Mirror Symmetry on K3 Surfaces via Fourier-Mukai Transform

Abstract

We use a relative Fourier-Mukai transform on elliptic K3 surfaces X to describe mirror symmetry. The action of this Fourier-Mukai transform on the cohomology ring of X reproduces relative T-duality and provides an infinitesimal isometry of the moduli space of algebraic structures on X which, in view of the triviality of the quantum cohomology of K3 surfaces, can be interpreted as mirror symmetry.From the mathematical viewpoint the novelty is that we exhibit another example of a Fourier-Mukai transform on K3 surfaces, whose properties are closely related to the geometry of the relative Jacobian of X.We denote byê : P 1 → X the canonical section; one hasê = ̟ • e. Moreover, we denote by π,π the projections of the fibred product X × P 1 X onto the factors.Remark 2.2. The Picard functor is also representable by an open dense subscheme J of X, the relative Jacobian J → P 1 of X → P 1 . If X s ⊂ X denotes the complement of the singular points of the fibres of π, then ̟ gives an isomorphism X s ∼ → J of schemes 3

Stable sheaves on elliptic fibrations

“…We write ι : X → X for the isomorphism mapping any rank-one torsion-free and zero-degree sheaf F on a fibre X s to its dual F * . Most of the results in [5] are also true in our more general setting, in some cases just with straightforward modifications.…”

“…In the first part we consider the "dual" elliptic fibrationp : X → B ( [5]) defined as the compactified relative Jacobian of X → B (actually, X turns out to be isomorphic with X) and we introduce the relative Fourier-Mukai transform and its properties. This allows for a nice description of the spectral cover construction.…”

“…The relative Jacobian J 0 → B of X as a B-scheme is the smooth locus X ′ ofp : X → B and if U ⊆ B is the open subset supporting the smooth fibres of p, one has J 0 U ≃ X U . As in [5], we denote byê : B ֒→ X the section ̟ • e and by Θ the divisorê(B) = ̟(H). We write ι : X → X for the isomorphism mapping any rank-one torsion-free and zero-degree sheaf F on a fibre X s to its dual F * .…”

“…Here we consider an elliptic fibration p : X → B as above and the associated "dual" fibrationp : X → B. We shall define a relative Fourier-Mukai in this setting by means of the relative universal Poincaré sheaf P on the fibred product X × B X normalized so that P |H× B X ≃ O X as in [5]. P is also flat over X, and P * enables us to identify p : X → B with a compactification of the relative Jacobian J 0 → B ofp : X → B.…”

“…Let L be a sheaf on X S , flat over S, whose restrictions to the fibres of p S are torsion-free and have rank one and degree zero. The universal property gives a morphism φ : S → X so that (1 × φ) * P ≃ L ⊗ p * S N for a certain line bundle N on S. Let Γ : S ֒→ X S be the graph of the morphism ι • φ : S → X. Lemma 2.11 and Corollary 2.12 of [5] now take the form:…”