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As the effective field theory of the superstring theory, ten-dimensional${\cal N}=1$ supersymmetric Yang-Mills theory is induced. We consider theten-dimensional space-time ${\cal M}_{10}$ as direct products of ourfour-dimensional space-time ${\cal M}_{4}$ and three two-dimensional compactspaces $X_i\ (i=1,2,3)$, i.e. ${\cal M}_{10} = {\cal M}_4 \times X_2 \times X_2\times X_3$. In particular, we consider $X_i$ as a torus, the torus orbifold,and also the blow-up manifold of the torus orbifold with background magneticfluxes. We discuss the modular symmetry in the magnetized torus and itsorbifold compactifications. We find that the modular flavor groups and therepresentations as well as modular weights of the chiral fields such as quarksand leptons obtained from the magnetized torus as well as the orbifoldcompactifications can be uniquely determined with the magnetic fluxes. We alsodiscuss the magnetized blow-up manifold compactification in which orbifoldsingularities are replaced by parts of sphere as a smooth manifold. We findthat the chiral zero mode numbers on the blow-up manifold as well as theorbifold can be determined only by the magnetic fluxes including localizedfluxes due to the Atiyah-Singer index theorem, and then the degree of freedomof localized fluxes gives new additional chiral zero modes. We also find thatthe new chiral zero modes correspond to localized modes around the orbifoldsingularities.

Flavor structure in magnetized orbifold and blow-up manifold compactification