Abstract
We prove a higher genus version of the genus
$0$
local-relative correspondence of van Garrel-Graber-Ruddat: for
$(X,D)$
a pair with X a smooth projective variety and D a nef smooth divisor, maximal contact Gromov-Witten theory of
$(X,D)$
with
$\lambda _g$
-insertion is related to Gromov-Witten theory of the total space of
${\mathcal O}_X(-D)$
and local Gromov-Witten theory of D.
Specializing to
$(X,D)=(S,E)$
for S a del Pezzo surface or a rational elliptic surface and E a smooth anticanonical divisor, we show that maximal contact Gromov-Witten theory of
$(S,E)$
is determined by the Gromov-Witten theory of the Calabi-Yau 3-fold
${\mathcal O}_S(-E)$
and the stationary Gromov-Witten theory of the elliptic curve E.
Specializing further to
$S={\mathbb P}^2$
, we prove that higher genus generating series of maximal contact Gromov-Witten invariants of
$({\mathbb P}^2,E)$
are quasimodular and satisfy a holomorphic anomaly equation. The proof combines the quasimodularity results and the holomorphic anomaly equations previously known for local
${\mathbb P}^2$
and the elliptic curve.
Furthermore, using the connection between maximal contact Gromov-Witten invariants of
$({\mathbb P}^2,E)$
and Betti numbers of moduli spaces of semistable one-dimensional sheaves on
${\mathbb P}^2$
, we obtain a proof of the quasimodularity and holomorphic anomaly equation predicted in the physics literature for the refined topological string free energy of local
${\mathbb P}^2$
in the Nekrasov-Shatashvili limit.
Numerical Verification of the Birch and Swinnerton-Dyer Conjecture for Hyperelliptic Curves of Higher Genus over ℚ up to Squares