TITLE: Topics in Differential geometry $\cap$ Algebraic geometry
QUESTION [16 upvotes]: I find (both: differential and algebraic) geometry fascinating. I'm just beginning my graduate studies, but I'd like to know some topics/theorems in the intersection of these two (since I don't really know where to go).

REPLY [4 votes]: In a completely different direction, you might wanna look at Arakelov Geometry.
Arakelov geometry is an area of geometry that is motly interrested in Diophantine problems. The idea behind it is that in order to grasp arithmetic properties, it won't be sufficient to only study the algebraic geometry of the objects over $\text{Spec } \mathbb Z$, because $\text{Spec } \mathbb Z$ is not proper (compact if you prefer), so we have to compactify the situation, and to do so, we need to add an "analytic fiber" (which is a smooth complex manifold), and endow it with hermitian data.
We then have to find analogues of statements and theorems in classical algebraic geometry in this new algebro-hermitian setting, which involves of course algebraic geometry on the variety itself, and complex differential geometry on the "fiber at infinity", and to treat them in an unified setting.
This relates to deep differential concepts and question, such as index theory for instance, or questions related to the curvature on specific bundles (e.g the quillen curvature on the determinant of cohomology) and of course analytic torsion.
Maybe to give you a taste of the mixing of the two styles of geometry, let me state an example. On a riemann surface, classical algebraic geometry tells you that is a good idea to study cycles over it. In the arakelov context you study so called "arithmetic cycles", which consists of pairs $(Z,g)$ where Z is a cycle on the surface (the geometric part), and g is a so called green function for $Z$, which you might think of as a potential for the electric field generated by the uniformly charged distribution of charges supported by $Z$ (you put charges at every point of $Z$ and you $g$ is then a potential for the electric field generated).
You can take a look at these notes, or those ones, by Soulé, but they're in French