Fourier-Mukai Transform in Algebraic Geometry

• Fourier-Mukai transform is a kernel-based integral functor that establishes derived equivalences between categories of coherent sheaves on varieties.
• It translates geometric, stability, and cohomological data through explicit matrix actions and Grothendieck-Riemann-Roch relations.
• Its applications range from moduli space birational correspondences and Bridgeland stability conditions to underpinning dualities in mirror symmetry.

The Fourier-Mukai transform is a pivotal construction in algebraic geometry and homological algebra, providing explicit derived equivalences between the bounded derived categories of coherent sheaves on varieties, and forming the categorical backbone for dualities such as T-duality in string theory and deep structural symmetries in moduli theory. Developed initially by Mukai for abelian varieties, the formalism has since been generalized to a broad class of varieties and stacks, playing a key role in modern developments such as Bridgeland stability, mirror symmetry, and the categorical Langlands program. As an integral transform defined by a kernel object on the product of two spaces, the Fourier-Mukai transform enables a functorial 'transport' of geometric and cohomological information between derived categories, intertwining stability, geometry, and moduli in a systematic way.

1. Formalism and Construction

Given smooth projective varieties (or, more generally, separated finite-type schemes) $X$ and $Y$ over a field (usually $\mathbb{C}$), an object $\mathcal{P} \in D^b(X \times Y)$ (the "kernel") determines an exact functor

$$\Phi_{\mathcal{P}}^{X \to Y} : D^b(X) \to D^b(Y), \quad \Phi_{\mathcal{P}}(E) = R p_{Y*}\left( p_X^* E \otimes^{\mathbf{L}} \mathcal{P} \right)$$

where $p_X$ and $p_Y$ are the projections from $X \times Y$ to $X$ and $Y$, respectively. This fundamental construction, called an "integral functor," encompasses all exact functors between derived categories of smooth projective varieties by Orlov’s representability theorem.

Many key attributes of $\Phi_{\mathcal{P}}$ (fully faithfulness, equivalence, adjoint existence) are determined by properties of the kernel $\mathcal{P}$, such as perfectness and geometric support properties. For abelian varieties and K3 surfaces, a normalized Poincaré bundle provides a canonical kernel inducing a derived equivalence.

On cohomology, such a functor induces a linear map via the Grothendieck-Riemann-Roch relation: $$\Phi_{\mathcal{P}}^H(\alpha) = p_{Y*}(p_X^*(\alpha) \cup \operatorname{ch}(\mathcal{P}))$$ for $\alpha \in H^*(X, \mathbb{Q})$. In the abelian/generic Calabi-Yau context, the Mukai vector construction and the induced isometries of "Mukai lattices" are central.

2. Fourier-Mukai on Product Elliptic Threefolds: Limit Tilt Stability

On the elliptic threefold $X = C \times S$, with $C$ an elliptic curve and $S$ a K3 surface of Picard rank 1, the Fourier-Mukai transform $\Phi$ is constructed by pulling back the Poincaré bundle from $C \times C$ fiberwise over $S$. The transform acts as an autoequivalence on $D^b(X)$: for any $E \in D^b(X)$,

$$\Phi(E) = R p_{13*} \left( p_{12}^* \mathcal{P} \otimes p_{23}^* E \right),$$

where $\mathcal{P}$ is the Poincaré bundle and $p_{12}, p_{23}, p_{13}$ are projections from $C \times C \times S$.

The action on Chern characters is explicitly described by a matrix: $$\operatorname{ch}(\Phi(E)) = \begin{pmatrix} a_{10} & a_{11} & a_{12} \ - a_{00} & - a_{01} & - a_{02} \end{pmatrix} = \begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix} \cdot \operatorname{ch}(E)$$ providing significant computational control for moduli problems.

Limit tilt stability is defined by considering the tilt of the heart of the bounded derived category at a family of polarizations $\omega = t H + s D$ subject to $ts = \alpha$, and taking the limit as $s \to \infty$. The resulting "limit heart" $\mathcal{B}$ and the induced central charge $Z_{\lim}$ capture the leading-order contribution in Laurent expansion, yielding a stability function with the Harder–Narasimhan property.

Key results under this transform:

• If $E \in \operatorname{Coh}(X)$ is $\mu_\omega$-stable for a suitable polarization and satisfies a codimension-2 vanishing condition ($$\operatorname{Hom}(W_{0,X} \cap \operatorname{Coh}^{\leq 1}(X), E[1]) = 0$$), then $\Phi(E)[1]$ is limit tilt stable in $\mathcal{B}$.
• Conversely, a limit tilt semistable object with nonzero fiber-degree is mapped, under an explicit decomposition, to a $\mu$-semistable torsion-free sheaf (up to possible modification by a codimension-2 sheaf).

The vanishing condition excludes unwanted lower-dimensional parts in the cohomology of transforms, ensuring good behavior under moduli functoriality.

3. Stability Conditions, Tilted Hearts, and Bridgeland Theory

In higher-dimensional settings (especially abelian threefolds as in (Piyaratne, 2017, Piyaratne, 2015, Maciocia et al., 2013)), the Fourier-Mukai transform is instrumental in constructing and transporting Bridgeland stability conditions. The critical elements are:

• Slope and tilt stability: Start with the classical slope $\mu$, tilt at slope to form a new abelian subcategory (the "tilted heart"), and possibly iterate (e.g., double tilt for threefolds).
• Central charge:

$$Z_{B, \omega}(E) = -\int_X e^{-B - i \omega} \operatorname{ch}(E)$$

This function provides the phase for defining semistability.

