Fourier-Mukai Duality

General Idea

Let $X$ and $Y$ be smooth projective varieties, and $\mathcal{E}$ a suitable object in the derived category $\mathcal{D}^b(X\times Y)$ of sheaves over $X\times Y$. There is a transformation $$ \mathcal{D}^b(X)\xrightarrow{(Rp_X)^\ast} \mathcal{D}^b(X\times Y)\xrightarrow{\cdot\otimes^L\mathcal{E}}\mathcal{D}^b(X\times Y)\xrightarrow{(Rp_Y)_\ast} \mathcal{D}^b(Y) $$ between derived categories of sheaves. $(Rp_X)_\ast$ is the derived inverse image functor and $(Rp_Y)_\ast$ ist the derived direct image functor for the coordinate projections $p_X$ and $p_Y$.

Mukai proved that if $T$ is an abelian variety and $\hat{T}$ its dual, with $\mathcal{E}$ the structure sheaf of sections of the Poincare bundle $\mathcal{P}$, then the transform $$ \mathcal{D}^b(T)\to \mathcal{D}^b(\hat{T}) $$ is an isomorphism, with inverse the map induced by the dual Fourier-Mukai transform obtained by flipping $T$ and $\hat{T}$. The isomorphism is an instance of T-duality, and is called Fourier-Mukai duality.

Baum-Connes map for free abelian groups

Let $[D]$ be the class of the $\mathbb{Z}^d$-equivariant Dirac operator on $\mathbb{R}^d$. Apply Kasparov’s descent and Fourier transform $$ j:\operatorname{KK}^{\mathbb{Z}^d}_\ast(C_0(\mathbb{R}^d),\mathbb{C}) \to \operatorname{KK}_{-d}(C(\mathbb{T}^d),C^\ast(\mathbb{Z}^d)) \cong \operatorname{KK}_{-d}(C(\mathbb{T}^d),C(\widehat{\mathbb{Z}}^d)) $$ then compose with the class of the standard Morita equivalence bimodule in $\operatorname{KK}_0(C(\mathbb{T}^d),C_0(\mathbb{R}^d)\rtimes \mathbb{Z}^d)$.

The composition of these maps sends $[D]\in \operatorname{KK}_{-d}(C_0(\mathbb{R}^d),\mathbb{C})$ to the class of the Fourier-Mukai correspondence $[\mathcal{F}_d]\in \operatorname{KK}_{-d}(\mathbb{T}^d,\widehat{\mathbb{Z}}^d)$. The Poincare bundle’s appearance is due to the Morita equivalence bimodule, which has a twisting effect.

Deformation quantization

Fourier-Mukai equivalence is an instance of derived equivalences of algebraic varieties.

Deformations of a scheme in a particular direction may correspond to deformations to a rather on the other side of a derived equivalence, typically twisted or noncommutative spaces. The deformed derived equivalences often give an interpretation of the deformations of the moduli spaces as moduli spaces in their own right…

See

D.Arinkin, J.Block, T.Pantev. $\ast$-Quantizations of Fourier-Mukai transforms. https://arxiv.org/abs/1101.0626

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