Why is it that mirror symmetry has many relations with algebraic geometry, rather than with complex geometry or differential geometry? (In other words, how is it that solutions to polynomials become relevant, given that these do not appear in the physics which motivates mirror symmetry?) I would especially appreciate nontechnical answers.

<ORIGINAL QUESTION: How does mirror symmetry has many relations with algebraic geometry, rather than complex geometry or differential geometry? (in more transparent terms , how does we consider the solution of polynomials ,then consider polynomial maps between two), or if it is really so, can anyone give some expository answers?>