Construction of Kähler submanifolds of Kähler manifolds

Question

I want to see how we can construct Kähler submanifolds of Kähler manifolds, i. e. what second fundamental form, metric, almost complex structure and connection satisfy in that case. I am searching for the proper reference but unsuccessfully, I guess that is proved long time ago. I would appreciate if you can recommend me a good reference.

Answer 1

Let $X$ be a Kähler manifold and $Y \subseteq X$ a complex submanifold. Let $J$ be the almost complex structure on $X$ corresponding to its complex structure, then $J|_Y$ is the almost complex structure on $Y$ (this is precisely what it means for $Y$ to be a complex submanifold of $X$). Furthermore, if $h$ is the Kähler metric on $X$ with Kähler form $\omega$, then $h|_Y$ is a Kähler metric on $Y$ with Kähler form $\omega|_Y$. That is:

Every complex submanifold of a Kähler manifold is Kähler.

In terms of references for this question and others related to Kähler geometry in general, here are some sources I recommend (in no particular order):

• Huybrechts, Complex Geometry: An Introduction,
• Moroianu, Lectures on Kähler Manifolds,
• Griffiths and Harris, Principles of Algebraic Geometry, and
• Demailly, Complex Analytic and Differential Geometry (pdf available here).

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