Fano 3-folds: mirror symmetry and classification

Fano varieties are projective algebraic varieties “with positive curvature”. They have a prominent role in algebraic geometry for many reasons, including the minimal model programme. It is known that in each dimension, the number of deformation classes of smooth Fano varieties is finite. Therefore it is natural to try to classify them. In dimension 2, smooth Fano varieties were classified by del Pezzo at the end of the 19th century. Smooth Fano 3-folds were studied by Fano in the 1930s, and finally classified by Mori and Mukai in the 1980s. At the moment, a complete classification of smooth Fano 4-folds seems out of reach. In this talk, I will introduce Fano varieties, outline their classification problem, and try to explain how mirror symmetry can help to study Fano varieties.