This e-book and the subsequent moment quantity is an creation into smooth algebraic geometry. within the first quantity the equipment of homological algebra, thought of sheaves, and sheaf cohomology are constructed. those tools are quintessential for contemporary algebraic geometry, yet also they are primary for different branches of arithmetic and of significant curiosity of their own.
within the final bankruptcy of quantity I those techniques are utilized to the idea of compact Riemann surfaces. during this bankruptcy the writer makes transparent how influential the tips of Abel, Riemann and Jacobi have been and that a few of the sleek tools were expected via them.
For this moment version the textual content was once thoroughly revised and corrected. the writer additionally further a brief part on moduli of elliptic curves with N-level buildings. This new paragraph anticipates a few of the suggestions of quantity II.

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