Apologies to start with if this question is 'too soft' or not really research level. I'm a graduate student interested in both algebraic and differential geometry, although very much a novice with the former. I'm hoping that people on this forum can give me some interesting examples of problems active in research if there are any in the following vein, so that I might get a sense of what kinds of stuff is out there, as I begin searching for a thesis adviser.
The thing I'm interested in is using differential geometric methods to solve problems in algebraic geometry or vice-versa. A possible example of this can be found in Cartan's Method of Moving Frames, which works for smooth projective varieties, since projective space is homogeneous. Smooth projective varieties are then submanifolds of projective space, and so we can use this tool from differential geometry to study algebraic objects. In the text 'Cartan for Beginners,' some propositions are proven as special cases where positive results can be made of Hartshorne's conjecture on complete intersections using tools of this type.
Another example might be the study of the Gauss-map of an algebraic variety. I know of examples where the Gauss map's image can be relatively small, but I don't know why this is of interest or what kinds of problems you can work on in this way.