One difficulty with using individual research papers to learn about quotients of Hermitian symmetric spaces from an algebraic geometry perspective is that there is so much background material to cover that it is not necessarily reasonable to expect an exhaustive review of all the prerequisites within one journal article. The result of this is that if the reader is not already ”in the know” about the subject at hand, one finds themselves on a veritable treasure hunt through a seemingly endless supply of references, constantly having to move two steps backward before taking one step forward. Michio Kuga’s text makes an effort to remedy this issue by providing a full treatise on the construction and examination of the eponymous Kuga fibre variety. That is not to say that this text explores every aspect on the forefront of research into the subject, but instead lays the groundwork for someone who is interested in the subject to gather the tools necessary to delve into the field.

The text itself is broken into two separate volumes, each consisting of two chapters. The first volume focuses on an analytic construction of Kuga varieties and their cohomology groups. In particular, the development of vector valued harmonic forms, which are crucial tools for studying these fibre varieties, is given great care. The author is also very explicit in providing proofs and references despite this being only the “developmental” part of the text. At the end of the first chapter, the author goes into a detailed example using SL(2, R) that serves to illustrate the numerous constructions discussed earlier in the chapter. This is a particularly useful introductory example as even a novice to the subject is likely already familiar with the upper half-plane and various aspects of the Lie group in question from studying modular forms at some point during their mathematical career. The second chapter explicitly details the construction of a Kuga variety as a fibre variety over a symmetric space. Much of the details of this construction mirror what may already be familiar to many readers who have studied Shimura varieties before (for example the Harish-Chandra embedding), but the level of detail should be greatly appreciated by anyone who is not an expert on the subject. Chapter 2 also provides explanations for a number of commonly used facts and results that are often taken for granted in papers, such as embeddings and isomorphisms among different cohomology groups of these fibre bundles as well as the realization of these analytic spaces as projective algebraic varieties.

The second volume of the text takes the construction of the fibre varieties from the first volume and studies the existence of Hecke operators on these spaces and how they interact with various harmonic forms and cohomology groups. There is a significantly more algebraic feel to this volume as questions of adelization, orders and l-adic constructions are used significantly, particularly in the fourth chapter. Although this change in perspectives is common when studying arithmetic quotients of a hermitian symmetric space, it is nice to have a smooth transition from one to the other within the same text. As is ideal, the topic of Hecke operators is presented from both the double coset as well as the algebraic correspondence perspective. Again the material is presented in such a way as to evoke connections with similar constructions that the reader may have already come across when studying Hecke operators for SL(2, Z) and its subgroups. The final chapter of the text can be thought of as one large example illustrating the application of Hecke operators using the familiar upper half plane as the base symmetric space but with an interesting arithmetic group lying inside a quaternion algebra. However, the material is not just presented as an example to illustrate the ideas of the previous chapter, but instead derives some very interesting results concerning the zeta and L-functions associated to the fibre variety. These results should be quite interesting to any number theorist and provide a strong motivation for trying similar constructions with different Lie groups and arithmetic subgroup types. It is a very nice change of pace from the more didactic style of the first three chapters and serves as an intriguing end to the text.

Overall, this text is a welcome departure from the myriad of research papers on the subject, if the reader is looking to just get into the field before wading into the deepest waters. I would certainly not classify this as a textbook as the number of exercises and examples are extremely limited, but this is about as close as one can expect for mathematic topics at this level. It should be warned that the amount of mathematics expected by the text is quite significant and on par for similar topics regarding quotients of symmetric domains. It would be extremely helpful if the reader already has a good amount of experience with various Shimura variety constructions as many of the ideas here are generalizations of similar ideas. I would not recommend this text for anyone below an experienced graduate student studying pure mathematics (and ideally one focused on algebraic geometry). The text can serve as either a platform for getting into the subject or as an all-in-one reference for more experienced researchers to look up results they need for their own problems.

Dr. Dylan Attwell-Duval works as a lecturer at Pennsylvania State University after graduating from McGill in 2015. His research is focused on orthogonal-type Shimura varieties and their compactification.