Non-algebraic theories in analytic geometry
The document in between algebraic and also analytic geometry is provocative. The GAGA makes this specific somewhat. Yet there is even more of this example. As I examined a write-up on analytic space, a lot of the ideas there appears to have an algebraic analogue. I had actually been asking yourself why individuals researched analytic geometry independently to the level it has actually been. It would certainly behave to find out about some theories in analytic geometry that have no analogue in algebraic geometry, and also this would certainly warrant the research of this subject independently. Below I suggest theories that do require to make use of the complete power of analytic geometry; not points like Hodge disintegration for which Kähler suffices.
I'm not exactly sure that the major inspiration for researching analytic geometry is that there are theorems real there that are not real in algebraic geometry ; if anything, the opposite holds true, given that any kind of intricate selection is additionally an intricate analytic room.
Yet there are analytic selections that are not algebraic, so analytic geometry is a more comprehensive topic than intricate algebraic geometry, which has its very own allure. Assuming analytically additionally offers a resource of strategies that are not readily available by totally algebraic approaches (and also possibly this is what you suggest by theories that hold true in analytic geometry yet not algebraic geometry).
Ultimately, allow me additionally state that really intriguing sensations can take place when you research the communication in between the algebraic and also analytic globes, as an example of the adhering to kind : the moduli room of analytic K3 surface areas is a linked 20 dimensional room. The moduli room of algebraic K3 surface areas is a shut subspace, which is the union of countably several 19 dimensional linked parts. This reveals that some troubles, like the research of moduli rooms, can come to be less complex in the analytic regimen than they are if you limit to the algebro - geometric setup.