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K3 Surfaces

Shigeyuki Kondō
European Mathematical Society
Publication Date: 
Number of Pages: 
EMS Tracts in Mathematics
[Reviewed by
McKenzie West
, on
K3 Surfaces is, unsurprisingly, a book about K3 surfaces. The book is particularly focused on the proof and applications of the Torelli-type theorem for K3 surfaces, which allow for isomorphism classification for these objects. As a translation of Kondō’s original publication on this topic to English from Japanese, some updates were made and two chapters were added to the work.  These objects, K3 surfaces, are considered to be 2-dimensional analogues of elliptic curves. Through results such as the Torelli-type theorem we have a rich understanding of the algebraic and geometric structure of K3 surfaces over the field of complex numbers. An interesting recent result is that K3 surfaces have been shown to connect to the Mathieu group and mathematical physics.
The text introduces the necessary content of lattice theory, reflection groups, and complex analytic surfaces in the first three chapters. Chapter 4 is all about defining K3 surfaces and sharing a variety of examples. Following this, chapter 5 includes a discussion on bounded symmetric domains of type IV, a generalization of the upper half plane for elliptic curves. The sixth chapter of the book introduces and proves the Torelli-type theorem for K3 surfaces. Chapters 7 and 8 discuss the period map of K3 surfaces and applications of the Torelli-type theorem to automorphisms of K3 surfaces. Next, chapters 9 and 10 include discussion on Enriques surfaces and the relationship of plane quartic curves to surfaces.
The first of the chapters new to the English version of the text discusses the relationship of K3 surfaces to the Mathieu group – the connection of K3 surfaces to mathematical physics. The second of these two chapters gives an account of the involutions that generate the automorphism group of a generic Kummer surface associated with a curve of genus 2.
In this book, Kondō restricts the content to complex surfaces, and it is an excellent resource for studying such objects. Kondō refers the reader to Huybrecht’s Lectures on K3 surfaces, and Dolgachev’s A brief introduction to Enriques surfaces for the case of positive characteristic. Over number fields, I recommend browsing the notes of Anthony Várilly-Alvarado from the 2015 Arizona Winter School.
I would recommend this book to any researcher with some experience in algebraic geometry interested in learning about K3 surfaces. It is an excellent resource on the basics of these objects and their classification. 
Mckenzie West is an assistant professor at the University of Wisconsin-Eau Claire. Her mathematical research is in the field of computational number theory and arithmetic geometry.