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The example I am aware of so far is quintic 3-fold equipped with $SU(3)$ holonomy. Why it is more natural to talk about $CY_3$ folds or Calibi-Yau 3 folds without talking about algebraic curves in string theory? Why it is not more natural to talk about curves first? Or why it is more intuitive to movtivate 3-fold first rather than motivate algebraic curves? Reminder: This is not a popular science question. I need mathematical and theoretical physics justification. Please give me the reference if possible.