Yoichi Miyaoka

Geometry of Higher Dimensional Algebraic Varieties

The subject of this book is the classification theory and geometry of higher dimensional varieties: existence and geometry of rational curves via characteristic p-methods, manifolds with negative Kodaira dimension, vanishing theorems, theory of extremal rays (Mori theory), and minimal models. The book gives a state-of-the-art intro- duction to a difficult and not readily accessible subject which has undergone enormous development in the last two decades. With no loss of precision, the volume focuses on the spread of ideas rather than on a deliberate inclusion of all proofs. The methods presented vary from complex analysis to complex algebraic geometry and algebraic geometry over fields of positive characteristics. The intended audience includes students in algebraic geometry and complex analysis as well as researchers in these fields and experts from other areas who wish to gain an overview of the theory.

Bounding codimension-one subvarieties and a general inequality between Chern numbers

We extend the Miyaoka-Yau inequality for a surface to an arbitrary nonuniruled normal complex projective variety, eliminating the hypothesis that the variety must be minimal. The inequality is sharp in dimension three and is also sharp among minimal varieties. For nonminimal varieties in dimension four or higher, an error term is picked up which can be controlled. As a consequence, we bound codimension one subvarieties in a variety of general type linearly in terms of their Chern numbers. In particular, we show that there are only a finite number of smooth Fano, Abelian and Calabi-Yau subvarieties of codimension one in any variety of general type.

Rational Curves on Algebraic Varieties

The aim of this article is to give a brief review on recent developments in the theory of embedded rational curves, which the author believes is a new, useful viewpoint in the study of higher dimensional algebraic varieties. By an embedded rational curve, or simply a rational curve, on a variety X, we mean the image of the projective line ℙ1 by a nontrivial morphism to X, hence complete, one-dimensional, but not necessarily smooth.

Stable Higgs bundles with trivial Chern classes. Several examples

We propose a new interpretation of Higgs bundles as vector bundles together with actions of the symmetric tangent tensor algebra. Via this interpretation, we construct new nontrivial examples of stable Higgs bundles with vanishing Chern classes.

Introduction: Why Rational Curves?

This note is based on a series of lectures given at the Mathematisches Forschungsinstitut at Oberwolfach, Germany, as a part of the DMV seminar “Mori Theory”. The construction of minimal models was discussed by T. Peternell, and my task was to give an overview of various aspects of the study of rational curves on algebraic varieties, including the following topics: (a) Techniques which enable us to find rational curves on certain classes of varieties; (b) Characterization of uniruled varieties (varieties that carry sufficiently many rational curves) in terms of canonical divisors; (c) Generic semipositivity of the cotangent bundle of non-uniruled varieties and its application to the “abundance conjecture” in dimension three; (d) Decomposition of a given variety into the “non-uniruled part” and the “rationally connected part”, (e) Application of the techniques above to the theory of Fano varieties.

Rationally Connected Fibrations and Applications

In this lecture, we study the finer structure of uniruled varieties. We show that a uniruled variety has a canonical structure (MRC-fibration) with maximally rationally connected fibres. This structure provides us a splitting of a uniruled variety into rationally connected varieties and a non-uniruled variety. Rational connectedness is a natural generalization of unirationality, and in dimension two or three, we can completely characterize rationally connected varieties in terms of global holomorphic differential forms. Thanks to the MRC-fibrations, we get a classification of the complex uniruled threefolds into three clearly distinguished classes.

Deformations and Rational Curves

In this lecture, we give a brief account on the fundamental notions such as Hilbert schemes, Chow schemes, deformation of morphisms, and intersection pairing between curves and divisors, along with their application to the construction of rational curves on a projective variety.

Construction of Non-Trivial Deformations via Frobenius

In the second lecture, we discuss a technique of S. Mori to construct deformation of morphisms via reduction modulo p, and show the existence of rational curves on smooth projective varieties whose canonical divisors are not nef.

Abundance for Minimal 3-Folds

In the first section, we give a quick introduction to the birational classification theory due to Enriques-Kodaira-Shafarevich-Iitaka, which divides the n-dimensional algebraic varieties into n + 2 disjoint classes according to a single invariant called the “Kodaira dimension” k. A nice thing of the theory is that a variety of intermediate Kodaira dimension has a canonical structure of a fibre space unique up to birational equivalence (“Iitaka fibration”).

Foliations and Purely Inseparable Coverings

In Lecture II, we gave a numerical characterization of uniruled varieties in terms of a certain numerical property of anti-canonical divisors. In this lecture, we discuss a refined characterization of such varieties in terms of the tangent bundle. Namely, a smooth projective variety in characteristic zero is uniruled unless its tangent bundle is almost everywhere seminegative.