In this paper, we define genus‐zero relative Gromov–Witten invariants with negative contact orders. Using this, we construct relative quantum cohomology rings and Givental formalism. A version of Virasoro constraints also follows from it.
We extend the definition of relative Gromov–Witten invariants with negative contact orders to all genera. Then we show that relative Gromov–Witten theory forms a partial CohFT. Some cycle relations on the moduli space of stable maps are also proved.
Given a smooth projective variety X with a smooth nef divisor D and a positive integer r, we construct an I-function, an explicit slice of Givental's Lagrangian cone, for Gromov-Witten theory of the root stack X D,r . As an application, we also obtain an I-function for relative Gromov-Witten theory following the relation between relative and orbifold Gromov-Witten invariants.
We derive a recursive formula for certain relative Gromov–Witten invariants with a maximal tangency condition via the Witten–Dijkgraaf–Verlinde–Verlinde equation. For certain relative pairs, we get explicit formulae of invariants using the recursive formula.
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.
An effective algorithm of determining Gromov-Witten invariants of smooth hypersurfaces in any genus (subject to a degree bound (1)) from Gromov-Witten invariants of the ambient space is proposed.
We prove that the Gromov-Witten theory (GWT) of a projective bundle can be determined by the Chern classes and the GWT of the base. It completely answers a question raised in a previous paper. Its consequences include that the GWT of the blow-up of X at a smooth subvariety Z is uniquely determined by GWT of X, Z plus some topological data.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.