An invitation to modern enumerative geometry (Q2160158)

The book under review is intended to be a gentle introduction to contemporary enumerative geometry. The leitmotif of the book follows from Okounkov's epigraph at the beginning of the book: \textit{Modern enumerative geometry is not so much about numbers as it is about deeper properties of the moduli spaces that parametrize the geometric objects being enumerated.} Enumerative geometry is a classic topic in algebraic geometry that still attracts many researchers. Every single algebraic geometer has touched a particularly nice problem, considered as a toy problem, namely the \(27\) lines on a smooth cubic surface in \(\mathbb{P}^{3}\) over an algebraically closed field. At the moment, enumerative geometry is becoming more and more interesting and exciting, mainly due to the fact that it is closely related to and influenced by many research areas in mathematics and physics, ranging from representation theory to string theory, touching on the theory of integrable systems, gauge theory, mirror symmetry, combinatorics, and last but not least the theory of moduli spaces. The main aim of the book is to provide an introduction (aimed at young graduate students) to enumerative geometry, emphasizing the differences and analogies between the modern approach to the subject, based on virtual fundamental classes, and the classical approach before the appearance of virtual fundamental classes. The bridge between modern and classical approach is given by the theory of torus localization and the cohomology theory. It is very visible that the author is focusing mostly on these two subjects looking both at the theory of equivariant cohomology and localizations. For example, the author presents the Atiyah-Bott localization formula and then he focuses on applications, for instance to compute the \(27\) lines on a smooth cubic surface (using equivariant cohomologies), or the \(2875\) lines on the quintic \(3\)-fold. On the other hand, following the classical approach, the author provides an outline on Grassmannians, relative Grassmannians, Hilb and Quot constructions, and then on the theory of Hilbert schemes of points. The most interesting part, according to my subjective viewpoint, is devoted to calculations involving \(\mathrm{Hilb}^{n}(X)\), where \(\dim X = 3\). On the classical side, the author recalls a purely combinatorial way to compute the Euler characteristic of the Hilbert scheme of points in arbitrary dimension. On the modern side, the author explains how to compute the degree \(0\) Donaldson-Thomas invariant of a toric Calabi-Yau \(3\)-fold obtaining that \(\mathrm{DT}_{n}^{X} = (-1)^{n}(\mathrm{Hilb}^{n}X)\). This result was chosen in order to explain how the virtual localization formula works. Of course, in the modern part, the author speaks also about the moduli problem and the Gromov-Witten theory looking also at the physical counterpart of the story. I would like to elaborate more about the content of the book, but then this review would take several pages to cover all the aspects. The book is intended to be suitable for young graduate students and to be self-contained enough to be read in one piece. For this reason, the author presents a very solid introduction to the basic concepts of algebraic geometry, which is mostly done in Chapter \(3\), and three detailed appendices, which are very useful since they deal with deformation theory, intersection theory and perfect obstruction theories, and virtual fundamental classes. Of course, sometimes it is not possible to provide all the necessary details, and some parts are used to clarify the terminology and notations (which is quite understandable). I like Chapter \(2\) very much, as it could be very useful for students, especially that it starts with a very fundamental question in enumerative geometry, namely how to ask a right question. To quote the author: \textit{How do we know how many constraints we should put on our objects in order to expect a finite answer?} I believe that the main advantage of the book is its ampleness (not geometrically speaking) and a plethora of very interesting topics covered by the author. The key asset is the bridging property since, as it was pointed out by the author, the enumerative geometry touches many areas of research, so it is impossible to speak about certain aspects in a very narrow perspective. I believe that the book is suitable for more advanced graduate students as it requires an understanding of many areas of the contemporary geometry. I can definitely recommend it as a rather advanced invitation to enumerative geometry.