Complex Analytic Cycles I

Barlet develops the theory of cycle spaces in complex geometry from first principles, then shows how the approach applies in several settings. The opening chapters cover the background of complex analysis, treated through a geometric lens. Along the way they establish the local parameterization theorem, Lelong's Theorem and Remmert's direct image theorem.

Methods based on cycle spaces have been part of complex geometry for roughly four decades. This book aims to present them systematically so that graduate students in mathematics and working researchers can follow them.

The final and most significant part turns to families of cycles. It examines their topology methodically and presents several basic, important applications of analytic families. The construction of the cycle space as a complex space, along with many further applications, is reserved for the second volume.

This edition is an English translation of the French original, which the French Mathematical Society published in 2014.