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Classical algebraic geometry begins by interpreting the solutions to polynomial equations as geometric objects. The solutions can then be studied geometrically, and a correspondence between their algebra and geometry is established. Through this correspondence, geometric properties can sometimes be derived algebraically.

In this case, however, we defined a geometric object starting from an algebraic object. My question is, what are some examples where the algebraic side of algebraic geometry can be applied to understand geometric objects which are not a priori defined based off of some algebraic object? I imagine, for example, that there are geometric objects that turned out to be algebraic varieties although they were originally motivated some other way. A more interesting example might be a geometric object that isn't an algebraic variety, but where algebraic techniques from algebraic geometry can still be applied.

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up vote 7 down vote accepted

There is the paper "On periodic points" by Artin and Mazur, in which they prove that if $M$ is a closed smooth manifold then there is a dense subspace of the space of $C^k$ maps from $M$ to itself such that for every member $f$ of the this dense subspace, there is a positive constant $c$ such that the number of isolated periodic points of $f$ of period $n$ is bounded by $c^n$.

So this is a result in the dynamical systems, but the argument is via methods of algebraic topology (and uses ideas of Nash that allow one to approximate an arbitrary manifold by a real algebraic variety).

Note also that the statement (and the idea that algebraic geometry could have something to say about dynamics) is motivated in part by the following analogue over a finite field: if $V$ is a $d$-dimensional variety over a finite field $\mathbb F_q$, and $f:V \to V$ is the Frobenius (i.e. the "raising to the $q$th power map"), then the number of fixed points of $f^q$ in $V(\overline{\mathbb F}_q)$ is bounded by some positive constant times $q^{dn}$.

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