[Paper Review] A Fourier-Mukai Transform for Stable Bundles on K3 Surfaces

Claudio Bartocci, Ugo Bruzzo+1 more|ArXiv.org|May 13, 1994

Algebraic Geometry and Number Theory6 references19 citations

TL;DR

This paper introduces a Fourier-Mukai transform for stable bundles on complex K3 surfaces using a moduli space of stable sheaves as the dual variety. It proves that the transform preserves polystability and stability for zero-degree bundles, is invertible when the dual moduli space is isomorphic to the original K3 surface, and preserves the Euler characteristic and Mukai vector degree.

ABSTRACT

We define a Fourier-Mukai transform for sheaves on K3 surfaces over $\C$, and show that it maps polystable bundles to polystable ones. The role of ``dual'' variety to the given K3 surface $X$ is here played by a suitable component $\hat X$ of the moduli space of stable sheaves on $X$. For a wide class of K3 surfaces $\hat X$ can be chosen to be isomorphic to $X$; then the Fourier-Mukai transform is invertible, and the image of a zero-degree stable bundle $F$ is stable and has the same Euler characteristic as $F$.

Motivation & Objective

• To define a Fourier-Mukai transform for stable vector bundles on complex K3 surfaces using a moduli space of sheaves as the dual variety.
• To establish conditions under which this transform preserves polystability and stability of vector bundles.
• To prove the transform is invertible when the dual moduli space is isomorphic to the original K3 surface.
• To show the transform preserves topological invariants such as the Euler characteristic and Mukai vector degree.
• To extend the correspondence between stable bundles and instanton bundles via the Hitchin-Kobayashi correspondence in the algebraic setting.

Proposed method

• Define the Fourier-Mukai transform via a kernel sheaf $Χ$ on $X \times \widehat{X}$, where $\widehat{X}$ is a component of the moduli space of stable sheaves on $X$.
• Use the derived category formalism: $\mathcal{S}_{X}(\mathcal{F}) = R\hat{\pi}_* (\mathcal{Q} \otimes^L \pi^* \mathcal{F})$ for sheaves $\mathcal{F}$ on $X$.
• Apply the Weak Index Theorem (WIT) and Index Theorem (IT) conditions to identify when the transform is concentrated in a single cohomological degree.
• Leverage Mukai’s result that $M_H(v)$ is a K3 surface when compact and two-dimensional, and isogenous to $X$.
• Use the Riemann-Roch theorem and Chern character computations to derive explicit formulas for the Mukai vector of the transform.
• Utilize the Hitchin-Kobayashi correspondence to relate algebraic stability to differential-geometric instanton solutions, ensuring stability is preserved.

Experimental results

Research questions

• RQ1 Under what conditions does the Fourier-Mukai transform preserve the stability of zero-degree vector bundles on a K3 surface?
• RQ2 When is the Fourier-Mukai transform invertible, and what conditions ensure the dual moduli space $\widehat{X}$ is isomorphic to the original K3 surface $X$?
• RQ3 How do the topological invariants—especially the Mukai vector and Euler characteristic—transform under this correspondence?
• RQ4 Can the transform be extended to preserve holomorphic symplectic structures on moduli spaces of stable sheaves?
• RQ5 What is the relationship between the moduli space of stable bundles on $X$ and the corresponding space on $\widehat{X}$ under the transform?

Key findings

• The Fourier-Mukai transform maps $\mu$-polystable sheaves of zero degree on $X$ to $\mu$-polystable sheaves on $\widehat{X}$, preserving stability.
• For a wide class of K3 surfaces, $\widehat{X} \cong X$, and the transform is invertible, with $\mathcal{F} \mapsto \widehat{\mathcal{F}}$ inducing a self-equivalence on the derived category.
• The transform preserves the Euler characteristic: $\chi(\widehat{\mathcal{F}}) = \chi(\mathcal{F})$ for WIT 1 sheaves.
• The Mukai vector transforms via explicit formulas: $\hat{\rho} = -3\rho + 2\sigma + \ell \cdot c_1$, $\hat{c}_1 = (\ell \cdot c_1 + 2d)\widehat{H} + (\rho + d - s)\hat{\ell} - \Psi^*(c_1)$, $\hat{\sigma} = 2\rho - 3\sigma - \ell \cdot c_1$.
• The transform preserves the degree of the Mukai vector: $\hat{u}^2 = u^2$, ensuring equal-dimensional moduli spaces.
• The map induces a birational correspondence between moduli spaces $M_H(u)$ and $M_{\widehat{H}}(\hat{u})$, and in special cases (e.g., $u = (1+2n, -n\hat{\ell}, 1-3n)$), $M_{\widehat{H}}(\hat{u})$ is biholomorphic to the punctual Hilbert scheme $\operatorname{Hilb}^n(X)$.