 
Summary: Donu Arapura
Algebraic Geometry over the
Complex Numbers
December 7, 2010
Springer
Preface
Algebraic geometry is the geometric study of sets of solutions to polynomial equa
tions over a field (or ring). These objects called algebraic varieties (or schemes or...)
can be studied using tools from commutative and homological algebra. When the
field is the field of complex numbers, these methods can be supplemented with tran
scendental ones, that is by methods from complex analysis, differential geometry
and topology. Much of the beauty of the subject stems from the rich interplay of
these various techniques and viewpoints. Unfortunately, this also makes it a hard
subject to learn. This book evolved from various courses in algebraic geometry that
I taught at Purdue. In these courses, I felt my job was to act as a guide to the vast
terain. I did not feel obligated to prove everything, since the standard accounts of
the algebraic and transcendental sides of the subject by Hartshorne [44] and Grif
fiths and Harris [37] are remarkably complete, and perhaps a little daunting as a
consequence. In this book I have tried to maintain a reasonable balance between
rigour, intuition and completeness; so it retains some of the informal quality of lec
