▁The ▁existence ▁of ▁some ▁am ple ▁line ▁bundle ▁on ▁X ▁is ▁equivalent ▁to ▁X ▁being ▁a ▁project ive ▁variety , ▁so ▁a ▁F ano ▁variety ▁is ▁always ▁project ive . ▁For ▁a ▁F ano ▁variety ▁X ▁over ▁the ▁complex ▁numbers , ▁the ▁K od ai ra ▁van ishing ▁theorem ▁implies ▁that ▁the ▁she af ▁coh om ology ▁groups ▁ ▁of ▁the ▁structure ▁she af ▁van ish ▁for ▁. ▁In ▁particular , ▁the ▁Tod d ▁genus ▁ ▁automatically ▁equals ▁ 1 . ▁The ▁ ▁cases ▁of ▁this ▁van ishing ▁statement ▁also ▁tell ▁us ▁that ▁the ▁first ▁Ch ern ▁class ▁indu ces ▁an ▁isomorphism ▁. ▁ ▁By ▁Y au ' s ▁solution ▁of ▁the ▁Cal abi ▁conject ure , ▁a ▁smooth ▁complex ▁variety ▁adm its ▁K äh ler ▁metrics ▁of ▁positive ▁Ric ci ▁curv ature ▁if ▁and ▁only ▁if ▁it ▁is ▁F ano . ▁My ers ' ▁theorem ▁ ▁therefore ▁tells ▁us ▁that ▁the ▁universal ▁cover ▁ ▁of ▁a ▁F ano ▁manifold ▁is ▁compact , ▁and ▁so ▁can ▁only ▁be ▁a ▁finite ▁covering . ▁However , ▁ ▁we ▁have ▁just ▁seen ▁that ▁the ▁Tod d ▁genus ▁of ▁a ▁F ano ▁manifold ▁must ▁equal ▁ 1 . ▁Since ▁this ▁would ▁also ▁apply ▁to ▁the ▁manifold ' s ▁universal ▁cover , ▁and ▁since ▁the ▁Tod d ▁genus ▁is ▁multip lic ative ▁under ▁finite ▁covers , ▁it ▁follows ▁that ▁any ▁F ano ▁manifold ▁is ▁simply ▁connected . ▁ ▁A ▁much ▁easier ▁fact ▁is ▁that ▁every ▁F ano ▁variety ▁has ▁K od ai ra ▁dimension ▁− ∞ . ▁