Mirror Symmetry

Two Calabi-Yau 3-folds X and X-check form a mirror pair. The quintic threefold X has Hodge numbers h^{1,1}(X) = 1 and h^{2,1}(X) = 101. What are the corresponding Hodge numbers of the mirror X-check?

Mirror symmetry exchanges the Kahler moduli (counted by h^{1,1}) and complex-structure moduli (counted by h^{2,1}) of a CY 3-fold: h^{1,1}(X) = h^{2,1}(X-check) and vice versa. So the quintic with (1, 101) mirrors to (101, 1). Option A would mean X is its own mirror, which holds only for self-mirror CYs (not the quintic). Option C would double both moduli spaces, which violates the swap. Option D would mean a rigid CY with no moduli, which contradicts the 101-parameter family of complex structures on the quintic.

What is the role of the mirror map t(z) = pi_1(z)/pi_0(z) in extracting GW invariants from periods?

The mirror map is the change of coordinates between the complex-structure modulus z on X-check and the Kahler modulus t on X. After exponentiating to q = exp(t), the period integral on X-check expanded in q has coefficients that match the genus-0 GW invariants N_d of X (e.g. N_1 = 2875, N_3 = 317206375 for the quintic). Option A confuses the map with the final invariant. Option C is wrong: Picard-Fuchs solutions involve hypergeometric series, not elementary functions. Option D misdescribes the mirror map - it is a coordinate identification, not a self-symmetry.

Kontsevich's Homological Mirror Symmetry conjecture asserts an equivalence between two categories attached to a mirror pair (X, X-check). Which equivalence does it state?

HMS pairs the symplectic A-side (Fukaya category, with Lagrangian submanifolds as objects and Floer cochain complexes as morphisms) with the complex-algebraic B-side (derived category of coherent sheaves on X-check), as A-infinity categories. Option B is far too weak: ordinary vector bundles miss derived/A-infinity structure. Option C is an old observation that holds at the level of Hodge numbers, but HMS is categorical and much deeper. Option D is false: there is no isomorphism between these large symmetry groups; HMS lives at the level of categories of objects, not groups of automorphisms.

According to the Strominger-Yau-Zaslow (SYZ) conjecture, how is the mirror X-check geometrically constructed from a Calabi-Yau n-fold X near the large complex structure limit?

SYZ gives mirror symmetry a geometric content: near the large complex structure limit, a CY n-fold admits a fibration over an n-dimensional base by special Lagrangian tori (the calibration condition Im(Omega|_L) = 0 picks out these fibers); the mirror is obtained by replacing each T^n fiber by its dual torus T-hat = H^1(T, R/Z). Option A would give a manifold with the same Hodge numbers as X, not the mirror. Option B is one explicit construction of mirror pairs (Greene-Plesser orbifold for the quintic family) but is not the general SYZ statement. Option D produces a birational modification, unrelated to mirror duality.