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People very often say that algebraic geometry is a hard subject and has many challenging problems to solve. I believe the hodge conjecture is the one of the most difficult in the field and you, which problems do you think are the most difficult in the realm of algebraic geometry?

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There's a discussion on this on MO. Also there are some notes on this in the MSRI held in 2009.

Also there are open problems in arithmetic algebraic geometry, which the Birch-Swinnerton-Dyer can be said to be a part of.

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