Riemann surfaces over algebraic surfaces

[tom.stoer]

Suppose we have one polynom

$P(r_1, r_2, \ldots, r_n) = 0$

in n complex variables. This defines a n-1 dimensional complex algebraic surface.

Suppose that for each variable we have

$r_i = e^{ip_i}$

with complex p.

In the case n=1 of one variable r this results in the complex logarithm

$p = -i\ln r$

and we have to deal with a Riemann surface (but only for a discrete set of solutions of P)

What happens in the case n>1?

Is there something like a Riemann surface (in more than one variable) over a (higher-dimensional) algebraic surface?

What can we say about the manifold defined as the solution of

$P\left( e^{ip_1}, e^{ip_2}, \ldots, e^{ip_n} \right) = 0$

in p-space?

 

[mathwonk]

this seems a rather odd question to an algebraic geometer. by definition you are looking at an infinite analytic but non algebraic cover of an algebraic hypersurface, but only over those points where no coordinate is zero.

i.e. you are taking an infinite analytic cover of the part of an algebraic hypersurface complementary to all the coordinate axes.

may i ask how this arises?

 

[tom.stoer]

The starting point is rather simple. We have

$c = \cos p$
$s = \sin p$

and a polynomial equation

$\tilde{P}(c,s) = 0$

Then we eliminate c,s via the new variable r and we get a new polynomial equation

$P(r) = 0$

But b/c we started with the variables c,s we are basically interested in

$P\left(e^{ip}\right) = 0$

Now you may want to look for solutions of equations formulated on higher dimensional spheres (where you need more angles) and you immediately get a higher-dimensional generalization of the polynomial equation. A rather simple example is the intersection of a sphere or an ellipsoid with a plane.

 

[tom.stoer]

Any new thoughts?

 

[blue_raver22]

In the case of n>1, the manifold defined by $P(e^{ip_1}, e^{ip_2}, \ldots, e^{ip_n}) = 0$ is not a Riemann surface, but rather a higher-dimensional complex manifold. This manifold can be thought of as a generalization of a Riemann surface, with each point on the algebraic surface corresponding to a higher-dimensional "patch" of the manifold.

This manifold has interesting topological and geometric properties, and can be studied using tools from algebraic and differential geometry. It is also closely related to the algebraic surface defined by the polynomial P, and the two can be thought of as different representations of the same mathematical object.

One important difference between Riemann surfaces and higher-dimensional complex manifolds is that the latter can have non-trivial higher cohomology groups, which can provide a richer structure and more complex behavior. This makes the study of these objects a fascinating and challenging area of research in mathematics and physics.

In summary, while the concept of a Riemann surface is limited to one variable, the idea of a complex manifold can be extended to higher dimensions, providing a powerful framework for understanding and studying the solutions of algebraic equations in multiple variables.