# Famous Fourier-Mukai Results I

A while back I (tried to) give a talk about some of the uses of the Fourier-Mukai transform. I’ve only learned about the Fourier-Mukai transform in the context of dealing with smooth projective varieties, so everything will be at least that. For , the Fourier-Mukai transform is via .

In my early readings these were the most prominent results I came across:

- If is an Abelian variety then the Poincare bundle gives an isomorphism .
- (Orlov’s Result) Any functor which is full, faithful and exact is represented by an object on the product.
- If or its inverse is ample then determines .
- For Abelian varieties ,

### Some Details for 1

The proof I plan to outline depends on the following results

1. If and then . See cohom of pic zero in this post.

2. ( Adjunction )For and set

.

Then represent left and right adjoints to respectively.

pf:

this isomorphism comes from the fact that and Grothendieck-Verdier duality:

Let be a morphism of smooth schemes over a field (lets say algebraically closed). There is an isomorphism

.

continuing the hom isomorphisms…

. QED.

3. if fully faithful iff

=

4. (purely category theory) If is a fully faithful, exact functor between triangulated categories and contains objects not isomorphic to and is indecomposable

roughly a triangulated category is decomposable if there subcategories s.t. $\forall O \in ob(C) \exists$ a distinguished triangle with and ; also both subcateogories have to contain objects not isomorphic to 0.

Then is an equivalence iff has left and right adjoints $G,H$ and for , .

A proof of this can be found in Huybrechts book on the Fourier-Mukai transform. This result gives away the idea of the proof: use algebraic geometry results to check the hypothesis of this assertion in the case at hand.

Putting it all together.

Using 2 it is easily checked that hence is exact with left and right adjoints.

Showing that is indecomposable is more category theoretic and I’m skipping that for now.

Unwrapping the definitions,

=

, =

so is fully faithful using 3 and 1.

I’m putting some details for Orlov’s result in a whole other post.

I don’t know how to prove result 3, but I know a few important points regarding its proof. The triangulated consists of all the categorical data plus the data of a Serre functor given by .

Central to the proof are the notions of point like objects and invertible objects. is point like if

- for
- is a field.

An object is invertible if point like $\exists n(L,P) \in \mathbb{Z}$ such that .

My understanding is roughly is picking out this information allows you to pick out and consequently the canonical ring which determines .

## About this entry

You’re currently reading “Famous Fourier-Mukai Results I,” an entry on Math Meandering

- Published:
- June 20, 2010 / 10:56 pm

- Category:
- alg. geo.

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