odge ▁conject ure ▁which ▁has ▁been ▁proved ▁for ▁all ▁K äh ler ▁manif olds . ▁ ▁Stat ement ▁of ▁the ▁theorem ▁ ▁Let ▁X ▁be ▁a ▁compact ▁K äh ler ▁manifold . ▁The ▁first ▁Ch ern ▁class ▁c 1 ▁gives ▁a ▁map ▁from ▁hol omorphic ▁line ▁bund les ▁to ▁. ▁By ▁H odge ▁theory , ▁the ▁de ▁R ham ▁coh om ology ▁group ▁H 2 ( X , ▁C ) ▁decom poses ▁as ▁a ▁direct ▁sum ▁, ▁and ▁it ▁can ▁be ▁proven ▁that ▁the ▁image ▁of ▁c 1 ▁lies ▁in ▁H 1 , 1 ( X ). ▁The ▁theorem ▁says ▁that ▁the ▁map ▁to ▁ ▁is ▁sur jective . ▁ ▁In ▁the ▁special ▁case ▁where ▁X ▁is ▁a ▁project ive ▁variety , ▁hol omorphic ▁line ▁bund les ▁are ▁in ▁bij ection ▁with ▁linear ▁equival ences ▁class ▁of ▁divis ors , ▁and ▁given ▁a ▁divis or ▁D ▁on ▁X ▁with ▁associated ▁line ▁bundle ▁O ( D ), ▁the ▁class ▁c 1 ( O ( D )) ▁is ▁Po inc ar é ▁dual ▁to ▁the ▁hom ology ▁class ▁given ▁by ▁D . ▁Thus , ▁this ▁establish es ▁the ▁usual ▁form ulation ▁of ▁the ▁H odge ▁conject ure ▁for ▁divis ors ▁in ▁project ive ▁vari eties . ▁ ▁Proof ▁using ▁normal ▁functions ▁ ▁Le f sch etz ' s ▁original ▁proof ▁worked ▁on ▁project ive ▁surfaces ▁and ▁used ▁normal ▁functions , ▁which ▁were ▁introduced ▁by ▁Po inc ar é . ▁Suppose ▁that ▁C t ▁is ▁a ▁p enc il ▁of ▁curves ▁on ▁X . ▁Each ▁of