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Riemann-Roch
Defining the Chern character:
See Chern class.
For Curves
For surfaces
General
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Defining the Chern character:
See Chern class.
See Riemann-Roch.
Manin proves the Mordell conjecture using the Gauss-Manin connection, generalized to Areklov theory. Uses Green functions on curves. Used in Faltings theorem. Imports tools like the adjunction formula, the Hodge index theorem, Riemann-Roch.
Proof: Riemann-Roch for surfaces.