Semi-infinite \(A\)-variations of Hodge structure over extended Kähler cone (Q2770884)

If \((X, \widehat{X})\) is a mirror pair of Calabi-Yau manifolds, Kontsevich's homological mirror conjecture implies that \(H^\ast (X, \wedge^\ast T_X) = H^\ast (\widehat{X}, \mathbb C)\). The left hand side of this relation can be identified with the tangent space at \(X\) to the extended moduli space \(\mathcal M_{\text{compl}}\) of complex structures. This moduli space is the base of semi-infinite \(B\)-variations \(\text{VHS}^B (X)\) of Hodge structures in \(H^\ast (\widehat{X}, \mathbb C)\). On the other hand, the right hand side of the relation is the tangent space to the extended moduli space \(\mathcal M_{\text{sympl}}\) of Kähler forms on \(\widehat{X}\). NEWLINENEWLINENEWLINEIn this paper the author constructs from semi-infinite \(A\)-variations of Hodge structures over \(\mathcal M_{\text{sympl}}\) a family of solutions of the WDVV equations parametrized by isotropic increasing filtrations which are complementary to the Hodge type filtration in \(\bigoplus_{i,j} H^i (X, \wedge^j T_X)\). He also establishes canonical isomorphisms \(\text{VHS}^A (X) = \text{VHS}^B (\widehat{X})\) and \(\text{VHS}^B (X) = \text{VHS}^A (\widehat{X})\) for dual torus fibrations.