# Blow-up formulae of high genus Gromov-Witten invariants in dimension six

Weiqiang He1 Department of Mathematics
Sun Yat-Sen University
Guangzhou, 510275
China<EMAIL_ADDRESS>,  Jianxun Hu2 Department of Mathematics
Sun Yat-Sen University
Guangzhou, 510275
China<EMAIL_ADDRESS>,  Hua-Zhong Ke  Department of Mathematics
Sun Yat-Sen University
Guangzhou, 510275
China<EMAIL_ADDRESS>and  Xiaoxia Qi  Sino-French Institute of
Nuclear and Technology
Sun Yat-sen University
Tang Jia Wan, Zhuhai 519082
China<EMAIL_ADDRESS>
###### Abstract.

Using the degeneration formula and absolute/relative correspondence, one
studied the change of Gromov-Witten invariants under blow-up for six
dimensional symplectic manifolds and obtained closed blow-up formulae for high
genus Gromov-Witten invariants. Our formulae also imply some relations among
generalized BPS numbers introduced by Pandharipande.

Key words: Gromov-Witten invariant, Blow-up, Degeneration formula,
Absolute/relative correspondence, Degenerate contribution

1Partially supported by China Scholarship Council

2Partially supported by NSFC Grant 11228101 and 11371381

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Formulae for Blow-up at a point
  4. 4 Formulae for Blow-up along a smooth curve
  5. 5 Generalized BPS numbers

##  1\. Introduction

Gromov-Witten invariants count stable pseudo-holomorphic curves in a
symplectic manifold. The Gromov-Witten invariants for semi-positive symplectic
manifolds were first defined by Ruan [R1] and Ruan-Tian [RT1, RT2]. Gromov-
Witten invariants can be applied to define a quantum product on the cohomology
groups of a symplectic manifold in [RT1] and have many applications in
symplectic geometry and symplectic topology, see [MS] and references therein.
Using the virtual moduli cycle technique, Li-Tian [LT1] defined the Gromov-
Witten invariants purely algebraically for smooth projective varieties. During
last two decades, there were a great deal of activities to remove the semi-
positivity condition, see [B, FO, R2, S, LT2]. After its mathematical
foundation was established, the study of Gromov-Witten theory focused on its
computation and applications. We now know a lot about genus zero invariants
of, say, toric manifolds, homogeneous spaces, etc. Some of the higher genus
computations have also been done, but the understanding of higher genus
Gromov-Witten invariants is still far from complete.

The computation of the Gromov-Witten invariants is known to be a difficult
problem in geometry and physics. There are two major techniques: the
degeneration formula and localization. Li-Ruan [LR] first obtained the
degeneration formula, see [IP] for a different version and [Li] for an
algebraical version. It used to be applied to the situations that a symplectic
or Kahler manifold $X$ degenerates into a union of two pieces $X^{\pm}$ glued
along a common divisor $Z$. The idea of degeneration formula is to express the
Gromov-Witten invariants of $X$ in terms of relative Gromov-Witten invariants
of the pairs $(X^{\pm},Z)$. Localization played a very important role in the
computation of Gromov-Witten invariants. Kontsevich [Ko2] first introduced
this technique into this field, then Givental [Gi] and Lian-Liu-Yau [LLY]
applied this technique to prove the mirror theorem in the genus zero case. So
far the computation of high genus invariants is still a difficult task. The
difficulty is that the localization technique often transfers the computation
of high genus invariants into that of some Hodge integrals over
$\bar{\mathscr{M}}_{g,n}$, which so far one does not have effective methods to
compute. To obtain some general structures or close formulae of Gromov-Witten
theory in many applications, we degenerate a symplectic or Kahler manifold
into two toric relative pairs $(X^{\pm},Z)$ and then use the localization
technique to compute the associated relative invariants, see [HLR, MP]. The
combination of the degeneration technique and localization technique has
proven to be very powerful.

Ruan [R3] speculated that there should be a deep relation between quantum
cohomology and birational geometry. The birational symplectic geometry program
requires a thorough understanding of blow-up type formula of Gromov-Witten
invariants and quantum cohomology, because blow-up is the elementary
birational surgery. Actually, it is rare to be able to obtain a general blow-
up formula. For the last twenty years, only a few limited case were known, see
[H1, H2, G]. Hu-Li-Ruan [HLR] studied the change of Gromov-Witten invariants
under blow-up and obtained a blow-up correspondence of absolute/relative
Gromov-Witten invariants. The second named author [H1, H2] obtained some blow-
up formulae for genus zero Gromov-Witten invariants. In this paper, we try to
apply the degeneration formula to study the change of Gromov-Witten invariants
under blow-ups and generalize a genus zero formula in [H1] to all genera case
in dimension six.

Throughout this paper, let $X$ be a connected, closed, smooth symplectic
manifold of real dimension six, and $p:\tilde{X}\rightarrow X$ the natural
projection of the symplectic blow-up $\tilde{X}$ of $X$ along a connected
smooth symplectic submanifold of $X$. Let $E$ be the exceptional divisor of
the blow-up, and $e\in H_{2}(\tilde{X},{\mathbb{Z}})$ the class of a line in
the fiber of $E$. Note that $p$ induces a natural injection via ’pullback’ of
$2$-cycles

$p^{!}=PD_{\tilde{X}}\circ p^{*}\circ PD_{X}:H_{2}(X,{\mathbb{Z}})\rightarrow
H_{2}(\tilde{X},{\mathbb{Z}}),$

where the image of $p^{!}$ is the subset of $H_{2}(\tilde{X},{\mathbb{Z}})$
consisting of $2$-cycles having intersection number zero with $E$.

We first consider blow-up at a point. Given a nonzero class $A\in
H_{2}(X,{\mathbb{Z}})$, from the viewpoint of geometry, we could express the
condition of counting curves with homology class $A$ passing through a generic
point in $X$ in two ways: adding a point class, or blowing up $X$ at the point
and counting curves in $\tilde{X}$ with homology class $p^{!}A-e$. One would
expect that the two methods give the same Gromov-Witten invariants, which was
proved by the second named author [H1] in all dimensions for $g=0$, and by the
fourth named author [Q] in real dimension four for all genera. In this paper,
we study the dimension six case for all genera:

######  Theorem 1.1.

Let $p:\tilde{X}\rightarrow X$ be the blow-up at a point. Suppose that
$\alpha_{1},\cdots,\alpha_{m}\in H^{>0}(X,{\mathbb{Q}})$, $1\leq i\leq m$, and
$d_{1},\cdots,d_{m}\in{\mathbb{Z}}_{\geqslant 0}$. Then for nonzero $A\in
H_{2}(X,{\mathbb{Z}})$ and $g\geq 0$, we have

$\langle[pt],\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}=\sum_{g_{1}+g_{2}=g}\frac{(-1)^{g_{1}}\cdot
2}{(2g_{1}+2)!}\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g_{2},p^{!}A-e}.$

######  Theorem 1.2.

Under the same assumptions as in Theorem 1.1, we have

$\displaystyle\langle\tau_{1}[pt],\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}$
$\displaystyle=$
$\displaystyle\sum_{g_{1}+g_{2}=g}\frac{(-1)^{g_{1}}}{(2g_{1}+1)!}\langle-E^{2},\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g_{2},p^{!}A-e}.$

Through studying the proof of Theorem 1.2 carefully, we obtain the following
result, which seems to be nontrivial when compared with divisor equation and
dilaton equation.

######  Theorem 1.3.

Under the same assumptions as in Theorem 1.1, we have

$\displaystyle\langle\tau_{1}E,\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}$
$\displaystyle=$ $\displaystyle
3\langle-E^{2},\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}$
$\displaystyle\qquad-2\sum_{g_{1}+g_{2}=g}\frac{(-1)^{g_{1}}}{(2g_{1}+1)!}\langle-E^{2},\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g_{2},p^{!}A-e}.$

We also consider the blow-up along a curve.

######  Theorem 1.4.

Let $p:\tilde{X}\rightarrow X$ be the blow-up along a smooth curve $C$ with
$\int_{C}c_{1}(X)>0$. Suppose that $\alpha_{1},\cdots,\alpha_{m}\in
H^{>2}(X,{\mathbb{Q}})$, $1\leq i\leq m$, support away from the curve $C$, and
$d_{1},\cdots,d_{m}\in{\mathbb{Z}}_{\geqslant 0}$. Then for nonzero $A\in
H_{2}(X,{\mathbb{Z}})$ and $g\geq 0$, we have

$\langle[C],\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}=\sum_{g_{1}+g_{2}=g}\frac{(-1)^{g_{1}}}{(2g_{1}+1)!\cdot
2^{2g_{1}}}\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g_{2},p^{!}A-e}.$

The above blow-up formulae relate Gromov-Witten invariants of $X$ and those of
$\tilde{X}$ in a nontrivial way. Theorem 1.1 and 1.4 imply the following
simple relations among generalized BPS numbers
$n^{X}_{g,A}(\alpha_{1},\dots,\alpha_{m})$ introduced by Pandharipande [P1,
P2].