• Generalized Bogomolov-Gieseker (BG) inequalities: For tilt-stable objects,

$$\operatorname{ch}_3^{B}(E) - \frac{\omega^2}{2} \operatorname{ch}_1^{B}(E) \leq 0,$$

whose verification (demonstrated for abelian threefolds) is necessary and sufficient for the existence of a Bridgeland stability condition.

The Fourier-Mukai transform, under these structures, provides explicit symmetries on the stability manifold: it carries Bridgeland stability conditions (hearts and central charges) on $X$ to those on $Y$, possibly up to shift.

Notably, the matrix action on twisted Chern characters under the transform exhibits anti-diagonal structure (e.g., $(1, -1, 1, -1)$ for threefolds), which is crucial for matching stability and cohomological data between equivalences (Piyaratne, 2017, Piyaratne, 2015).

4. Applications to Moduli Spaces and Wall-Crossing

The preservation of stability under Fourier-Mukai transforms has critical consequences for moduli theory:

• Moduli spaces of stable sheaves (or complexes) with fixed invariants on derived-equivalent varieties are related by explicit birational correspondences, as isometries of the Mukai lattice match Bridgeland–stability conditions. In particular, under suitable isometry of Mukai vectors, the moduli spaces $M_H(v)$ and $M_{H'}(v')$ are birational, a result that is established by tracking wall-crossing in stability space and using properties preserved by the transform (Minamide et al., 2011).
• For elliptic and Calabi–Yau threefolds, limit tilt stability provides new methods of constructing sheaves and complexes with desired stability, moduli-theoretic properties, and invariants, especially relevant for Donaldson–Thomas theory and the computation of DT/PT invariants.
• In the setting of integrable systems (e.g., Hitchin fibrations), variants such as the partial Fourier-Mukai transform yield fully faithful functors between categories associated to relative Picard varieties and the total space, central to categorical approaches to the geometric Langlands program (Arinkin et al., 2014).

5. Deformation, Quantization, and Homological Mirror Symmetry

The Fourier-Mukai formalism extends beyond classical algebraic geometry:

• Deformation Theory: The ability to deform Fourier-Mukai equivalences across families is governed by the behavior of the Chern character of the kernel under the Hodge (or crystalline) filtration. Full faithfulness is preserved under deformation if, and only if, the filtered Chern classes remain of Hodge type, with the obstruction class formulated as the cup product with the derived Kodaira–Spencer class (Rienks, 12 Aug 2024).
• Quantization: Given a formal $\star$-quantization of $X$, there exists a unique quantization of $Y$ so that the Fourier-Mukai transform deforms to an equivalence between quantized derived categories, with deformations governed by $D$-comodule theory and vanishing of obstruction groups in the context of stacks of algebroids (Arinkin et al., 2011).
• Generalized Categories and Mirror Symmetry: On K3 and Calabi–Yau manifolds, the Fourier-Mukai transform interchanges deformation spaces of (generalized) complex and symplectic structures, aligning with predictions from homological mirror symmetry, and provides explicit transport of deformation and Hochschild cohomological data (Sawon, 2012).

6. Generalizations and Structural Results

• Relative and Twisted Settings: For families of varieties over a base, the relative Fourier-Mukai transform is constructed using kernels on the fiber product over the base and is sensitive to both geometric and group-theoretical structures (SL$_2(\mathbb{Z})$, unitary automorphism groups). Isometric isomorphism criteria relate the existence of relative partners and give finiteness results for derived partners (Martín et al., 2010).
• KR-theory: In Real and twisted settings, the Fourier-Mukai transform generalizes to isomorphisms between (twisted) $KR$-theory groups of $T$-dual torus bundles, requiring the careful construction of Real Poincaré bundles and gerbe-theoretic data. This framework provides new insights into Real $T$-duality and applications to the indices of families of Real Dirac operators (Baraglia, 29 Sep 2025).
• Noncommutative/Families/Arithmetic: Extensions to arithmetic $\mathcal{D}^{(0)}$-modules and formal schemes retain crucial features (involutivity, essential surjectivity on objects) despite the partial failure of the Kashiwara theorem in positive characteristic, preserving the overall functional and categorical framework (Viguier, 2022).

7. Cohomological and Computational Aspects

The central computational techniques exploit the explicit action of the transform on Chern classes, Mukai vectors, and cohomological generators. Key matrix formulas, spectral sequences (e.g., the Mukai spectral sequence), and identification of ample/polarizing classes under derived equivalence ground many structural results and applications to moduli and birational geometry.

Tables such as the action on twisted Chern vectors or cohomological generators make the calculation of invariants and the transport of stability data tractable in explicit moduli problems.

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The Fourier-Mukai transform thus forms the backbone of a wide swath of modern algebraic geometry, connecting derived category techniques to moduli spaces, stability conditions, arithmetic, and even mathematical physics. The explicit kernel-based methodology provides both categorical and computational control, underpinning the deep symmetry properties of the derived category and its role in dualities, deformation, quantization, and mirror symmetry.