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I am in middle with my grad school, and as you know most of the years as a doctorate student focuses on a very specific topic and a very single problem. I am coming from the world of moduli spaces of sheaves, and my background in algebraic geometry is what you find in Hartshorne, plus whatever similar topics in moduli spaces of sheaves. Unfortunately, I don't have a complex algebraic geometer in my department who works in Hodge conjuctere related stuff. After graduation, I will have a small job in a small university where I will not have the pressure of publishing papers, and I want to devote my time to work on Hodge conjecture. Not only I hope to resolve it, but also I find the topic and related math very nice and attractive. my questions:

with someone with experience in algebraic geometry as in Hartshorne, and the book Geometry of moduli space of sheaves, How far someone is from handling complex geometry topics?

Can you please help me on where to start, i,e. what are the books that cover the basics?

what are the books that come after the books of the basics? including anything in deferential geometry, or any other geometry.

what are the good starting research Papers to look at?

how to keep up with improvments? what are the mathematicians names who are famous for the subjects and can be approached?.

Any advise is appreciated. Thank you all .

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    I suggest you start with something like Griffiths and Harris (perhaps you'll need to fill in more background in topology/geometry) and learn the basics of complex geometry. You don't know about $(p,q)$ forms from working in the Hartshorne world. :) – 2017-02-27
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    Also, Voisin's books are quite good. – 2017-02-27
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    Thank you so much. I am till hoping for more information regarding papers that someone could look at. – 2017-02-27

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I would recommend the following:

  • "Topics in transcendental algebraic geometry" -- is a must to read, although not so easy to read.
  • "Principles of algebraic geometry" -- the "bible" of complex algebraic geometry
  • "A survey of the Hodge conjecture" -- really good and easy to read.
  • "Algebraic Cycles and Hodge Theory" -- the so called "Torino Volume".
  • "Hodge's general conjecture is false for trivial reasons"
  • "Nodes and the Hodge conjecture"
  • "Hodge classes on self-products of a variety with an automorphism"