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Lecture

Albanese Morphism: Kodaira Dimension Zero

Description

This lecture discusses the Albanese morphism of varieties with Kodaira dimension zero in positive characteristic, exploring the application of generic vanishing techniques to understand the geometry of such varieties. The talk aims to explain the theory in positive characteristic and provide insights on proving Ueno's conjecture in this context.

Official source

Ontological neighbourhood

Mathematics

Geometry: Algebraic geometry

Related concepts (32)

Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined over that field.

Algebraic variety

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly.

Projective variety

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .

Albanese variety

In mathematics, the Albanese variety , named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve. The Albanese variety is the abelian variety generated by a variety taking a given point of to the identity of . In other words, there is a morphism from the variety to its Albanese variety , such that any morphism from to an abelian variety (taking the given point to the identity) factors uniquely through .

Kodaira dimension

In algebraic geometry, the Kodaira dimension κ(X) measures the size of the canonical model of a projective variety X. Igor Shafarevich in a seminar introduced an important numerical invariant of surfaces with the notation κ. Shigeru Iitaka extended it and defined the Kodaira dimension for higher dimensional varieties (under the name of canonical dimension), and later named it after Kunihiko Kodaira. The canonical bundle of a smooth algebraic variety X of dimension n over a field is the line bundle of n-forms, which is the nth exterior power of the cotangent bundle of X.

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