Relative Gromov-Witten invariants of projective completions of vector bundles

Abstract

It was proved by Fan-Lee and Fan that the absolute Gromov-Witten invariants of two projective bundles (V i ) → X are identified canonically when the total Chern classes c(V 1 ) = c(V 2 ) for two bundles V 1 and V 2 over a smooth projective variety X . In this note we show that for the two projective completions (V i ⊕ ) of V i and their infinity divisors (V i ), the relative Gromov-Witten invariants of ( (V i ⊕ ), (V i )) are identified canonically when c(V 1 ) = c(V 2 ). STATEMENT OF THE MAIN RESULTLet V i → X , i = 1, 2 be two rank n vector bundles over a smooth projective variety X . Denote the total Chern classes of V i , i = 1, 2 bySuppose that c(V 1 ) = c(V 2 ). Let Y i := (V i ⊕ X ) be the projective completion (or projectification) of V i . Denote the projection maps by π i : Y i → X . By Leray-Hirsch we get