By Kenji Ueno

ISBN-10: 0821808621

ISBN-13: 9780821808627

ISBN-10: 0821813579

ISBN-13: 9780821813577

It is a stable e-book on vital rules. however it competes with Hartshorne ALGEBRAIC GEOMETRY and that's a difficult problem. It has approximately an identical must haves as Hartshorne and covers a lot a similar rules. the 3 volumes jointly are literally a piece longer than Hartshorne. I had was hoping this may be a lighter, extra simply surveyable booklet than Hartshorne's. the topic includes a major volume of fabric, an total survey displaying how the components healthy jointly may be very valuable, and the IWANAMI sequence has a few fantastic, short, effortless to learn, overviews of such subjects--which supply evidence strategies yet refer in other places for the main points of a few longer proofs. however it seems that Ueno differs from Hartshorne within the different path: He offers extra particular nuts and bolts of the elemental structures. total it's more straightforward to get an summary from Hartshorne. Ueno does additionally provide loads of "insider info" on easy methods to examine issues. it's a stable e-book. The annotated bibliography is especially fascinating. yet i need to say Hartshorne is better.If you get caught on an workout in Hartshorne this ebook can assist. when you are operating via Hartshorne by yourself, you can find this replacement exposition precious as a better half. it's possible you'll just like the extra huge straightforward therapy of representable functors, or sheaves, or Abelian categories--but you'll get these from references in Hartshorne as well.Someday a few textbook will supercede Hartshorne. Even Rome fell after adequate centuries. yet this is my prediction, for what it really is worthy: That successor textbook usually are not extra simple than Hartshorne. it's going to benefit from growth due to the fact that Hartshorne wrote (almost 30 years in the past now) to make a similar fabric speedier and less complicated. it is going to comprise quantity conception examples and may deal with coherent cohomology as a unique case of etale cohomology---as Hartshorne himself does in brief in his appendices. it is going to be written via a person who has mastered each element of the maths and exposition of Hartshorne's ebook and of Milne's ETALE COHOMOLOGY, and prefer either one of these books it is going to draw seriously on Grothendieck's really good, unique, yet thorny parts de Geometrie Algebrique. in fact a few humans have that point of mastery, significantly Deligne, Hartshorne, and Milne who've all written nice exposition. yet they cannot do every little thing and nobody has but boiled this all the way down to a textbook successor to Hartshorne. should you write this successor *please* enable me comprehend as i'm demise to learn it.

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**Additional resources for Algebraic Geometry 2: Sheaves and Cohomology (Translations of Mathematical Monographs) (Vol 2)**

**Example text**

F correspond to a r-invariant holomorphic section of the homogeneous vector bundle E = SP2g{R) x Vp . 1) Holomorphtcity condition at cusps is automatic when g>l by the so called Koecher's principle. 2) In the special case p = (det)k for some kEN, the transformation law becomes f{y~ ) = det{ C~+O)k . f{~) Vy = 'A B1 r, V~ lc oj E E Hg . and we recover the usual definiton for Siegel modular forms of weight k. The space of Siegel modular forms of weight k for r will be denoted by Rk{r). 8. 1) The Geometric Meaning of Siegel Modular Forms Let Xg= Zg\Hg x Cg ~Hg be the universal holoE morphic family of principally polarized abelian varieties, the rank 9 vector bundle n * (~lXg/Hg)= E*~lXg/Hg can be canonically identified with Hg x (Cdz,ij)·· ..

2) (a) e[~] (O,nn) of 9 -1 9 9 defines an immersion a£n Z IZ in pn -1 31 if 41n . { e[ab](O,~)} (b) r 2g _1 in P -1 9 defines an immersion of a,bEr Z r(r 2 ,2r 2 ) \H g if 21r. e[~](o,~)la,b E 6- 1z9/z 9 }, (b) The Satake compactification H* r(c,2c~ 9 normalization of the closure of 8c(r(c,2c~g) N = 9 IT i=l is isomorphic to the in pN, where di - 1. When c = (n,···,n), 21n, we can restate this lemma in more familiar form: R(r(n,2n)) = integral closure of the C-a1gebra generated by {e[g](O,n~). e[~](o,n~)la,b E n- 1Zg/Z 9}.

Is known to be finitely it is usually difficult to give explicit generators, not to mention relations among a given set of generators. This fact makes a purely algebraic study of the modular varieties difficult. Fortunately, works of Igusa [I2J teaches us that if we make a careful choice of level structure, then the classical theta constants produce the moduli. 1 Here is the story. 1) For each even integer n E N, define a congruence subgroup. AB (C D) Clearly, r(2n} c r(n,2n} E c Sp(2g,Z} r(n}.

### Algebraic Geometry 2: Sheaves and Cohomology (Translations of Mathematical Monographs) (Vol 2) by Kenji Ueno

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