######  Proposition 1.5.

Suppose that $\alpha_{1},\cdots,\alpha_{m}\in H^{>2}(X,{\mathbb{Q}})$, $A\in
H_{2}(X,{\mathbb{Z}})$ is nonzero and $g\in{\mathbb{Z}}_{\geqslant 0}$.

  * (a)

If $p:\tilde{X}\rightarrow X$ is the blow-up at a point, then we have

$n_{g,A}^{X}([pt],\alpha_{1},\cdots,\alpha_{m})=n_{g,p^{!}A-e}^{\tilde{X}}(p^{*}\alpha_{1},\cdots,p^{*}\alpha_{m}).$

  * (b)

If $p:\tilde{X}\rightarrow X$ is the blow-up along a smooth curve $C$ with
$\int_{C}c_{1}(X)>0$, then we have

$n_{g,A}^{X}([C],\alpha_{1},\cdots,\alpha_{m})=n_{g,p^{!}A-e}^{\tilde{X}}(p^{*}\alpha_{1},\cdots,p^{*}\alpha_{m}).$

Our proof of the above blow-up formulae is inspired by the absolute/relative
correspondence obtained by Hu-Li-Ruan [HLR], which is a generalization of the
idea of Maulik-Pandharipande [MP]. This correspondence partially describes the
change of Gromov-Witten invarians under blow-ups. We first use degeneration
formula to obtain comparison results between absolute and relative Gromov-
Witten invariants, and then use these comparison results to prove our blow-up
formulae.

The rest of the paper is organized as follows. In Section 2, we briefly review
basic materials of absolute/relative Gromov-Witten invariants and the
degeneration formula. In Section 3, we consider the case of blow-up at a point
and prove Theorem 1.1, 1.2 and 1.3. In Section 4, we consider the case of
blow-up along a smooth curve and prove Theorem 1.4. In Section 5, we review
the definition of generalized BPS numbers and prove Corollary 1.5.

##  2\. Preliminaries

In this section, we briefly review absolute/relative Gromov-Witten invariants
and the degeneration formula and fix notations throughout. We use [LR] as our
general reference.

Recall that we always let $X$ be a connected compact smooth symplectic
manifold of real dimension six. For $A\in H_{2}(X,{\mathbb{Z}})$, let
$\overline{\mathscr{M}}_{g,m}(X,A)$ be the moduli space of connected
$m$-pointed stable maps to $X$ of arithmetic genus $g$ and degree $A$. Let
$e_{i}:\overline{\mathscr{M}}_{g,m}(X,A)\longrightarrow X$ be the evaluation
map at the $i^{th}$ marked point. The Gromov-Witten invariants of $X$ are
defined as

$\langle\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}:=\int_{[\overline{\mathscr{M}}_{g,m}(X,A)]^{vir}}\prod\limits_{i=1}^{m}\psi_{i}^{d_{i}}e_{i}^{*}\alpha_{i},$

where $\alpha_{1},\cdots,\alpha_{m}\in H^{*}(X,{\mathbb{Q}})$,
$d_{1},\cdots,d_{m}\in{\mathbb{Z}}_{\geqslant 0}$, $\psi_{i}$ is the first
Chern class of the cotangent line bundle, and
$[\overline{\mathscr{M}}_{g,m}(X,A)]^{vir}$ is the virtual fundamental cycle.

The degeneration formula [LR, IP, Li] provides a rigorous formulation about
the change of Gromov-Witten invariants under semi-stable degeneration, or
symplectic cutting. The formula relates the absolute Gromov-Witten invariant
of $X$ to the relative Gromov-Witten invariants of two smooth pairs.

Now we recall the relative invariants of a smooth relative pair $(X,Z)$ with
$Z\hookrightarrow X$ a connected smooth symplectic divisor. Let $A\in
H_{2}(X,{\mathbb{Z}})$ with $A\cdot Z\geqslant 0$, and $\mu$ a partition of
$A\cdot Z$. We customarily use relative graphs to describe the topological
type of relative stable maps. A connected relative graph $\Gamma=(g,m,A,\mu)$
is defined to be a connected decorated graph consisting of the following data:

  1. (1)

a vertex decorated by $A$ and genus $g$;

  2. (2)

$m$ tails with no decoration;

  3. (3)

$\ell(\mu)$ tails decorated by entries of $\mu$.

A connected relative stable map has topological type $\Gamma$ if it has
arithmetic genus $g$, degree $A$, $m$ absolute marked points and $\ell(\mu)$
relative marked points with contact order given by $\mu$. Let
$\overline{\mathscr{M}}_{\Gamma}(X,Z)$ be the moduli space of connected
relative stable maps with topological type $\Gamma$. Let
$e_{i}:\overline{\mathscr{M}}_{\Gamma}(X,Z)\longrightarrow X$ be the
evaluation map at the $i^{th}$ absolute marked point, and
$e_{j}^{Z}:\overline{\mathscr{M}}_{\Gamma}(X,Z)\longrightarrow Z$ the
evaluation map at the $j^{th}$ relative marked point. The relative Gromov-
Witten invariants of $(X,Z)$ are of the form

$\langle\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\mid\delta_{1},\cdots,\delta_{\ell(\mu)}\rangle_{\Gamma}^{X,Z}:=\int_{[\overline{\mathscr{M}}_{\Gamma}(X,Z)]^{vir}}\prod_{i=1}^{m}\psi_{i}^{d_{i}}e_{i}^{*}\alpha_{i}\cdot\prod\limits_{j=1}^{\ell(\mu)}(e^{Z}_{j})^{*}\delta_{i},$

where $\alpha_{1},\cdots,\alpha_{m}\in
H^{*}(X,{\mathbb{Q}}),d_{1},\cdots,d_{m}\in{\mathbb{Z}}_{\geqslant
0},\delta_{1},\cdots,\delta_{\ell(\mu)}\in H^{*}(Z,{\mathbb{Q}})$, and
$[\overline{\mathscr{M}}_{\Gamma}(X,Z)]^{vir}$ is the virtual fundamental
cycle of dimension:

$\dim[\overline{\mathscr{M}}_{\Gamma}(X,Z)]^{vir}=2\int_{A}c_{1}(X)+2m+2\ell(\mu)-2|\mu|.$

The relative invariants with disconnceted domains are defined by the usual
product rule, and the invariants will be denoted by
$\langle\cdots|\cdots\rangle_{\Gamma}^{\bullet X,Z}$.

Next, we shall introduce the degeneration formula. Let
$\pi:\chi\longrightarrow D$ be a connected, smooth symplectic manifold of real
dimension eight over a disk $D$ such that $\chi_{t}=\pi^{-1}(t)\cong X$ for
$t\not=0$ and $\chi_{0}$ is a union of two connected compact smooth symplectic
manifolds $X_{1}$ and $X_{2}$ intersecting transversally along a symplectic
divisor $Z$. We write $\chi_{0}=X_{1}\cup_{Z}X_{2}$.

Consider the natural inclusion maps

$i_{t}:X=\chi_{t}\longrightarrow\chi,\,\,\,\,\,\,\,\,i_{0}:\chi_{0}\longrightarrow\chi,$

and the gluing map

$g=(j_{1},j_{2}):X_{1}\coprod X_{2}\longrightarrow\chi_{0}.$

We have

$H_{2}(X,{\mathbb{Z}})\stackrel{{\scriptstyle
i_{t*}}}{{\longrightarrow}}H_{2}(\chi,{\mathbb{Z}})\stackrel{{\scriptstyle
i_{0_{*}}}}{{\longleftarrow}}H_{2}(\chi_{0},{\mathbb{Z}})\stackrel{{\scriptstyle
g_{*}}}{{\longleftarrow}}H_{2}(X_{1},{\mathbb{Z}})\oplus
H_{2}(X_{2},{\mathbb{Z}}),$

where $i_{0*}$ is an isomorphism since there exists a deformation retract from
$\chi$ to $\chi_{0}$(see [C]). Also, since the family $\chi\longrightarrow D$
comes from a trivial family, it follows that each $\alpha\in
H^{*}(X,{\mathbb{Q}})$ has global liftings such that the restriction
$\alpha(t)$ on $\chi_{t}$ is defined for all $t$.

