This book gives an introduction to modern geometry. Starting from an elementary level, the author develops deep geometrical concepts that play an important role in contemporary theoretical physics, presenting various techniques and viewpoints along the way. This second edition contains two additional, more advanced geometric techniques: the modern language and modern view of Algebraic Geometry and Mirror Symmetry.
From the reviews of the second edition: "As the title suggests, this book is an introduction to Riemann surfaces, with the target audience being students of string theory. ... an excellent book to use to become familiar with these concepts, and as a result the book is able to touch on a wide variety of concepts which are not broached by more traditional treatments of the subject. ... I would certainly recommend the book for anyone who wants an enjoyable conceptual introduction to what can be a highly technical subject." (Mark Gross, Mathematical Reviews, Issue 2008 k) "This book is an introduction to the language of modern algebraic geometry, designed mainly for Physics students who are interested in string theory. ... Overall the book is very readable and it fulfills its goal remarkably well ... that are useful to physicists interested in string theory, with all the necessary references for further reading. This book will be an excellent addition to the bookshelf of any physics student or researcher who wants to learn about the mathematical aspects of string theory." (Valentino Tosatti, Zentrablatt MATH, Vol. 1153, 2009)