Kodaira embedding theorem

Okay, so let's imagine you want to put a really big toy on a shelf. But the shelf is kinda small and the toy is really big. What can you do? One option is to squish the toy so it fits on the shelf, but that would make it look weird and not pretty. Another option is to find a bigger shelf that can accommodate the toy without squishing it. That's kinda what the Kodaira embedding theorem does, but instead of toys and shelves, it's about complex manifolds and projective spaces.

So, a complex manifold is like a shape that has a bunch of points that are connected together and that look like they're part of a bigger picture. It's kinda like a puzzle piece that can fit together with other puzzle pieces to form a bigger puzzle. The projective space is like a special kind of puzzle that has a really nice shape, like a sphere or a cube. It's super easy to put things in a projective space because everything fits nicely.

But what if your complex manifold is too big to fit inside a projective space? That's where the Kodaira embedding theorem comes in. It says that if you have a big enough complex manifold, then you can always find a way to put it inside a projective space without squishing or distorting it. It's like finding a bigger shelf for your toy. And this is important because projective spaces are really nice and easy to work with, so it makes it easier for mathematicians to do math on the complex manifold.

So, to sum up, the Kodaira embedding theorem is like finding a bigger shelf for your toy, but instead of toys and shelves, it's about complex manifolds and projective spaces. It helps mathematicians do math on complex manifolds by finding a nice way to put them in a projective space without distorting them.