One important use of PDEs in algebraic geometry is in so-called "Hitchin-Kobayashi correspondences". The original example of this is the following theorem.

**Theorem (Donaldson, Uhlenbeck-Yau)**
Let $L \to X$ be an ample line bundle over a compact complex manifold and $\omega$ a Kähler metric representing $c_1(L)$. A holomorphic vector bundle $E \to X$ is slope polystable with respect to $L$ if and only if it admits a Hermitian-Einstein metric with respect to $\omega$.

To spell this out in detail: the slope of a bundle $\mu(E)$ is its degree divided by its rank, where degree is $\langle c_1(E) \cup c_1(L)^{n-1}, [X] \rangle$. A bundle is slope stable if all proper coherent subsheaves have strictly smaller slope. A bundle is called slope polystable if it is the sum of stable bundles of equal slope. Meanwhile, a Hermitian metric in $E$ is called Hermitian-Einstein if its curvature $F$ satisfies the equation $(F,\omega) = c \cdot \mathrm{Id}$, for a constant $c$. (Here we are taking the innerproduct on 2-forms, the result being an endomorphism of $E$.) This is a non-linear PDE on the Hermitian metric.

Notice that the slope polystability is purely algebraic - it makes no mention whatsoever of the metric $\omega$. What is remarkable is that this is equivalent to the existence of a solution of a non-linear PDE. The story behind this theorem is quite a long one. It can be seen as an infinite dimensional example of the equivalence between quotients via GIT and symplectic quotients. The PDE plays the role of the moment map.

Since this result was proved there have been many other versions involving bundles with additional data, say, e.g., a Higgs field, which appears both in the definition of stability and the corresponding PDE. The corresponding "Hitchin-Kobayashi correspondences" play important roles in the study of the moduli of the algebraic objects. For example the fact that one can always solve the relevant PDE leads directly to an interesting Kähler metric on the moduli space of stable objects. In the case of Higgs fields, this is a hyperkähler metric. This metric is one of the starting points of the approach to geometric Langlands proposed by Kapustin and Witten. There are also other applications of the PDE point of view here leading to, amongst other things, strong restrictions on the fundamental groups of Kähler manifolds. This subject sometimes goes by the name "Non-abelian Hodge theory". (I should stress that this is a long way from my expertise!)

In a different vein, one version of the Hitchin-Kobayashi correspondence (which is still conjectural) concerns not metrics on bundles but rather metrics on the manifold itself. Many people in Kähler geometry are currently working on understanding both the conjecture and its ramifications. The idea (originally due to Yau, later refined by Donaldson and Tian) is that given an ample line bundle $L \to X$, one should be able to find a so-called "extremal" Kähler metric in $c_1(L)$ if and only if the polarised variety is "stable". Here a metric is "extremal" if it's a critical point of the $L^2$-norm of the curvature tensor, restricted to metrics in $c_1(L)$. This turns out to be equivalent to the gradient of the scalar curvature being holomorphic, a sixth-order fully non-linear PDE. "Simple" examples are Kähler-Einstein metrics (when $L$ is a multiple of the canonical bundle). The correct definition of stability, known as K-polystability, is a little too involved to give neatly here, but it is important to mention that, just as for slope polystabilty of a vector bundle, it is a purely algebraic concept. This whole subject is vast, and I could write about it for pages and pages, but I've probably already said too much for one answer!

sortedlist of answers to this particular question is a right thing to do. $\endgroup$