Elliptic Gromov - Witten invariants and the generalized mirror conjecture

A conjecture expressing genus 1 Gromov-Witten invariants in mirror-theoretic terms of semi-simple Frobenius structures and complex oscillating integrals is formulated. The proof of the conjecture is given for torus-equivariant Gromov - Witten invariants of compact K\"ahler manifolds with isolated fixed points and for concave bundle spaces over such manifolds. Several results on genus 0 Gromov - Witten theory include: a non-linear Serre duality theorem, its application to the genus 0 mirror conjecture, a mirror theorem for concave bundle spaces over toric manifolds generalizing a recent result of B. Lian, K. Liu and S.-T. Yau. We also establish a correspondence (see the extensive footnote in section 4) between their new proof of the genus 0 mirror conjecture for quintic 3-folds and our proof of the same conjecture given two years ago.