, ▁Z ) ▁and ▁that ▁the ▁class ▁of ▁ ν Γ ▁is ▁the ▁fundamental ▁class ▁of ▁ Γ . ▁Furthermore , ▁he ▁proved ▁that ▁a ▁class ▁in ▁H 2 ( X , ▁Z ) ▁is ▁the ▁class ▁of ▁a ▁normal ▁function ▁if ▁and ▁only ▁if ▁it ▁lies ▁in ▁H 1 , 1 . ▁T ogether ▁with ▁Po inc ar é ' s ▁existence ▁theorem , ▁this ▁proves ▁the ▁theorem ▁on ▁( 1 , 1 )- classes . ▁ ▁Proof ▁using ▁she af ▁coh om ology ▁ ▁Because ▁X ▁is ▁a ▁complex ▁manifold , ▁it ▁adm its ▁an ▁exponential ▁she af ▁sequence ▁ ▁T aking ▁she af ▁coh om ology ▁of ▁this ▁exact ▁sequence ▁gives ▁maps ▁ ▁The ▁group ▁ ▁of ▁line ▁bund les ▁on ▁X ▁is ▁is omorphic ▁to ▁. ▁The ▁first ▁Ch ern ▁class ▁map ▁is ▁c 1 ▁by ▁definition , ▁so ▁it ▁suff ices ▁to ▁show ▁that ▁i * ▁is ▁zero . ▁ ▁Because ▁X ▁is ▁K äh ler , ▁H odge ▁theory ▁implies ▁that ▁. ▁However , ▁i * ▁factors ▁through ▁the ▁map ▁from ▁H 2 ( X , ▁Z ) ▁to ▁H 2 ( X , ▁C ), ▁and ▁on ▁H 2 ( X , ▁C ), ▁i * ▁is ▁the ▁restriction ▁of ▁the ▁projection ▁onto ▁H 0 , 2 ( X ). ▁It ▁follows ▁that ▁it ▁is ▁zero ▁on ▁, ▁and ▁consequ ently ▁that ▁the ▁cycle ▁class ▁map ▁is ▁sur jective . ▁ ▁References ▁ ▁Bibli ography ▁▁▁▁▁ ▁Re print ed ▁in ▁▁ ▁Category : The or ems