I'm beginner of Complex Geometry. Please Understand.
I'm learning the Daniel Huybrechts's Complex Geometry. I want to understand the Kodaira Embedding Theorem as soon as possible since it provoke surprise to me. But I stucked from the beginning. If possible, can someone guide a fast load map or crash course to understand the proof of Kodaira Embedding Theorem?
Anyway, I now try to understand the next underlined statement.
Anyone can explain the underlined statement more detail?
p.s. I studied Huybrechts's Complex Geometry, Chapter 3.2 Hodge Theory on Kahler Manifolds, Chapter 4.2 Connections, 4.3 Curvature..but I don't know where I should supplement to understand the underlined statement. Can anyone help me?
$L$ being ample by definition implies that $L^k$ induces an embedding $f:X\to \mathbb P^n$. And $L^k=f^*(\mathcal O(1))$ by proposition 2.3.26 in Huybrechts' book. Therefore $k\cdot c_1(L)=c_1(\mathcal O(1)|_X$, this is what he meant by $c_1(L)$ is up to positive scalar the restriction Fubini-Study form. Positivity follows from a local description of $c_1(\mathcal O(1))$, it boils down to checking directly that the form is positive-definite at each point.