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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
4
votes
Down-to-earth expositions of Hodge theory
If I recall correctly, a fairly readable and straightforward introduction to the basics of Hodge theory (in the differential-geometric setting) can be found in Chapter 6 aptly entitled "Hodge theorem" …
1
vote
Best algebraic geometry textbook? (other than Hartshorne)
Another nice introductory AG book that, I believe, was not mentioned here yet is Hassett.