I want to learn the book, but it seems that I should have some background on harmonic analysis, Lie Groups and measure theory. Can you give some references?
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4$\begingroup$ I think you have the wrong title for the book. $\endgroup$– Felipe VolochCommented Jun 6, 2011 at 11:17
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$\begingroup$ (edited title ) $\endgroup$– QfwfqCommented Jun 6, 2011 at 11:31
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5$\begingroup$ I am having trouble remembering where harmonic analysis or measure theory comes up in Mumford's Abelian Varieties, even implicitly. How did you determine that these were prerequisites? $\endgroup$– Pete L. ClarkCommented Jun 6, 2011 at 11:53
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2$\begingroup$ As suggested by Charles Matthews's answer below, once you get past the analytic theory of the first chapter, the main prerequisites are a solid understanding of EGA style algebraic geometry (such as you might learn from Hartshorne Chapters II and III). It doesn't seem that realistic to me to read these later chapters without that background. The first chapter is different; for this some analytic background is necessary. Perhaps that is what you want to read, given your analytically oriented take on the prerequisites? $\endgroup$– EmertonCommented Jun 6, 2011 at 13:02
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3$\begingroup$ i encourage all young people to simply open these great books and start reading. prerequisites be damned. i have personally benefited from this book and don't know much about the prerequisites you list. $\endgroup$– roy smithCommented Jun 15, 2011 at 6:42
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1 Answer
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To discuss generally first, the book was written up by C. P. Ramanujam, and he was more conscientious than usual in trying to tie down Mumford's lectures to existing references. Still, it is quite hard to sort out the exact prerequisites.
The analytic theory and theta-functions required in the first chapter stands rather aside from the rest, which is mostly in the EGA/SGA vein. You do need some Fourier theory for Chapter I; the style given is a bit Bourbaki-oriented, I recall, but there are alternatives (for example, take certain things for granted, and/or read another text on the analytic theory).
As for the rest, I remember finding the homological algebra prerequisites steep. Hartshorne's text will take you some way; but the spectral sequences and mapping cylinders rather expect expertise. Since the "Great American Sheaf Theory Book" isn't yet written, I'm not quite sure what to recommend. Probably a more modern treatment of the duality theory post-Mukai would be good to look at anyway, rather than obsessing over each of Mumford's proofs.
answered Jun 6, 2011 at 12:06
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1$\begingroup$ "Since the "Great American Sheaf Theory Book" isn't yet written..." Very well put, Charles, and a regrettable lack, too. I am sure many here would share my gratitude if some MathOverflower(s) closed that gap in the literature. $\endgroup$ Commented Jun 6, 2011 at 19:56