Topology of Algebraic Varieties

It is classical to study the geometry of projective varieties over algebraically closed fields through the properties of various positive cones of divisors or curves. Several counterexamples have shifted attention from the higher (co)dimensional case. They show that the analogous definitions do not lead to analogous positivity properties. To correct the negative outlook, we look at stronger positivity conditions on numerical classes. A sample result is that the pseudoeffective cone, the closure of the cone of effective \(k\)-dimensional cycle classes is pointed. We impose no restrictions on the characteristic of the field, or the singularities of the projective variety. This is joint work with Brian Lehmann.