A smooth real-valued function f on a complex manifold X is said to be strongly q-convex if its Levi form L(f) has at most q-1 nonpositive eigenvalues at each point. The manifold X is called strongly q-convex if X admits an exhaustion function which is strongly q-convex on the complement of some compact subset K (X is q-complete if one may take K to be empty). There are also well-known notions of q-convexity for complex spaces. If g is a Hermitian metric on X, then f is of class SP(g,q) if the trace of the restriction of L(f) to any complex vector subspace of dimension q in the tangent space at any point in X is positive. Such functions are strongly q-convex. The principal investigator and Mohan Ramachandran have applied such functions to obtain results concerning the Levi problem and the structure of complete Kaehler manifolds. They have also developed and applied analogous classes on complex spaces. The principal investigator plans to apply such functions to extend some results for Kaehler manifolds to singular Kaehler spaces and to study q-convexity properties of coverings; in particular, covering spaces of (quasi)projective varieties. Some or all of the proposed work will probably involve collaboration with Michael Fraboni, Cezar Joita, or Mohan Ramachandran.
Complex spaces (in particular, Stein spaces and projective varieties), are the fundamental objects of study in several complex variables and algebraic geometry. The notion of q-convexity is one of the many useful generalizations of geometric convexity (for example, a region in the plane is geometrically convex if any line segment connecting two points in the region lies entirely within the region). The convexity properties of a complex space (for example, a covering space of a projective variety) are intimately connected with the space's holomorphic function theory and geometry. Holomorphic functions on complex spaces (for example, on regions in the complex number plane) are the natural analogues of differentiable functions on the real number line from differential calculus. Thus the study of convexity properties is an essential element in the study of complex spaces. Moreover, notions of convexity (like those of symmetry) appear in almost every field of mathematics, science, and engineering. Hence the study of convexity properties of complex spaces provides further evidence of the importance of convexity.
