Algebraic Geometry is the study of  algebraic varieties. These are the solutions of a finite collection  of polynomials such as x2+y2-1 or x+y-z. The solutions to the first
equation form  a circle and to the second, a plane in three space.
In complex geometry one looks at the complex solutions. Then the world becomes more democratic.  For example one cannot solve of x2+y2=-1 with real numbers, but in complex
coordinates there is no difference with a real circle: just make the substitutions x--> ix, y--> iy and remember that i2 =-1 to find back the equation x2+y2=1.
We want to make everything even more democratic: there should be no difference between a circle and a parabola x2=y. You do this by making first these equations homogeneous.
The circle becomes x2+y2=z2 and the parabola x2 =yz and you see that replacing y by -y+z and z by y+z transforms the first equation in the second. Making homogeneous amounts to replacing coordinates (x,y) by ratios (x:y:z).  If we do this we have  added points "at infinity", i.e. where z=0. For the circle those are the two ``circle points'' (1:i:0) and (1:-i:0) but
for the parabola there is just one point,  (0:1:0). This shows that the similarity between a circle and a parabola never becomes visible in the finite plane since the line at infinity, z=0, cuts
the circle in two points while it is tangent to the parabola.
This should be enough motivation to add these points at infinity; the modified objects are now the zero sets of homogeneous polynomials, which are called projective varieties.
These sets sometimes have self-crossings such as for two intersectiong lines: these create singularities and if we avoid these, we get the so called  complex projective manifolds.

Although  we described these  as algebraic objects (solution sets of equations), they can also be viewed as geometric or topological objects so that algebraic, topological, differential geometric and complex-analytic tools can be applied.

The first basic invariant for a complex projective variety is the number of free parameters we need to describe points on them, the dimension: our example, the circle is 1-dimensional, essentially described by the angle parameter (in complex geometry this must be replaced by a complex parameter). A circle, viewed as a complex solution set,  is an example of an algebraic curve or Riemann surface (its dimension is 1). A variety of dimension 2 is called an algebraic surface. More generally, a complex surface is a complex manifold of dimension 2, i.e it locally looks like an open subset of C2. Within the (large) field of complex algebraic geometry my specialization comprizes several subfields, the two most important of which are the fields of complex surfaces and Hodge theory.
• The classification of algebraic surfaces is largely due to Castelnuovo (1865-1952) and Enriques(1871-1946). Their classification has been extended to include the non-algebraic surfaces by Kodaira (1915-1995). There is much recent activity, including by physicists who are using detailed results about K3-surfaces for string theory (for instance this prepublication).
• One of the goals of Hodge theory is to measure and describe what happens if you introduce extra ``parameters'' in the equations. In our example, one might want to vary the radius of the circle. A related question is to describe the subsets of a given variety which themselves are given by supplementary polynomial equations; these are called ``subvarieties, or more generally ``algebraic cycles''. The theory has been named after Sir William Hodge (1903-1975) who boosted research in this area after he formulated a conjecture. In fact it is one of the millenium prize questions posed by the Clay Foundation.

Hodge theory has links to other fields. It has for instance been discovered that Hodge theory is a natural tool to attack the so called ``mirror conjecture'', a conjecture posed by mathematical physicists. See for instance this book.

• Webpages related to my research: