This monograph treats classical complex analysis problems considered for toric varieties. These problems were originally posed and solved on complex n-spaces; and they are very difficult if considered for an arbitrary complex manifold. The author describes the Hartogs and the Hartogs-Bochner extension phenomena in smooth toric varieties and their connection with the first cohomology group with compact supports. The main problem with compact sets, which appear in the Hartogs phenomena, is that a compact set cut into a particular coordinate patch does not have to remain compact. Therefore, a global view of compact sets is absolutely necessary for this problem. The affirmative and negative results are proved using topological, analytic, and algebraic methods. The monograph is suitable for junior and senior mathematicians interested in the theory of several complex variables, toric varieties, and ends of topological spaces.

Malgorzata Aneta Marciniak, Ph.D., studied Mathematics at theUniversity of Warsaw and at the Missouri University of Scienceand Technology. She has given multiple presentations formathematical societies including research talks during specialsessions of AMS meetings. She has been a member of AMS and MAAsince January 2006.