Fix a basis $\\{\delta_{i}\\}$ of $H^{*}(Z,{\mathbb{Q}})$ and denote by
$\\{\delta^{i}\\}$ its dual basis. The degeneration formula expresses the
absolute invariants of $X$ in terms of the relative invariants of the two
smooth pairs $(X_{1},Z)$ and $(X_{2},Z)$:

$\displaystyle\langle\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}$
$\displaystyle=$
$\displaystyle\sum_{\mu}{\mathfrak{z}}(\mu)\sum\limits_{i_{1},\cdots,i_{\ell(\mu)}}\sum_{\eta\in\Omega_{\mu}}\langle\tau_{d_{i^{-}_{1}}}j_{1}^{*}\alpha_{i^{-}_{1}}(0),\cdots,\tau_{d_{i^{-}_{k_{1}}}}j_{1}^{*}\alpha_{i^{-}_{k_{1}}}(0)\mid\delta_{i_{1}},\cdots,\delta_{i_{\ell(\mu)}}\rangle^{\bullet
X_{1},Z}_{\Gamma_{1}}$
$\displaystyle\,\,\cdot\,\,\langle\tau_{d_{i^{+}_{1}}}j_{2}^{*}\alpha_{i^{+}_{1}}(0),\cdots,\tau_{d_{i^{+}_{k_{2}}}}\alpha_{i^{+}_{k_{2}}}(0)\mid{\delta}^{i_{1}},\cdots,\delta^{i_{\ell(\mu)}}\rangle^{\bullet
X_{2},Z}_{\Gamma_{2}},$

where
${\mathfrak{z}}(\mu)=|\mbox{Aut}\mu|\prod\limits_{i=1}^{\ell(\mu)}\mu_{i}$,
and $\eta=(\Gamma_{1},\Gamma_{2},I)$ is an admissible triple, which consists
of (possibly disconnected) topological types $\Gamma_{1},\Gamma_{2}$ with the
same partition $\mu$ under the identification $I$ of relative marked points,
satisfying the following requirements:

  * (1)

the gluing of $\Gamma_{1}$ and $\Gamma_{2}$ under $I$ is connected;

  * (2)

let $g_{i}$ be the total genus of $\Gamma_{i}$, and we have
$g=g_{1}+g_{2}+\ell(\mu)+1-|\Gamma_{1}|-|\Gamma_{2}|$, where $|\Gamma_{i}|$ is
the number of connected components of $\Gamma_{i}$;

  * (3)

let $A_{i}\in H_{2}(X_{i},{\mathbb{Z}})$ be the total degree of $\Gamma_{i}$,
and we have $i_{t*}A=i_{0*}(j_{1*}A_{1}+j_{2*}A_{2})$ and $|\mu|=A_{1}\cdot
Z=A_{2}\cdot Z$;

  * (4)

the absolute marked points of $\Gamma_{1},\Gamma_{2}$ are indexed by
$\\{i_{1}^{-},\cdots,i_{k_{1}}^{-}\\}$ and $\\{i_{1}^{+},\cdots$,
$i_{k_{2}}^{+}\\}$ respectively, the disjoint union of which is exactly
$\\{1,2,\cdots,m\\}$.

We denote by $\Omega_{\mu}$ the equivalence class of all admissible triples
with fixed partition $\mu$. For $\eta\in\Omega_{\mu}$ having nonzero
contribution in the degeneration formula, we have the following important
dimension constraint (Theorem 5.1 in [LR]):

(2)
$\dim\overline{\mathscr{M}}_{\Gamma_{1}}(X_{1},Z)+\dim\overline{\mathscr{M}}_{\Gamma_{2}}(X_{2},Z)=\dim\overline{\mathscr{M}}_{g,m}(X,A)+4\ell(\mu).$

######  Remark 2.1.

Symplectic cutting is a kind of surgery in symplectic geometry which is
suitable for the above degeneration formula (see [LR]). Suppose that
$X_{0}\subset X$ is an open codimension zero submanifold with Hamiltonian
$S^{1}$-action. Let $H:X_{0}\longrightarrow{\mathbb{R}}$ be a Hamiltonian
function with $0$ as a regular value. If $H^{-1}(0)$ is a separating
hypersurface of $X_{0}$, then we obtain two connected manifolds $X^{\pm}_{0}$
with boundary $\partial X^{\pm}_{0}=H^{-1}(0)$, where the $+$ side corresponds
to $H<0$. Suppose further that $S^{1}$ acts freely on $H^{-1}(0)$. Then the
symplectic reduction $Z=H^{-1}(0)/S^{1}$ is canonically a symplectic manifold.
Collapsing the $S^{1}$-action on $\partial X^{\pm}=H^{-1}(0)$, we obtain two
closed smooth manifolds $\bar{X}^{\pm}$ containing respectively real
codimension 2 submanifolds $Z^{\pm}=Z$ with opposite normal bundles.
Furthermore $\bar{X}^{\pm}$ admits a symplectic structure $\bar{\omega}^{\pm}$
which agrees with the restriction of $\omega$ away from $Z$, and whose
restriction to $Z^{\pm}$ agrees with the canonical symplectic structure
$\omega_{Z}$ on $Z$ from symplectic reduction. The pair of symplectic
manifolds $(\bar{X}^{\pm},\bar{\omega}^{\pm})$ is called the symplectic cut of
$X$ along $H^{-1}(0)$.

Suppose that $Y\subset X$ is a submanifold of $X$ of codimension $2k$. Denote
by $N_{Y}$ the normal bundle. By the symplectic neighborhood theorem ,and by
possibly taking a smaller $\epsilon_{0}$, a tubular neighborhood
${\mathscr{N}}_{\epsilon_{0}}(Y)$ of $Y$ in $X$ is symplectomorphic to the
disc bundle $N_{Y}(\epsilon_{0})$ of $N_{Y}$. Denote by
$\phi:{\mathscr{N}}_{\epsilon_{0}}(Y)\longrightarrow N_{Y}(\epsilon_{0})$ be
such a symplectomorphism. Consider the Hamiltonian $S^{1}$-action on
$X_{0}={\mathscr{N}}_{\epsilon_{0}}(Y)$ by complex multiplication. Fix
$\epsilon$ with $0<\epsilon<\epsilon_{0}$ and consider the moment map

$H(u)=|\phi(u)|^{2}-\epsilon,\,\,\,\,\,u\in{\mathscr{N}}_{Y}(\epsilon_{0}),$

where $|\phi(u)|$ is the norm of $\phi(u)$ considered as a vector in a fiber
of the Hermitian bundle $N_{Y}$. We cut $X$ along $H^{-1}(0)$ to obtain two
closed symplectic manifolds $\bar{X}^{\pm}$. Notice that
$\bar{X}^{+}\cong{\mathbb{P}}_{Y}(N_{Y}\oplus{\mathscr{O}}_{Y})$.
$\bar{X}^{-}$ is called the blow-up of $X$ along $Y$, denoted by $\tilde{X}$.

##  3\. Formulae for Blow-up at a point

In this section, we prove Theorem 1.1, 1.2 and 1.3. We always assume that
total degrees of insertions match the virtual dimension of the moduli spaces,
since otherwise the required equalities are trivial.

First of all, we will divide the proof of Theorem 1.1 into some comparison
theorems of Gromov-Witten invariants as follows.

######  Lemma 3.1.

Under the same assumptions as in Theorem 1.1, we have

(3)
$\displaystyle{\langle[pt],\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle}^{X}_{g,A}$
$\displaystyle=$
$\displaystyle\sum_{g^{+}+g^{-}=g}{\langle[pt]|[pt]\rangle}^{\mathbb{P}^{3},H}_{g^{+},L,(1)}{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\mathbbm{1}\rangle}^{\tilde{X},E}_{g^{-},p^{!}A-e,(1)},$

where $H$ is the hyperplane at infinity, and $L\in
H_{2}(\mathbb{P}^{3};{\mathbb{Z}})$ is the class of a line.

###### Proof.

We first perform the symplectic cutting along a point as in Remark 2.1. Here
we have assumed that the class $[pt]$ has support in $X^{+}$ and $\alpha_{i}$
has support in $X^{-}$. By the degeneration formula (2),we have

$\displaystyle\langle[pt],\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}$
$\displaystyle=$
$\displaystyle\sum{\mathfrak{z}}(\mu)\langle[pt]|\delta_{j_{1}},\cdots,\delta_{j_{\ell(\mu)}}\rangle^{\mathbb{P}^{3},H}_{g^{+},A^{+},\mu}$
$\displaystyle\cdot\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\delta^{j_{1}},\cdots,\delta^{j_{\ell(\mu)}}\rangle^{\tilde{X},E}_{g^{-},A^{-},\mu}.$

By our assumption that total degrees of insertions match the virtual dimension
of the moduli space, we have

$\dim\bar{\mathscr{M}}_{g,m+1}(X,A)=\sum_{i=1}^{m}\deg\alpha_{i}+2\sum_{i=1}^{m}d_{i}+6.$

Suppose that $(\Gamma^{+},\Gamma^{-})$ has nonzero contribution in the
degeneration formula. Then

$\displaystyle\dim\bar{\mathscr{M}}_{\Gamma^{+}}(\mathbb{P}^{3},H)$
$\displaystyle=$ $\displaystyle
2\int_{A^{+}}c_{1}({\mathbb{P}}^{3})+2+2\ell(\mu)-2|\mu|,$
$\displaystyle\dim\bar{\mathscr{M}}_{\Gamma^{-}}(\tilde{X},E)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{m}\deg\alpha_{i}+2\sum_{i=1}^{m}d_{i}+\sum\limits_{i=1}^{\ell(\mu)}\deg\delta^{j_{i}}.$

So by the dimension constraint (2),

$\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}+\int_{A^{+}}c_{1}({\mathbb{P}}^{3})-|\mu|=2+\ell(\mu).$

Note that $A^{+}\cdot H=|\mu|$, and hence $A^{+}=|\mu|L$, which implies that

$\displaystyle\int_{A^{+}}c_{1}({\mathbb{P}}^{3})=4|\mu|.$

Now the dimension constraint becomes

$\displaystyle\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}+3|\mu|=2+\ell(\mu).$

So the dimension constraint holds only if

$\displaystyle\mu=(1),\quad deg\delta^{j_{1}}=0,$

which implies the required equality. ∎

######  Lemma 3.2.

Under the same assumptions as in Theorem 1.1, we have

$\displaystyle{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle}^{\tilde{X}}_{g,p^{!}A-e}$
$\displaystyle=$
$\displaystyle\sum_{g^{+}+g^{-}=g}{\langle\,\,|[pt]\rangle}_{g^{+},F,(1)}^{\tilde{\mathbb{P}}^{3},H}$
$\displaystyle\cdot{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\mathbbm{1}\rangle}^{\tilde{X},E}_{g^{-},p^{!}A-e,(1)},$

where $F\in H_{2}(\tilde{\mathbb{P}}^{3},{\mathbb{Z}})$ is the class of a
fiber in
$\tilde{\mathbb{P}}^{3}\cong{\mathbb{P}}_{{\mathbb{P}}^{2}}(\mathscr{O}\oplus\mathscr{O}(-1))$.

###### Proof.

We perform symplectic cutting along $E$ in $\tilde{X}$ as in Remark 2.1. Here
we also assumed that the class $p^{*}\alpha_{i}$ has support away from $E$. By
the degeneration formula (2), we have

$\displaystyle{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle}^{\tilde{X}}_{g,p^{!}A-e}$
$\displaystyle=$
$\displaystyle\sum{\mathfrak{z}}(\mu){\langle\,\,|\delta_{j_{1}},\cdots,\delta_{j_{\ell(\mu)}}\rangle}_{g^{+},(p!(A)-e)^{+},\nu}^{\tilde{\mathbb{P}}^{3},H}$
$\displaystyle\cdot{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\delta^{j_{1}},\cdots,\delta^{j_{\ell(\mu)}}\rangle}^{\tilde{X},E}_{g^{-},(p^{!}A-e)^{-},\mu}.$

By our assumption that degrees match the virtual dimension, we have

$\displaystyle\dim_{\mathbb{C}}\bar{\mathscr{M}}_{g,m}(\tilde{X},p^{!}A-e)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{i=1}^{m}deg\alpha_{i}+\sum_{i=1}^{m}d_{i}.$

Suppose that a term with $(\Gamma^{+},\Gamma^{-})$ has nonzero contribution in
RHS of the degeneration formula (3). Then

$\dim_{\mathbb{C}}\bar{\mathscr{M}}_{\Gamma^{+}}(\tilde{\mathbb{P}}^{3},H)=\int_{(p^{!}A-e)^{+}}c_{1}(\tilde{{\mathbb{P}}}^{3})+\ell(\mu)-|\mu|,$
$\dim_{\mathbb{C}}\bar{\mathscr{M}}_{\Gamma^{-}}(\tilde{X},E)=\frac{1}{2}\sum_{i=1}^{m}deg\alpha_{i}+\sum_{i=1}^{m}d_{i}+\frac{1}{2}\sum_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}.$

So by the dimension constraint (2),

$\displaystyle\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}+\int_{(p^{!}A-e)^{+}}c_{1}(\tilde{{\mathbb{P}}}^{3})-|\mu|=\ell(\mu).$

Let $L\in H_{2}(\tilde{\mathbb{P}}^{3},{\mathbb{Z}})$ be the class of the
total transform of a line in ${\mathbb{P}}^{3}$. Then we have the following
natural decomposition

$\displaystyle
H_{2}(\tilde{\mathbb{P}}^{3},{\mathbb{Z}})={\mathbb{Z}}F\oplus{\mathbb{Z}}L.$

We have the following constraints for $(p^{!}A-e)^{+}$:

${(p!(A)-e)}^{+}\cdot H=|\mu|,\,\,\,\,\,(p^{!}(A)-e)^{+}\cdot E=1.$

So we have $(p^{!}A-e)^{+}=F+(|\mu|-1)L$, and hence
$\int_{(p^{!}A-e)^{+}}c_{1}(\tilde{\mathbb{P}}^{3})=4|\mu|-2$. Now the
dimension constraint becomes

$\displaystyle\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}+3|\mu|=2+\ell(\mu).$

So the dimension constraint holds only if

$\displaystyle\mu=(1),\quad deg\delta^{j_{1}}=0,$

which implies the required equality. ∎

Using the above comparison results we may obtain the following
absolute/relative correspondence for Gromov-Witten invariants under blow-up.

######  Lemma 3.3.

Under the same assumptions as in Theorem 1.1, denote
$\langle[pt],\tau_{d_{1}}\alpha_{1},$
$\cdots,\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}$ and
$\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}$
by $H_{g}$ and $P_{g}$ respectively. Then

(6) $H_{g}=\sum_{g_{1}+g_{2}=g}C_{g_{1}}P_{g_{2}},$

where $C_{g}$’s can be determined by relative invariants
${\langle[pt]|[pt]\rangle}^{\mathbb{P}^{3},H}_{g,L,(1)}$ and
${\langle\,\,|[pt]\rangle}_{g,F,(1)}^{\tilde{\mathbb{P}}^{3},H}$.

###### Proof.

Denote
$\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\mathbbm{1}\rangle^{\tilde{X},E}_{g,p!(A)-e,(1)}$,
${\langle[pt]|[pt]\rangle}^{\mathbb{P}^{3},H}_{g,L,(1)}$ and
${\langle\,\,|[pt]\rangle}_{g,F,(1)}^{\tilde{\mathbb{P}}^{3},H}$ by $K_{g}$,
$I_{g}$ and $J_{g}$ respectively. Then for $g\geq 0$, we may rewrite our
comparison results (3) and (3.2) as

$\displaystyle H_{g}$ $\displaystyle=$ $\displaystyle
I_{g}K_{0}+I_{g-1}K_{1}+\cdots+I_{0}K_{g}$ $\displaystyle P_{g}$
$\displaystyle=$ $\displaystyle J_{g}K_{0}+J_{g-1}K_{1}+\cdots+J_{0}K_{g},$

or in matrix form

$\left(\begin{array}[]{l}H_{0}\\\ H_{1}\\\ \vdots\\\
H_{g}\end{array}\right)=\left(\begin{array}[]{llll}I_{0}&&\lx@intercol\hfil
0\hfil\lx@intercol\\\ I_{1}&I_{0}&&\\\ \vdots&&\ddots&\\\
I_{g}&I_{g-1}&\cdots&I_{0}\end{array}\right)\left(\begin{array}[]{l}K_{0}\\\
K_{1}\\\ \vdots\\\ K_{g}\end{array}\right),$

$\left(\begin{array}[]{l}P_{0}\\\ P_{1}\\\ \vdots\\\
P_{g}\end{array}\right)=\left(\begin{array}[]{llll}J_{0}&&\lx@intercol\hfil
0\hfil\lx@intercol\\\ J_{1}&J_{0}&&\\\ \vdots&&\ddots&\\\
J_{g}&J_{g-1}&\cdots&J_{0}\end{array}\right)\left(\begin{array}[]{l}K_{0}\\\
K_{1}\\\ \vdots\\\ K_{g}\end{array}\right).$

This is a special form of absolute/relative correspondence for Gromov-Witten
invariants (Theorem 5.15 in [HLR]). In particular, $I_{0}\neq 0$ and
$J_{0}\neq 0$, which implies that both matrices with entries $I_{g}$ and
$J_{g}$ are invertible (one can also use virtual localization [GP] to check
that $I_{0}=J_{0}=1$). Write

$\left(\begin{array}[]{llll}C_{0}&&\lx@intercol\hfil 0\hfil\lx@intercol\\\
C_{1}&C_{0}&&\\\ \vdots&&\ddots&\\\
C_{g}&C_{g-1}&\cdots&C_{0}\end{array}\right)=\left(\begin{array}[]{llll}I_{0}&&\lx@intercol\hfil
0\hfil\lx@intercol\\\ I_{1}&I_{0}&&\\\ \vdots&&\ddots&\\\
I_{g}&I_{g-1}&\cdots&I_{0}\end{array}\right)\left(\begin{array}[]{llll}J_{0}&&\lx@intercol\hfil
0\hfil\lx@intercol\\\ J_{1}&J_{0}&&\\\ \vdots&&\ddots&\\\
J_{g}&J_{g-1}&\cdots&J_{0}\end{array}\right)^{-1},$

and we obtain the required equality. ∎

To get Theorem 1.1, we need to compute $C_{g}$’s in (6). A crucial observation
from the proof of Lemma 3.3 is that $C_{g}$’s are determined by relative
invariants ${\langle[pt]|[pt]\rangle}^{\mathbb{P}^{3},H}_{g,L,(1)}$ and
${\langle\,\,|[pt]\rangle}_{g,F,(1)}^{\tilde{\mathbb{P}}^{3},H}$, and are
independent of the choice of $X,m,\alpha_{i},A$. Therefore, to compute these
universal coefficients, we may choose
$X=\mathbb{P}^{3},m=1,\alpha_{1}=[pt],A=L$. Then (6) becomes

(7)
$\langle[pt],[pt]\rangle_{g,L}^{\mathbb{P}^{3}}=\sum_{g_{1}+g_{2}=g}C_{g_{1}}\cdot\langle[pt]\rangle_{g_{2},F}^{\tilde{\mathbb{P}}^{3}},$

where $F$ is the class of a fiber in
$\tilde{\mathbb{P}}^{3}\cong{\mathbb{P}}_{{\mathbb{P}}^{2}}({\mathscr{O}}\oplus{\mathscr{O}}(-1))$.

To get $C_{g}$’s by solving the equation (7), we need to compute the absolute
Gromov-Witten invariants $\langle[pt],[pt]\rangle_{g,L}^{\mathbb{P}^{3}}$ and
$\langle[pt]\rangle_{g_{2},F}^{\tilde{\mathbb{P}}^{3}}$. From this, we have

######  Lemma 3.4.

$\displaystyle\langle[pt],[pt]\rangle_{g,L}^{\mathbb{P}^{3}}$ $\displaystyle=$
$\displaystyle\frac{(-1)^{g}\cdot 2}{(2g+2)!},$
$\displaystyle\langle[pt]\rangle_{g,F}^{\tilde{\mathbb{P}}^{3}}$
$\displaystyle=$ $\displaystyle\delta_{g,0}.$

These equalities can be proved either directly by virtual localization [GP] or
by degenerate contribution computation [P2]. In fact, Theorem 3 in [P2] may
specialize to the case of ${\mathbb{P}}^{3}$ and obtain these invariants. Here
we omit the proof.

Proof of Theorem 1.1: We first perform symplectic cutting at a point in $X$
and get equation (3). Then we perform symplectic cutting along the exceptional
divisor $E$ in $\tilde{X}$ and get (3.2). Finally, we can solve the equation
(7) to get the universal coefficients

$C_{g}=\frac{(-1)^{g}\cdot 2}{(2g+2)!}.$

This proves Theorem 1.1.

######  Remark 3.5.

One can relax the requirement in Theorem 1.1 to $m\geqslant 0$, which can be
checked by going through the proof of Lemma 3.1, 3.2 and 3.3. This also holds
for Theorem 1.2 and 1.3.

It is illuminating to rephrase this using a genus $g$ gravitational Gromov-
Witten generating function. Suppose that $T_{0}=1,T_{1},\cdots,T_{m}$ is a
basis for $H^{*}(X,{\mathbb{Q}})$. We introduce supercommuting variables
$t_{d}^{j}$ for $d\geq 0$ and $0\leq j\leq m$ with $\deg t^{j}_{d}=\deg
T_{j}$. Set

$\gamma=\sum_{d=0}^{\infty}\sum_{j=1}^{m}t^{j}_{d}\tau_{d}T_{j}.$

Define the genus $g$ gravitational Gromov-Witten generating function as

$F_{g}^{X}(t_{d}^{j})=\sum_{n=0}^{\infty}\sum_{A\in
H_{2}(X,{\mathbb{Z}})}\frac{1}{n!}\langle\gamma^{n},[pt]\rangle_{g,A}^{X}q^{A},$
$F_{g}^{\tilde{X}}(t_{d}^{j})=\sum_{n=0}^{\infty}\sum_{A\in
H_{2}(X,{\mathbb{Z}})}\frac{1}{n!}\langle(p^{*}\gamma^{n}\rangle_{g,p^{!}(A)-e}^{\tilde{X}}q^{p^{!}(A)-e}.$

Set

$F^{X}(u,t_{d}^{j})=\sum_{g\geq 0}u^{2g-2}F_{g}^{X}(t_{d}^{j})$

and

$F^{\tilde{X}}(u,t_{d}^{j})=\sum_{g\geq
0}u^{2g-2}F_{g}^{\tilde{X}}(t_{d}^{j}).$

Then from Theorem 1.1, we have

######  Corollary 3.6.

$F^{X}(u,\gamma)=\bigg{(}\frac{\sin\frac{u}{2}}{\frac{u}{2}}\bigg{)}^{2}\cdot
F^{\tilde{X}}(u,p^{*}\gamma),$

where we need to change the variable $q^{A}$ to $q^{p^{!}(A)-e}$.

Similar to the proof of Theorem 1.1 above, we may divide the proof of Theorem
1.2 into the following Lemma 3.7, 3.8 and 3.9, the proof of which is analogous
to that of Lemma 3.1, 3.2 and 3.3 respectively.

######  Lemma 3.7.

Under the same assumptions as in Theorem 1.1, we have

$\displaystyle{\langle\tau_{1}[pt],\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle}^{X}_{g,A}$
$\displaystyle=$
$\displaystyle\sum_{g^{+}+g^{-}=g}{\langle\tau_{1}[pt]|\xi\rangle}^{\mathbb{P}^{3},H}_{g^{+},L,(1)}{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\xi\rangle}^{\tilde{X},E}_{g^{-},p^{!}A-e,(1)},$

where $H$ is the hyperplane at infinity, $L$ is the class of a line in
${\mathbb{P}}^{3}$, and $\xi$ is the cohomology class of a line in $H\cong
E\cong{\mathbb{P}}^{2}$.

###### Proof.

The same argument as in the proof of Lemma 3.1 leads to the dimension
constraint

$\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}\deg\delta^{j_{i}}+3|\mu|=3+\ell(\mu).$

This constraint holds only if

$\mu=(1),\,\,\,\,\deg\delta_{j_{1}}=2.$

This implies Lemma 3.7. ∎

######  Lemma 3.8.

Under the same assumptions as in Theorem 1.1, we have

$\displaystyle{\langle-E^{2},\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle}^{\tilde{X}}_{g,p^{!}A-e}$
$\displaystyle=$
$\displaystyle\sum_{g^{+}+g^{-}=g}{\langle-E^{2}|\xi\rangle}_{g^{+},F,(1)}^{\tilde{\mathbb{P}}^{3},H}\cdot{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\xi\rangle}^{\tilde{X},E}_{g^{-},p^{!}A-e,(1)},$

where $F\in H_{2}(\tilde{\mathbb{P}}^{3},{\mathbb{Z}})$ is the class of a
fiber in
$\tilde{\mathbb{P}}^{3}\cong{\mathbb{P}}_{{\mathbb{P}}^{2}}(\mathscr{O}\oplus\mathscr{O}(-1))$,
and $\xi$ is the cohomology class of a line in $H\cong
E\cong{\mathbb{P}}^{2}$.

###### Proof.

The same dimension calculation as in the proof of Lemma 3.1 gives rise to the
dimension constraint

$\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}\deg\delta^{j_{i}}+3|\mu|=3+\ell(\mu).$

This also implies

$\mu=(1),\,\,\,\,\deg\delta_{j_{1}}=2,$

which proves Lemma 3.8. ∎

######  Lemma 3.9.

Under the same assumptions as in Theorem 1.1, denote
$\langle\tau_{1}[pt],\tau_{d_{1}}\alpha_{1}$,
$\cdots,\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}$ and
$\langle-E^{2},\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}$
by $H_{g}$ and $P_{g}$ respectively. Then

(8) $H_{g}=\sum_{g_{1}+g_{2}=g}C_{g_{1}}P_{g_{2}},$

where $C_{g}$’s can be determined by relative invariants
${\langle\tau_{1}[pt]|\xi\rangle}^{\mathbb{P}^{3},H}_{g,L,(1)}$ and
${\langle-E^{2}|\xi\rangle}_{g,F,(1)}^{\tilde{\mathbb{P}}^{3},H}$. Here $\xi$
is the cohomology class of a line in $H\cong E\cong{\mathbb{P}}^{2}$.

The proof of Lemma 3.9 is identical to that of Lemma 3.3 with Lemma 3.1 and
3.2 replaced by Lemma 3.7 and 3.8 respectively.

Proof of Theorem 1.2: Similar to the proof of Theroem 1.1, we only need to
compute the universal coefficients $C_{g}$’s in (8). Similarly, we choose
$X=\mathbb{P}^{3},m=1,\alpha_{1}=[L],A=L$. Then (8) becomes

(9)
$\langle\tau_{1}[pt],[L]\rangle_{g,L}^{\mathbb{P}^{3}}=\sum_{g_{1}+g_{2}=g}C_{g_{1}}\cdot\langle-E^{2},[L]\rangle_{g_{2},F}^{\tilde{\mathbb{P}}^{3}}.$

By virtual localization [GP], we have

$\langle\tau_{1}[pt],[L]\rangle_{g,L}^{\mathbb{P}^{3}}=\frac{(-1)^{g}}{(2g+1)!},$
$\langle-E^{2},[L]\rangle_{g,F}^{\tilde{\mathbb{P}}^{3}}=\delta_{g,0}.$

We solve (9) to obtain the universal coefficients:

$C_{g}=\frac{(-1)^{g}}{(2g+1)!},$

which gives Theorem 1.2.

In the rest of this section, we will prove Theorem 1.3. Similar argument to in
the proof of Lemma 3.2 and 3, we may prove the following Lemmas.

######  Lemma 3.10.

Under the same assumptions as in Theorem 1.1, we have

$\displaystyle{\langle\tau_{1}E,\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle}^{\tilde{X}}_{g,p^{!}A-e}$
$\displaystyle=$
$\displaystyle\sum_{g^{+}+g^{-}=g}{\langle\tau_{1}E|\xi\rangle}_{g^{+},F,(1)}^{\tilde{\mathbb{P}}^{3},H}\cdot{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\xi\rangle}^{\tilde{X},E}_{g^{-},p^{!}A-e,(1)},$

where $F\in H_{2}(\tilde{\mathbb{P}}^{3},{\mathbb{Z}})$ is the class of a
fiber in
$\tilde{\mathbb{P}}^{3}\cong{\mathbb{P}}_{{\mathbb{P}}^{2}}(\mathscr{O}\oplus\mathscr{O}(-1))$,
and $\xi$ is the cohomology class of a line in $H\cong
E\cong{\mathbb{P}}^{2}$.

######  Lemma 3.11.

Under the same assumptions as in Theorem 1.1, denote
$\langle\tau_{1}E,\tau_{d_{1}}p^{*}\alpha_{1}$,
$\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}$ and
$\langle-E^{2},\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}$
by $H_{g}$ and $P_{g}$ respectively. Then

$H_{g}=\sum_{g_{1}+g_{2}=g}C_{g_{1}}P_{g_{2}},$

where $C_{g}$’s can be determined by relative invariants
${\langle\tau_{1}E|\xi\rangle}_{g,F,(1)}^{\tilde{\mathbb{P}}^{3},H}$ and
${\langle-E^{2}|\xi\rangle}_{g,F,(1)}^{\tilde{\mathbb{P}}^{3},H}$. Here $\xi$
is the cohomology class of a line in $H\cong E\cong{\mathbb{P}}^{2}$.

Proof of Theorem 1.3: Similar to the proof of Theorem 1.2, we choose
$X=\tilde{\mathbb{P}}^{3},m=1,\alpha_{1}=[L],A=F$ and obtain:

$\langle\tau_{1}E,L\rangle_{g,F}^{\tilde{\mathbb{P}}^{3}}=\sum_{g_{1}+g_{2}=g}C_{g_{1}}\cdot\langle-E^{2},L\rangle_{g_{2},F}^{\tilde{\mathbb{P}}^{3}}.$

By virtual localization [GP], we have

$\langle\tau_{1}E,L\rangle_{g,F}^{\tilde{\mathbb{P}}^{3}}=\delta_{g,0}\cdot
3-\frac{(-1)^{g}\cdot 2}{(2g+1)!}.$

So

$C_{g}=\delta_{g,0}\cdot 3-\frac{(-1)^{g}\cdot 2}{(2g+1)!},$

which gives Theorem 1.3.

##  4\. Formulae for Blow-up along a smooth curve

In this section, we give a detailed proof of Theorem 1.4. We always assume
that total degrees of insertions match the virtual dimension of the moduli
spaces, since otherwise the required equalities are trivial.

######  Lemma 4.1.

Under the same assumptions as in Theorem 1.4, we have

$\displaystyle\langle[C],\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle_{g,A}^{X}$
$\displaystyle=$
$\displaystyle\sum\limits_{g_{1}+g_{2}=g}\langle[C]|[pt]\rangle^{\bar{X}^{+},Z}_{g_{1},F,(1)}\cdot{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\mathbbm{1}\rangle}^{\tilde{X},E}_{g_{2},p^{!}A-e,(1)},$

where $F\in H_{2}(\bar{X}^{+},{\mathbb{Z}})$ is the class of a line in the
fiber of $\bar{X}^{+}={\mathbb{P}}_{C}(N_{C}\oplus{\mathscr{O}}_{C})$.

###### Proof.

We first perform symplectic cutting along $C$ and assume that the support of
$[C]$ is in $X^{+}$ and the support of $\alpha_{i}$ is away from $C$. By the
degeneration formula (2), we have:

$\displaystyle\langle[C],\tau_{d_{1}}\alpha_{1},\cdots,\tau_{d_{m}}\alpha_{m}\rangle_{g,A}^{X}$
$\displaystyle=$
$\displaystyle\sum\limits\mathfrak{z}(\mu)\langle[C]|\delta_{j_{1}},\cdots,\delta_{j_{\ell(\mu)}}\rangle_{\Gamma_{+}}^{\bullet,\bar{X}^{+},Z}$
$\displaystyle\qquad\cdot\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\delta^{j_{1}},\cdots,\delta^{j_{\ell(\mu)}}\rangle_{\Gamma_{-}}^{\bullet,\tilde{X},E}.$

Recall that we have assumed that

$\dim\overline{M}_{g,m+1}(X,A)=\sum\limits_{i=1}^{m}deg\alpha_{i}+2\sum\limits_{i=1}^{m}d_{i}+4.$

Assume that A term with $(\Gamma^{+},\Gamma^{-})$ in RHS of (4) has nonzero
contribution, and then

$\displaystyle\dim\overline{M}_{\Gamma_{+}}(\bar{X}^{+},Z)$ $\displaystyle=$
$\displaystyle 2\int_{A^{+}}c_{1}(X^{+})+2+2\ell(\mu)-2|\mu|,$
$\displaystyle\dim\overline{M}_{\Gamma_{-}}(\tilde{X},E)$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{m}deg\alpha_{i}+2\sum\limits_{i=1}^{m}d_{i}+\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}.$

So by the dimension constraint (2) for the degeneration formula, we have

$\displaystyle\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}+\int_{A^{+}}c_{1}(X^{+})-|\mu|=1+\ell(\mu).$

Let $\xi^{+}$ be the tautological line bundle of
$\bar{X}^{+}={\mathbb{P}}_{C}(N_{C}\oplus{\mathscr{O}}_{C})$, and we have

$\displaystyle c_{1}(\bar{X}^{+})=\pi^{*}c_{1}(X)|_{C}-3c_{1}(\xi^{+}),$

where $\pi:\bar{X}^{+}\rightarrow C$ is the natural projection. Note that
$-c_{1}(\xi^{+})$ is the Poincaré dual of the divisor $Z$ in $\bar{X}^{+}$.
Since $|\mu|=A^{+}\cdot Z$, it follows that

$\displaystyle\int_{A^{+}}c_{1}(\bar{X}^{+})=\int_{\pi_{*}A^{+}}c_{1}(X)|_{C}+3|\mu|.$

Therefore, dimension constraint becomes

$\displaystyle\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}+\int_{\pi_{*}A^{+}}c_{1}(X)|_{C}+2|\mu|=1+\ell(\mu).$

Since $m\geqslant 1$, it follows that $\mu\neq\emptyset$ by the connectedness
of the stable maps to $X$, and the dimension constraint holds only if

$\displaystyle\mu=(1),\quad\textrm{deg}\delta^{j_{1}}=0,\quad\int_{\pi_{*}A^{+}}c_{1}(X)|_{C}=0,$

which implies Lemma 4.1. ∎

######  Lemma 4.2.

Under the same assumptions as in Theorem 1.4, we have

$\displaystyle\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle_{g,p^{!}A-e}^{\tilde{X}}$
$\displaystyle=$
$\displaystyle\sum\limits_{g_{1}+g_{2}=g}\langle|[pt]\rangle^{\bar{\tilde{X}}^{+},Z}_{g_{1},F,(1)}{\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\mathbbm{1}\rangle}^{\tilde{X},E}_{g_{2},p^{!}A-e,(1)},$

where $F\in H_{2}(\tilde{X}^{+},{\mathbb{Z}})$ is the class of a line in the
fiber of $\bar{\tilde{X}}^{+}={\mathbb{P}}_{E}(N_{E}\oplus{\mathscr{O}}_{E})$.

###### Proof.

We first degenerate $\tilde{X}$ along $E$, and assume that the support of
$p^{*}\alpha_{i}$ is away from $E$. By the degeneration formula (2), we have

$\displaystyle{\langle
E,\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle}^{\tilde{X}}_{g,p^{!}A-e}$
$\displaystyle=$ $\displaystyle\sum\limits\mathfrak{z}(\mu)\langle
E|\delta_{j_{1}},\cdots,\delta_{j_{\ell(\mu)}}\rangle_{\Gamma_{+}}^{\bullet,\bar{\tilde{X}}^{+},Z}$
$\displaystyle\qquad\cdot\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}|\delta^{j_{1}},\cdots,\delta^{j_{\ell(\mu)}}\rangle_{\Gamma_{-}}^{\bullet,\tilde{X},E}.$

Recall that we have assumed that

$\dim\overline{M}_{g,m+1}(\tilde{X},p^{!}A-e)=\sum\limits_{i=1}^{m}deg\alpha_{i}+2\sum\limits_{i=1}^{m}d_{i}+2.$

Assume that a term with $(\Gamma^{+},\Gamma^{-})$ in RHS of (4) has nonzero
contribution. Then

$\displaystyle\dim\overline{M}_{\Gamma_{+}}(\bar{\tilde{X}}^{+},Z)$
$\displaystyle=$ $\displaystyle
2\int_{(p^{!}A-e)^{+}}c_{1}(\bar{\tilde{X}}^{+})+2+2\ell(\mu)-2|\mu|,$
$\displaystyle\dim\overline{M}_{\Gamma_{-}}(\tilde{X},E)$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{m}deg\alpha_{i}+2\sum\limits_{i=1}^{m}d_{i}+\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}.$

So by the dimension constraint (2) for the degeneration formula, we have

$\displaystyle\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}\textrm{deg}\delta^{j_{i}}+\int_{(p^{!}A-e)^{+}}c_{1}(\tilde{X}^{+})-|\mu|=\ell(\mu).$

Let $\xi^{+}$ be the tautological line bundle of
$\bar{\tilde{X}}^{+}={\mathbb{P}}_{E}(N_{E}\oplus{\mathscr{O}}_{E})$. Then
Euler exact sequence gives

$\displaystyle
c_{1}(\bar{\tilde{X}}^{+})=\pi^{*}c_{1}(E)+\pi^{*}c_{1}(N_{E})-2c_{1}(\xi^{+}),$

where $\pi:\bar{\tilde{X}}^{+}\rightarrow E$ is the natural projection. Note
that $N_{E}$ is the tautological line bundle of
$E\cong{\mathbb{P}}_{C}(N_{C})$, and so

$\displaystyle c_{1}(E)=\pi_{E}^{*}c_{1}(X)|_{C}-2c_{1}(N_{E}),$

where $\pi_{E}:E\rightarrow C$ is the natural projection. Therefore,

$\displaystyle
c_{1}(\bar{\tilde{X}}^{+})=(\pi_{E}\circ\pi)^{*}c_{1}(X)|_{C}-\pi^{*}c_{1}(N_{E})-2c_{1}(\xi^{+}).$

Note that we have the following natural decomposition

$\displaystyle H_{2}(\bar{\tilde{X}}^{+},{\mathbb{Z}})\cong{\mathbb{Z}}F\oplus
H_{2}(E,{\mathbb{Z}}),$

and we can write

$\displaystyle(p^{!}A-e)^{+}=aF+\pi_{*}(p^{!}A-e)^{+},\textrm{ for some
}a\in{\mathbb{Z}}_{\geqslant 0}.$

We have the following constraints for $(p^{!}A-e)^{+}$:

$\displaystyle\left\\{\begin{array}[]{ccl}(p^{!}A-e)^{+}\cdot Z&=&|\mu|,\\\
(p^{!}A-e)^{+}\cdot E&=&(p^{!}A-e)\cdot E=1,\end{array}\right.$

and this gives

$\displaystyle\pi_{*}(p^{!}A-e)^{+}\cdot E=-(|\mu|-1).$

Note that $-c_{1}(\xi^{+})$ is the Poincaré dual of the divisor $Z$ in
$\bar{\tilde{X}}^{+}$, and therefore

$\displaystyle\int_{(p^{!}A-e)^{+}}c_{1}(\bar{\tilde{X}}^{+})=\int_{(\pi_{E}\circ\pi)_{*}(p^{!}A-e)^{+}}c_{1}(X)|_{C}+3|\mu|-1.$

Hence the dimension constraint becomes

$\displaystyle\frac{1}{2}\sum\limits_{i=1}^{\ell(\mu)}deg\delta^{j_{i}}+\int_{(\pi_{E}\circ\pi)_{*}(p^{!}A-e)^{+}}c_{1}(X)|_{C}+2|\mu|=1+\ell(\mu).$

Since $m\geqslant 1$, it follows that $\mu\neq\emptyset$ by the connectedness
of the stable maps to $\tilde{X}$. So the dimension constraint holds only if

$\displaystyle\mu=(1),\quad
deg\delta^{j_{1}}=0,\quad\int_{(\pi_{E}\circ\pi)_{*}(p^{!}A-e)^{+}}c_{1}(X)|_{C}=0,$

which implies Lemma 4.2. ∎

Using the above comparison results, the same argument as in the proof of Lemma
3.3 shows that the following lemma holds.

######  Lemma 4.3.

Under the same assumptions as in Theorem 1.4, denote
$\langle[C],\tau_{d_{1}}\alpha_{1},\cdots$ ,
$\tau_{d_{m}}\alpha_{m}\rangle^{X}_{g,A}$ and
$\langle\tau_{d_{1}}p^{*}\alpha_{1},\cdots,\tau_{d_{m}}p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}$
by $H_{g}$ and $P_{g}$ respectively. Then

(13) $H_{g}=\sum_{g_{1}+g_{2}=g}C_{g_{1}}P_{g_{2}},$

where $C_{g}$’s can be determined by relative invariants
${\langle[C]|[pt]\rangle}^{\bar{X}^{+},Z}_{g,F,(1)}$ and
${\langle|[pt]\rangle}_{g,F,(1)}^{\bar{\tilde{X}}^{+},Z}$.

Similar to Theorem 1.1, we only need to determine the universal coefficients
$C_{g}$’s in (13). For this, we choose
$X={\mathbb{P}}_{C}(N_{C}\oplus{\mathscr{O}}_{C}),m=1,\alpha_{1}=[pt],A=F$ and
rewrite (13) as follows

(14)
$\langle[C],[pt]\rangle_{g,F}^{{\mathbb{P}}_{C}(N_{C}\oplus{\mathscr{O}}_{C})}=\sum_{g_{1}+g_{2}=g}C_{g_{1}}\cdot\langle[pt]\rangle_{g_{2},F}^{{\mathbb{P}}_{E}(N_{E}\oplus{\mathscr{O}}_{E})}.$

About the two absolute invariants in (14), we have

######  Lemma 4.4.

$\displaystyle\langle[C],[pt]\rangle_{g,F}^{{\mathbb{P}}_{C}(N_{C}\oplus{\mathscr{O}}_{C})}$
$\displaystyle=$ $\displaystyle\frac{(-1)^{g}}{(2g+1)!\cdot 2^{2g}},$
$\displaystyle\langle[pt]\rangle_{g,F}^{{\mathbb{P}}_{E}(N_{E}\oplus{\mathscr{O}}_{E})}$
$\displaystyle=$ $\displaystyle\delta_{g,0}.$

###### Proof.

In the first equality, let
$C={\mathbb{P}}_{C}(\\{0\\}\oplus{\mathscr{O}}_{C})$ and
$P_{0}\in{\mathbb{P}}_{C}(N_{C}\oplus\\{0\\})$ be the Poincaré duals of $[C]$
and $[pt]$ respectively. There is a unique connected smooth embedded curve
with homology class $F$ passing through $C$ and $P_{0}$, which is the line, in
the fiber containing $P_{0}$, passing through $P_{0}$ and the intersection of
$C$ and the fiber. Now LHS is equal to the degenerate contribution of the
line. So Theorem 1.5 in [Z] can be specialized to the first equality. The
proof of the second equality is similar. ∎

######  Remark 4.5.

Theorem 1.5 in [Z] is the symplectic version of degenerate contribution
computation for Fano case in [P1].

Proof of Theorem 1.4: Using Lemma 4.4 and solving the equaiton (14), we obtain
the universal coefficients $C_{g}=\frac{(-1)^{g}}{(2g+1)!\cdot 2^{2g}}$, which
gives Theorem 1.4.

######  Remark 4.6.

If $\int_{A}c_{1}(X)>1$, then we can relax the condition $m>0$ to $m\geqslant
0$. One can check this by going through the proof of Lemma 4.1 and 4.2.

##  5\. Generalized BPS numbers

Gromov-Witten invariants are only rational numbers in general, and hidden
integrality for these invariants of projective $3$-folds has been studied
since the very beginning of Gromov-Witten theory. For example, the
mathematically non-rigorous computation of genus zero invariants of quintic
$3$-folds [COGP] inspired the famous multiple covering formula [AM]. Based on
M-theory consideration, Gopakumar and Vafa [GV1, GV2] conjectured that
countings of BPS states give hidden integrality for Gromov-Witten invariants
of Calabi-Yau $3$-folds in all genera, which are multiplicities of certain
representations of $SL(2)$ in the cohomology of moduli space of sheaves. Based
on degenerate contribution computation, Pandharipande [P1, P2] generalized the
working definition of BPS numbers to arbitrary $3$-folds, and he also
conjectured that these generalized BPS numbers are integers which are counts
of curves satisfying incidence conditions given by insertions.

Let us review the definition of generalized BPS numbers and Pandharipande’s
conjecture. Let $X$ be a connected closed symplectic manifold of real
dimension $6$ and $A\in H_{2}(X,{\mathbb{Z}})$ a nonzero class. Note that by
dimension consideration, $A$ carries nonzero Gromov-Witten invariants only if
$\int_{A}c_{1}(X)\geqslant 0$. Suppose that $\alpha_{1},\cdots,\alpha_{m}\in
H^{>2}(X,{\mathbb{Q}})$. When $\int_{A}c_{1}(X)>0$, the generalized BPS number
$n_{g,A}^{X}(\alpha_{1},\cdots,\alpha_{m})$ (at least one insertion) is given
by

$\displaystyle\sum\limits_{g=0}^{\infty}u^{2g}\langle\alpha_{1},\cdots,\alpha_{m}\rangle_{g,A}^{X}=\sum\limits_{g=0}^{\infty}u^{2g}n_{g,A}^{X}(\alpha_{1},\cdots,\alpha_{m})\cdot(\frac{\sin\frac{u}{2}}{\frac{u}{2}})^{2g-2+\int_{A}c_{1}(X)},$

and when $\int_{A}c_{1}(X)=0$, then generalized BPS number $n_{g,A}^{X}$ (no
insertion) is given by

$\displaystyle\sum\limits_{g=0}^{\infty}u^{2g}\langle\rangle_{g,A}^{X}=\sum\limits_{g=0}^{\infty}u^{2g}n_{g,A}^{X}\cdot\sum\limits_{\begin{subarray}{c}d\in{\mathbb{Z}}_{>0}\\\
\frac{A}{d}\in
H_{2}(X,{\mathbb{Z}})\end{subarray}}\frac{1}{d}(\frac{\sin\frac{du}{2}}{\frac{u}{2}})^{2g-2}.$

In general, the generalized BPS numbers are defined to satisfy the divisor
equation and defined to vanish if degree $0$ and $1$ classes are inserted. So
these invariants can be extended to include all cohomology classes.

If $X$ is a projective $3$-fold, and $\alpha_{i}$ is the Poincaré dual of a
subvariety $X_{i}\subset X$ in general position, then Pandharipande
conjectured that $n_{g,A}^{X}(\alpha_{1},\cdots,\alpha_{m})$ is the number of
irreducible embedded curves in $X$ of geometric genus $g$, with homology class
$A$ and intersecting all $X_{i}$’s. An important corollary of this conjecture
is the integrality of generalized BPS numbers, which was proved by Zinger in
the Fano case [Z], and by Ionel and Parker in the Calabi-Yau case [IP3].

Proof of Proposition 1.5: We first prove Part (a) in Proposition 1.5.

From Corollary 3.6, we have

$\displaystyle\sum_{g=0}^{\infty}u^{2g-2}\langle[pt],\alpha_{1},\cdots,\alpha_{m}\rangle^{X}_{g,A}=(\frac{\sin(u/2)}{u/2})^{2}\sum_{g=0}^{\infty}u^{2g-2}\langle
p^{*}\alpha_{1},\cdots,p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}.$

Now by the definition of generalized BPS numbers, we have

$\displaystyle\sum_{g\geq
0}u^{2g-2}n^{X}_{g,A}([pt],\alpha_{1},\cdots,\alpha_{m})(\frac{\sin(u/2)}{u/2})^{2g-2+\int_{A}c_{1}(X)}$
$\displaystyle=$
$\displaystyle\sum_{g=0}^{\infty}u^{2g-2}\langle[pt],\alpha_{1},\cdots,\alpha_{m}\rangle^{X}_{g,A}$
$\displaystyle=$
$\displaystyle(\frac{\sin(u/2)}{u/2})^{2}\sum_{g=0}^{\infty}u^{2g-2}\langle
p^{*}\alpha_{1},\cdots,p^{*}\alpha_{m}\rangle^{\tilde{X}}_{g,p^{!}A-e}$
$\displaystyle=$
$\displaystyle(\frac{\sin(u/2)}{u/2})^{2}\sum_{g=0}^{\infty}u^{2g-2}n^{\tilde{X}}_{g,p^{!}A-e}(p^{*}\alpha_{1},\cdots,p^{*}\alpha_{m})(\frac{\sin(u/2)}{u/2})^{2g-2+\int_{p^{!}A-e}c_{1}(\tilde{X})}$
$\displaystyle=$
$\displaystyle\sum_{g=0}^{\infty}u^{2g-2}n^{\tilde{X}}_{g,p^{!}A-e}(p^{*}\alpha_{1},\cdots,p^{*}\alpha_{m})(\frac{\sin(u/2)}{u/2})^{2g-2+\int_{A}c_{1}(X)}.$

This gives Part (a) of Corollary 1.5. The proof of Part (b) is analogous.

######  Remark 5.1.

In the proof above, we only consider the case $m\geqslant 1$. The case $m=0$
can be treated similarly.

Acknowledgements. The authors would like to thank Prof. Pandharipande for
pointing out his early results about the degenerate contributions to us. We
would also like to thank Prof. Yongbin Ruan and Pedro Acosta for their useful
comments on earlier drafts. Huazhong would like to thank Prof. Jian Zhou for
sharing his ideas on Gromov-Witten theory generously, and Xiaowen Hu and
Hanxiong Zhang for helpful discussions. Weiqiang and Huazhong would like to
thank Department of Mathematics of University of Michigan for its hospitality
during their visiting.

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# YMIR: A Rapid Data-centric Development Platform for Vision Applications

Phoenix X. Huang1 Wenze Hu1 William Brendel2 Manmohan Chandraker3
Li-Jia Li4 Xiaoyu Wang1

1 Lighthouse Co.Ltd 2 Heali AI 3 University of California, San Diego 4
Stanford University
{phoenix.huangx<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper introduces an open source platform to support the rapid development
of computer vision applications at scale. The platform puts the efficient data
development at the center of the machine learning development process,
integrates active learning methods, data and model version control, and uses
concepts such as projects to enable fast iterations of multiple task specific
datasets in parallel. This platform abstracts the development process into
core states and operations, and integrates third party tools via open APIs as
implementations of the operations. This open design reduces the development
cost and adoption cost for ML teams with existing tools. At the same time, the
platform supports recording project development histories, through which
successful projects can be shared to further boost model production efficiency
on similar tasks. The platform is open source and is already used internally
to meet the increasing demand for different real world computer vision
applications.

##  1 Introduction

This paper introduces YMIR111YMIR denotes You Mine In Recursion. The project
is available at: http://www.viesc.com. Code is available at:
https://github.com/industryessentials/ymir., a data-centric ML development
platform that aims to support developing vision applications at scale.

YMIR uses data labeling, model training, and active learning based data mining
as primitive operations to build datasets and models. It embeds version
control into the iterative process to track and manage data annotations and
models, much like code version control in software engineering. The platform
also introduces the project concept, to isolate different jobs and enable
parallel development of datasets for multiple applications.

In terms of system design, YMIR treats dataset and model as the key inputs and
outputs of the primitive operations, and stores the specification of each
operation into metadata. This allows a decoupled implementation of each
operation, which has the following advantages: 1) It allows YMIR to flexibly
integrate the best third party tools readily available in the open source
community; 2) It reduces the adoption cost for experienced teams, as they can
quickly integrate their existing tools into the platform.

YMIR adopts existing version control tools to manage datasets and models, but
manages the actual data payload outside the tool to better support frequent
export of images to operators as well as frequent updates to the annotations
in the dataset development process.

To summarize, the highlights of the YMIR platform are as follows:

  1. 1.

The platform prioritizes data instead of model development in its product
design and system architecture.

  2. 2.

The platform adopts a modular open design approach to easily integrate third
party machine learning tools, and to leverage the contribution of the open
source community.

  3. 3.

The platform embeds version control of data and model into the system to track
the development process.

In the following sections, we will analyze one of the use cases to point out
the necessity of prioritizing the dataset development in ML development
process, introduce the details of the platform in the aspects of product
design and system design, show the key product functions and discuss potential
future works.

##  2 Background

Computer vision technology is becoming ready to be used in many industries.
For example, object detection alone can help a manufacturing factory to
enhance its digital transformation process in many aspects: detect if any
worker in the factory is not properly wearing protection gears; check whether
all the tools are back into cabinets after a work shift; if the evacuation
routes are unintentionally blocked, etc.

Although from many different application verticals, many potential
applications share the following traits: the problems can be cast into
standard computer vision tasks; the customer has positive image examples, but
these are usually rare events buried in their data storage with no
annotations; customers do not have a clear cut on the scope of the interested
events, so algorithm developing experts need to iterate with end users on task
definition through a trial and error process; these applications need to be
developed at low cost, as the users are usually hesitate to heavily invest in
yet to be verified new technologies.

Due to the challenges described above, a matured tool to semi-automate the
active learning based data development process is the key to a viable
solution, making project opportunities into real applications. