Could you please recommend any good texts on algebraic geometry (just over the complex numbers rather than arbitrary fields) and on complex geometry including Kahler manifolds that could serve as an informal introduction to the subject for a theoretical physicist (having in mind the applications in physics, e.g. in the string theory)?
What I want for a moment is to get some informal picture of the subject rather than being dug up into the gory details of the proofs and lost in higher and higher layers of abstraction of commutative algebra and category theory. The texts I have found so far are all rather dry and almost completely lack this informal streak, and all of them are geared towards pure mathematicians, so if there exists something like "Algebraic geometry for physicists" and "Kahler manifolds for physicists" (of course, they would probably have different titles :)), I would greatly appreciate the relevant references.
Best Answer
Griffiths and Harris' "Principles of Algebraic Geometry" (Wiley) is the best for your purposes (read only the parts on Kahler geometry). The sections on algebraic geometry in "Mirror Symmetry" (Clay/AMS) are essentially a Crib Notes version of that paper and some of the classic CY and special geometry papers referred to above.
What you should keep in mind going in is the following:
Kahler manifolds are complex manifolds with a hermitian inner product on tangent vectors which have a metric that is determined (locally) by a single function. It is the geometry in which the metric and the complex structure "get along very nicely." This simplifies lots of calculations and adds new symmetries. That's why we know so much about them.