Modern algebraic geometry is a wonder. It is no exaggeration to say that after Zariski passed the baton to Serre and Grothendieck and the subject’s foundation evolved from commutative algebra à la Zariski-Samuel to Tôhoku + SGA style homological algebra, the subject’s novel methodology soon burst beyond its once parochial borders. Any number of soon-to-be allied areas of mathematics proceeded to evolve accordingly and the last half-century has seen homological algebraic revolutions in such disparate areas as PDE (the microlocal analysis of Mikio Sato and Masaki Kasjiwara) and number theory (as is borne out by the recent emergence of arithmetic geometry). The concomitant growth in these areas has been dramatic, too, leading to major breakthroughs. I can imagine no better example of this than the establishment of the Shimura-Taniyama-Weil Conjecture (and therefore Fermat’s Last Theorem) by Andrew Wiles. To abuse a popular idiom: we’re all algebraic geometers now.
Well, not quite. The subject is both gorgeous and demanding and doing it justice is no trivial task, especially if one’s line of approach is tangential, i.e. if one comes at it from another initial direction (mine being number theory). It is accordingly of critical importance to have the right sources at hand. Which brings me to the book under review: it qualifies — in spades.
But The Geometry of Moduli Spaces of Sheaves is no easy page-turner: this is dense and serious stuff; gorgeous, yes, but oh so demanding. The preface to the first edition recommends Hartshorne’s Algebraic Geometry as preparation, and this is essentially non-negotiable. Don’t crack the book’s spine unless you have this, or its equivalent under your belt (Griffiths-Harris’ Principle of Algebraic Geometry, I suppose, or a hodgepodge (Hodge-podge?) including, say, MacDonald’s Introduction to Algebraic Geometry, Mumford’s Red Book of Varieties and Schemes (now garbed in Springer yellow), and a good book on sheaves and their cohomology like Bredon’s Sheaf Theory, or if you read French, Serre’s incomparable Faisceaux Algébriques Cohérents). Once this hurdle has been cleared, however, this book is for you!
Huybrechts and Lehn state that
[t]he topic of this book is the theory of semistable coherent sheaves on a smooth algebraic surface and of moduli spaces of such sheaves. The content ranges from the definition of semistable sheaf and its basic properties over the construction of moduli spaces to the birational geometry of these moduli spaces.
They go on to enumerate three reasons why this is important business: (1) moduli spaces of sheaves on surfaces “provide examples of higher dimensional algebraic varieties with a rich and interesting geometry”; (2) “moduli spaces are varieties naturally attached to any surface” [!]; and (3) Simon Donaldson’s results on “the relation between certain intersection numbers on the moduli spaces and the differentiable structure of the four manifold … throw a bridge from algebraic geometry to gauge theory and differential geometry.”
The book itself is laid out in two parts, Part I being “General Theory,” and Part II being “Sheaves on Surfaces.” Part I starts with “Some Homological Algebra,” namely coherent sheaves on Nötherian schemes, Hilbert polynomials and semistability, filtrations (e.g. Harder-Narasimhan), and boundedness, and ends with compact discussions of seven examples of moduli spaces of the indicated sort. In between we encounter a LOT of hard-core algebraic geometry, all in the service of explicating the inner life of moduli spaces of sheaves. And that’s only Part I.
Well, my propaganda notwithstanding, why would anyone other than an algebraic geometer care? Here’s Huybrecht-Lehn’s answer (from their Introduction):
It is one of the deep problems in algebraic geometry to determine which cohomology classes on a projective variety can be realized as Chern classes of vector bundles. In low dimensions the answer is known. On a curve … any [2-cocycle with integer coefficients] can be realized as the first Chern class of a vector bundle of prescribed rank … In dimension two [Well, here it gets a bit sticky: read p. xiii] … The next step in the classification of bundles aims at a deeper understanding of the set of all bundles with fixed rank and Chern classes. This naturally leads to the concept of moduli spaces … The case [rank] = 1 is a model for the theory … This book is devoted to the analogous questions for bundles of rank greater than one…
This second edition of The Geometry of Moduli Spaces of Sheaves is literally a “back by popular demand” affair, with the authors pointing out that “the book [is] still … a useful source for the main techniques, results and open problems in this area and … has been appreciated by newcomers wanting to learn the material from scratch.”
Rightly so. The Geometry of Moduli Spaces of Sheaves is very well written, the discussion is compact and elegant, and the proofs are complete and clear. It’s an important part of the literature, indeed.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.