# Polishchuk Ex. 4,6 pg. 149

### Exercise 4

Prop. a complete variety, and is a map into an abel. var. s.t. is trivial. Then is a point.

proof: Consider the pull back of the normalized poincare bundle

on

To calculate I use the diagram

I have and . By the see-saw principle this implies for some line bundle on . Then for all , but using the poincare bundle, I have

That is is constant and since characterizes line bundles upto isomorphism, it follows that is constant. QED

Corollary: is determined by upto translation, i.e upto replacing with .

proof: Suppose are such that , then for

So by the proposition, , hence for . QED.

### Exercise 6

For this problem, here are two big results that I’ll use heavily but wont prove. For the first, see Abelian Varieties by Mumford pg. 150.

Thm (Riemman-Roch) Let be an invertible sheaf on an abelian variety . If , then for , and is the g-fold self intersection number. [_]

Polishchuk ex 5 pg 149: where is an isogeny of abelian varieties and is a coherent sheaf. [_]

Ex 6: Let be a line bundle on an abel. var. . Then for , for . Also the multiplication by n map has degree .

proof: For the first statement apply the Riemann-Roch theorem. If then and

For the second statement, note that every abelian variety is projective, so has a very ample line bundle , as is an isomorphism, is also ample and is symmetric. Indeed,

By proposition 8.5 on page 103 of Polishuck, , so using ex 5,

Hence .QED.

## About this entry

You’re currently reading “Polishchuk Ex. 4,6 pg. 149,” an entry on Math Meandering

- Published:
- June 27, 2009 / 12:29 am

- Category:
- Teleman

- Tags:
- deg of [n], Riemann-Roch

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