Hodge theory of singular algebraic curves

On Hodge Theory of Singular Plane Curves

The dimensions of the graded quotients of the cohomology of a plane curve complement U = P \ C with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail. We also give a precise numerical estimate for the difference between the Hodge filtration and the pole order filtration on H(U,C).

Enumeration of singular algebraic curves

We enumerate plane algebraic curves with one singular point of any (prescribed) singularity type. It is shown how to generalize the method to the singular hypersurfaces and some cases of enumeration of singular hypersurfaces are solved.

Algebraic aspects of higher nonabelian Hodge theory

The Hodge - Arakelov Theory of Elliptic Curves

The Intrinsic Hodge Theory of Hyperbolic Curves

In elementary mathematics, the simplest variety – i.e., geometric object defined by polynomial equations – that one encounters is a line, i.e., the set of points in the Euclidean plane R defined by an equation of the form aX + bY = c (where a, b, c ∈ R). After translation, rotation, and dilation, such an equation may be written in the form X = 0. In this case, the variety in question passes thr...