MFV3D Book Archive > Differential Geometry > Download Fourier-Mukai and Nahm Transforms in Geometry and by Claudio Bartocci, Ugo Bruzzo, Daniel Hernández Ruipérez PDF

Download Fourier-Mukai and Nahm Transforms in Geometry and by Claudio Bartocci, Ugo Bruzzo, Daniel Hernández Ruipérez PDF

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.

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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.

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