## Are algebraic varieties manifolds?

Table of Contents

Many algebraic varieties are manifolds, but an algebraic variety may have singular points while a manifold cannot. Algebraic varieties can be characterized by their dimension. Algebraic varieties of dimension one are called algebraic curves and algebraic varieties of dimension two are called algebraic surfaces.

### What is the difference between manifold and variety?

In English, “manifold” refers to spaces with a differentiable or topological structure, while “variety” refers to spaces with an algebraic structure, as in algebraic varieties.

**What is a projective manifold?**

A projectively flat manifold (orbifold) is a manifold (orbifold) with an atlas of charts to the projective space with transition maps in the projective automorphism group. These objects are closely related to the representations of groups into the projective groups PGL(n + 1, R).

**What is zariski closure?**

The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of the variety. In the case of an algebraic variety over the complex numbers, the Zariski topology is thus coarser than the usual topology, as every algebraic set is closed for the usual topology.

## How difficult is algebraic geometry?

1) Algebraic geometry is indeed vast and difficult. But don’t be discouraged: professors and experts only know parts of it and you would be surprised to discover how little they know outside of their narrow domain of expertise.

### Is algebraic geometry interesting?

The first few items concern classical algebraic geometry, which studies polynomials over the reals and complex numbers. Classical algebraic geometry is the ‘next step’ after linear algebra. Linear algebra allows only addition and multiplication by scalars.

**What is manifold theory?**

The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions.

**Are all varieties manifolds?**

So to reiterate, some varieties are manifolds (if the defining polynomials satisfy a certain condition on partial derivatives) and some are not.

## What is number theory?

Number theory is the study of the integers (e.g. whole numbers) and related objects. Topics studied by number theorists include the problem of determining the distribution of prime numbers within the integers and the structure and number of solutions of systems of polynomial equations with integer coefficients.