s):  
Characteristic Zero ◽  
General Type ◽  
Positive Characteristic ◽  
Algebraic Surfaces ◽  
Surfaces Of General Type ◽  
Arbitrary Characteristic ◽  
Numerical Invariants ◽  
Characteristic 2

AbstractWe establish Noether’s inequality for surfaces of general type in positive characteristic. Then we extend Enriques’ and Horikawa’s classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we show that Horikawa surfaces lift to characteristic zero.

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.

Algebraic Surfaces and the Miyaoka-Yau Inequality

2017 ◽  
Author(s):  
Paula Tretkoff
Keyword(s):  
Differential Geometry ◽  
Algebraic Surface ◽  
Smooth Surface ◽  
General Type ◽  
Kodaira Dimension ◽  
Algebraic Surfaces ◽  
Surfaces Of General Type ◽  
Smooth Complex ◽  
Dimension 2

This chapter discusses complex algebraic surfaces, with particular emphasis on the Miyaoka-Yau inequality and the rough classification of surfaces. Every complex algebraic surface is birationally equivalent to a smooth surface containing no exceptional curves. The latter is known as a minimal surface. Two related birational invariants, the plurigenus and the Kodaira dimension, play an important role in distinguishing between complex surfaces. The chapter first provides an overview of the rough classification of (smooth complex connected compact algebraic) surfaces before presenting two approaches that, in dimension 2, give the Miyaoka-Yau inequality. The first, due to Miyaoka, uses algebraic geometry, whereas the second, due to Aubin and Yau, uses analysis and differential geometry. The chapter also explains why equality in the Miyaoka-Yau inequality characterizes surfaces of general type that are free quotients of the complex 2-ball.

scholarly journals Equality in the Bogomolov–Miyaoka–Yau inequality in the non-general type case

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Feng Hao ◽  
Stefan Schreieder
Keyword(s):  
General Type ◽  
Universal Constant ◽  
Kodaira Dimension ◽  
Chern Number ◽  
Minimal Models

Abstract We classify all minimal models X of dimension n, Kodaira dimension n - 1 {n-1} and with vanishing Chern number c 1 n - 2 ⁢ c 2 ⁢ ( X ) = 0 {c_{1}^{n-2}c_{2}(X)=0} . This solves a problem of Kollár. Completing previous work of Kollár and Grassi, we also show that there is a universal constant ϵ > 0 {\epsilon>0} such that any minimal threefold satisfies either c 1 ⁢ c 2 = 0 {c_{1}c_{2}=0} or - c 1 ⁢ c 2 > ϵ {-c_{1}c_{2}>\epsilon} . This settles completely a conjecture of Kollár.

ENRIQUES’ CLASSIFICATION IN CHARACTERISTIC 0$" height="12pt">: THE -THEOREM

2018 ◽  
Vol 235 ◽  
pp. 201-226
Author(s):  
FABRIZIO CATANESE ◽  
BINRU LI
Keyword(s):  
Positive Characteristic ◽  
Kodaira Dimension ◽  
Algebraic Surfaces ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Precise Version

The main goal of this paper is to show that Castelnuovo–Enriques’ $P_{12}$ - theorem (a precise version of the rough classification of algebraic surfaces) also holds for algebraic surfaces $S$ defined over an algebraically closed field $k$ of positive characteristic ( $\text{char}(k)=p>0$ ). The result relies on a main theorem describing the growth of the plurigenera for properly elliptic or properly quasielliptic surfaces (surfaces with Kodaira dimension equal to 1). We also discuss the limit cases, i.e., the families of surfaces which show that the result of the main theorem is sharp.

scholarly journals Projective normality and syzygies of algebraic surfaces

1999 ◽  
Vol 1999 (506) ◽  
pp. 145-180 ◽  
Author(s):  
F. J Gallego ◽  
B. P Purnaprajna
Keyword(s):  
General Type ◽  
Kodaira Dimension ◽  
Algebraic Surfaces ◽  
List Type ◽  
Singular Surfaces ◽  
Surfaces Of General Type ◽  
New Techniques ◽  
Uniform Bounds ◽  
Projective Normality ◽  
Koszul Cohomology

Abstract In this work we develop new techniques to compute Koszul cohomology groups for several classes of varieties. As applications we prove results on projective normality and syzygies for algebraic surfaces. From more general results we obtain in particular the following: Mukai's conjecture (and stronger variants of it) regarding projective normality and normal presentation for surfaces with Kodaira dimension 0, and uniform bounds for higher syzygies associated to adjoint linear series,effective bounds along the lines of Mukai's conjecture regarding projective normality and normal presentation for surfaces of positive Kodaira dimension, and,results on projective normality for pluricanonical models of surfaces of general type (recovering and strengthening results by Ciliberto) and generalizations of them to higher syzygies. In addition, we also extend the above results to singular surfaces.

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