By Claudio Bartocci, Ugo Bruzzo, Daniel Hernández Ruipérez

Integral transforms, comparable to the Laplace and Fourier transforms, were significant instruments in arithmetic for a minimum of centuries. within the final 3 many years the advance of a couple of novel principles in algebraic geometry, classification concept, gauge concept, and string concept has been heavily on the topic of generalizations of critical transforms of a extra geometric character.

*Fourier–Mukai and Nahm Transforms in Geometry and Mathematical Physics* examines the algebro-geometric strategy (Fourier–Mukai functors) in addition to the differential-geometric structures (Nahm). additionally integrated is a large amount of fabric from current literature which has no longer been systematically prepared right into a monograph.

Key features:

* simple structures and definitions are awarded in initial historical past chapters

* Presentation explores purposes and indicates a number of open questions

* vast bibliography and index

This self-contained monograph presents an creation to present examine in geometry and mathematical physics and is meant for graduate scholars and researchers simply getting into this field.

**Read Online or Download Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics PDF**

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**Extra resources for Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics**

**Example text**

Moreover, the Kodaira-Spencer map at s for the family FU is the composition of the tangent map Ts α with the Kodaira-Spencer map for the universal family Q. Since the latter is an isomorphism because of the universality of Q (cf. Eq. 20)), KSs (F) is injective. Let X, Y be proper varieties and F a coherent sheaf on X × Y , flat over X. The flatness of F implies Fx ΦF X→Y (Ox ) for every closed point x ∈ X. 24. 21) which coincides with the Kodaira-Spencer morphism for the family F. Proof. 8) one has the identification Ext1X (Ox , Ox ) = Tx X.

Then the class Σ = {Pi }i∈Z ⊂ Ob(Db (A)) is a spanning class for Db (A). Proof. Let A• be an object of Db (A) and assume that HomjDb (A) (Pi , A• ) = 0 for every i and j. 8, A• is isomorphic to an object A in A. 7, one has A = 0. Now take a complex A• in Db (A) such that HomjDb (A) (A• , Pi ) = 0 for every i and j. If S is the Serre functor of the category Db (A), then HomjDb (A) (Pi , S(A• )) HomDb (A) (Pi , S(A• [j])) HomDb (A) (A• [j], Pi )∗ Hom−j (A• , Pi )∗ = 0 for every i and j. Then by the D b (A) • previous argument we have S(A [j]) = 0 for every j, so that A• = 0.

Moreover, this morphism is the only one satisfying the condition F (ψ) ◦ hA• = hB• ◦ ψ, as HomDb (A) (A• , F (Pi⊕k )[−q]) HomDb (A) (A• , Pi⊕k [−q]) HomqDb (A) (A• , Pi⊕k ) = 0 38 Chapter 2. Fourier-Mukai functors Again, one easily proves that the morphism hA• does not depend on the choice of the morphism φ : Pi⊕k [−q] → A• ≤q A• . Finally, we prove that this construction is functorial. Let us take a morphism : A• → C • with (C • ) ≤ N . 2) is commutative. Let q be as above the maximum of the integers such that H q (A• ) = 0 and p the maximum of the integers such that H p (C • ) = 0.