This article is a geometric application of polarized logarithmic Hodge theory of Kazuya Kato and Sampei Usui. We prove generic Torelli theorem for the well-known quintic-mirror family in two ways by using different logarithmic points at the boundary of the fine moduli of polarized logarithmic Hodge structures.
References
P. Candelas, C. de la Ossa, P. S. Green and L. Parks, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), no. 1, 21–74. MR1115626 10.1016/0550-3213(91)90292-6P. Candelas, C. de la Ossa, P. S. Green and L. Parks, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), no. 1, 21–74. MR1115626 10.1016/0550-3213(91)90292-6
D. R. Morrison, Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians, J. Amer. Math. Soc. 6 (1993), no. 1, 223–247. MR1179538 10.1090/S0894-0347-1993-1179538-2D. R. Morrison, Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians, J. Amer. Math. Soc. 6 (1993), no. 1, 223–247. MR1179538 10.1090/S0894-0347-1993-1179538-2
B. Szendröi, Calabi-Yau threefolds with a curve of singularities and counterexamples to the Torelli problem, Internat. J. Math. 11 (2000), no. 3, 449–459. MR1769616B. Szendröi, Calabi-Yau threefolds with a curve of singularities and counterexamples to the Torelli problem, Internat. J. Math. 11 (2000), no. 3, 449–459. MR1769616
C. Voisin, A generic Torelli theorem for the quintic threefold, in New trends in algebraic geometry (Warwick, 1996), 425–463, Cambridge Univ. Press, Cambridge. MR1714833 10.1017/CBO9780511721540.017C. Voisin, A generic Torelli theorem for the quintic threefold, in New trends in algebraic geometry (Warwick, 1996), 425–463, Cambridge Univ. Press, Cambridge. MR1714833 10.1017/CBO9780511721540.017