The conference “Recent Advances in Algebraic Geometry” was held between May 16 and 19, 2013, at the University of Michigan, Ann Arbor, to honor Robert K. Lazarsfeld (known as “Rob” among friends and colleagues) on the occasion of his 60th birthday. The conference honored Rob's outstanding contributions to algebraic geometry and the mathematical community, bringing together a large crowd, including many of his former students, collaborators, colleagues, and friends. It was a happy occasion for many of us who have known Rob and have been touched by his influence as students and peers, or simply as members of the algebraic geometry world.

From a personal point of view, we cannot even begin to discuss Rob's career without mentioning one of its most distinguished aspects, namely the unique influence he has had on the younger generations through teaching and mentoring. His style as a doctoral advisor and as an expositor is famous throughout the algebraic geometry community. He has been the advisor of more than 20 students, has numerous other mathematical descendants, and has mentored successful postdoctoral fellows. Many of these are now established mathematicians helping to expand the boundaries of Rob's mathematical vision. His generosity and ability to generate good problems, and his active support of the careers of his students, have been for many of us some of the most crucial aspects of our mathematical lives.

We highlight a few reference points in Rob's mathematical career. He received his B.A. from Harvard in 1975, and his Ph.D. from Brown in 1980, under the direction of William Fulton. He then went back to Harvard as a Benjamin Peirce Assistant Professor until 1983. During the 1981–82 academic year, Rob was awarded a postdoctoral fellowship from the American Mathematical Society, which he used to visit the Institute for Advanced Study in Princeton while on leave from Harvard. There he met Lawrence Ein, with whom he would later develop a long-lasting collaboration, resulting in over 25 joint papers. In 1983, Rob moved as an Assistant Professor to UCLA, where he became a Professor in 1987.

1. Introduction

From the point of view of the Minimal Model Program, Fano varieties constitute the building blocks of uniruled varieties. Important information on the biregular and birational geometry of a Fano variety is encoded, via Mori theory, in certain combinatorial data corresponding to the Neron-Severi space of the variety. It turns out that, even when there is actual variation in moduli, much of this combinatorial data remains unaltered, provided that the singularities are “mild” in an appropriate sense. One should regard any statement of this sort as a rigidity property of Fano varieties.

This paper gives an overview of Fano varieties, recalling some of their most important properties and discussing their rigid nature under small deformations. We will keep a colloquial tone, referring the reader to the appropriate references for many of the proofs. Our main purpose is indeed to give a broad overview of some of the interesting features of this special class of varieties. Throughout the paper, we work over the complex numbers.

2. General properties of Fano varieties

A Fano manifold is a projective manifold X whose anticanonical line bundle is ample (here n = dim X).

The simplest examples of Fano manifolds are given by the projective spaces. In this case, in fact, even the tangent space is ample. (By [Mori 1979], we know that projective spaces are the only manifolds with this property.)

In dimension two, Fano manifolds are known as del Pezzo surfaces. This class of surfaces has been widely studied in the literature (it suffices to mention that several books have been written just on cubic surfaces), and their geometry is quite well understood. There are ten families of del Pezzo surfaces. The following theorem, obtained as a result of a series of papers [Nadel 1990; 1991; Campana 1991; 1992; Kollár et al. 1992a; 1992b], shows that this is a general phenomenon.

Theorem 2.1. For every n, there are only finitely many families ofFano manifolds of dimension n.