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Mihnea Popa works on algebraic geometry, including the birational classification of varieties and the geometry of families. His Harvard course on Kodaira dimension documents the connection with positivity and Hodge theory.^[ref]

His Kodaira dimension of algebraic varieties by Mihnea Popa explains how sections of powers of the canonical bundle measure the growth of pluricanonical maps. For a smooth projective curve, the trichotomy between genus zero, genus one and higher genus appears as Kodaira dimension $-\infty$, zero and one. This gives a concrete entry into a classification invariant that remains meaningful in higher dimensions.^[ref]