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 improvement of a few novel principles in algebraic geometry, type conception, gauge conception, and string concept has been heavily with regards to generalizations of necessary transforms of a extra geometric character.

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

Key features:

* simple buildings and definitions are provided in initial heritage chapters

* Presentation explores functions and indicates a number of open questions

* vast bibliography and index

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

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Example text

We can iterate this process to define the right convolution of a bounded complex of objects of the derived category. Actually, we do not need to work with a derived category, since any triangulated category will do. 3) a complex of objects of B (that is, the composition of any two consecutive morphisms vanishes). Assume also that one has HomB (a−p [r], a−q ) = 0 , for every p > q and r > 0. 3 as the pair formed by the object a of B and the morphism d0 : a0 → a constructed by induction on the length m as follows: • If m = 0, then a = a0 and d0 is the identity.

Since HomhD(X) (F • , E • ) = HomD(X) (F • , E • [h]), the first statement holds true because Φ is fully faithful. As for the second, we use the first formula together with Φ(F) F[−i], Φ(E) E[−j] and Yoneda’s formula (cf. 68). 4 The equivariant case If an algebraic (typically, finite) group G acts on an algebraic variety X, one may define the equivariant derived category of coherent sheaves on X (cf. [39]). This is defined in terms of coherent sheaves carrying a linearized action of G, compatible with the action on X.

To see that r = 1, it is enough to prove that ΦM X→X (Ox ) one closed point x ∈ X. Let us consider the exact sequence 0 → Px → ΦM X→X (Ox ) → Ox → 0 where the last morphism is the adjunction and Px is the kernel. We want to prove that for some point x the sheaf Px is zero. Since Px is supported at x, it suffices to see that HomX (Px , Ox ) = 0. 22), we have an exact sequence 0 → HomX (Px , Ox ) → Hom1D(X) (Ox , Ox ) → Hom1D(X) (ΦM X→X (Ox ), Ox ) , so that we have to show that Hom1D(X) (Ox , Ox ) → Hom1D(X) (ΦM X→X (Ox ), Ox ) is injective.

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