Many times I have heard string theorists say that string theory has a lot of algebraic geometry, but physicists seem to have identified complex differential geometry with algebraic geometry and obliviously those two fields are not the same.
Having searched some string theory notes and some papers and I saw that the techniques that are mostly used in string theory are cohomology, Chern classes, some category theory (which is not algebraic geometry, it's just used in it) and other techniques from differential geometry, but I wasn't able to find actual algebraic geometry techniques. Perhaps there are some and I wasn't able to find them, so if you know of any, please give some references.
When I think algebraic geometry, I think of ideals, K-algebras, modules,algebraic stacks and other algebraic structures.
I understand that complex manifolds are also projective varieties and hence this could be the source of confusion but the techniques used in physics are purely from a differential geometry perspective not an algebraic one. And although I am not a physicist, I can't imagine how would things like ideals could be used in physics, but maybe that is due to my lack of understanding of physics.
So I want to ask why do physicists identify algebraic geometry with complex differential geometry?
Are techniques from algebraic geometry actually used in physics besides of derived categories?