"In the paper under review the authors define an analogue of the relative Fourier-Mukai transform for a minimal K3 surface equipped with an elliptic fibration p: X --> P^1 which is assumed to have a section. The compactification of the dual fibration is the moduli space M of stable sheaves on X of pure dimension 1 and Chern character (0, μ, 0) where μ is the cohomology class of the fibres of p. In fact, one has a natural isomorphism of X and M. The authors also show that M is isomorphic to the compactified relative Jacobian J(X) of X --> P^1. Further, it is shown that the action of the Fourier transform on the cohomologies of X reproduces relative T-duality and provides an isomorphism \psi: H^{1,1}(J(X) ,C)/Pic(J(X))\otimes C --> H^{1,1}(X,C)/Pic(X)\otimes C, where the LHS [resp. RHS] can be interpreted as the tangent space to the space of deformations of algebraic structures on J(X) [resp. Kähler structures on X] preserving the Picard lattice. Thus, can be regarded as the tangent map to a mirror map at J(X)." Alexander E. Polishchuk, MR1637405 (99f:14046)
Mirror symmetry on K3 surfaces via Fourier-Mukai transform
BARTOCCI, CLAUDIO;
1998-01-01
Abstract
"In the paper under review the authors define an analogue of the relative Fourier-Mukai transform for a minimal K3 surface equipped with an elliptic fibration p: X --> P^1 which is assumed to have a section. The compactification of the dual fibration is the moduli space M of stable sheaves on X of pure dimension 1 and Chern character (0, μ, 0) where μ is the cohomology class of the fibres of p. In fact, one has a natural isomorphism of X and M. The authors also show that M is isomorphic to the compactified relative Jacobian J(X) of X --> P^1. Further, it is shown that the action of the Fourier transform on the cohomologies of X reproduces relative T-duality and provides an isomorphism \psi: H^{1,1}(J(X) ,C)/Pic(J(X))\otimes C --> H^{1,1}(X,C)/Pic(X)\otimes C, where the LHS [resp. RHS] can be interpreted as the tangent space to the space of deformations of algebraic structures on J(X) [resp. Kähler structures on X] preserving the Picard lattice. Thus, can be regarded as the tangent map to a mirror map at J(X)." Alexander E. Polishchuk, MR1637405 (99f:14046)
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