Peter spots two preprints that give two different proofs of a major open problem in algebraic geometry: the finite generation of the pluricanonical ring:

• Existence of minimal models for varieties of log general type
• A General Non-Vanishing Theorem and an Analytic Proof of the Finite Generation of the Canonical Ring

By a standard construction in algebraic geometry, this would give a particular embedding of a variety as a subvariety of projective space. Apparently for varieties of general type, this would give something even better: minimal model of the variety.