This is an announcement for the paper "The complex Busemann-Petty problem on sections of convex bodies" by A.Koldobsky, H.Koenig, and M.Zymonopoulou. Abstract: The complex Busemann-Petty problem asks whether origin symmetric convex bodies in $\C^n$ with smaller central hyperplane sections necessarily have smaller volume. We prove that the answer is affirmative if $n\le 3$ and negative if $n\ge 4.$ Archive classification: math.FA math.MG Mathematics Subject Classification: 52A20 Remarks: 18 pages The source file(s), complexbp.tex: 46749 bytes, is(are) stored in gzipped form as 0707.3851.gz with size 14kb. The corresponding postcript file has gzipped size 101kb. Submitted from: koldobsk@math.missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0707.3851 or http://arXiv.org/abs/0707.3851 or by email in unzipped form by transmitting an empty message with subject line uget 0707.3851 or in gzipped form by using subject line get 0707.3851 to: math@arXiv.org.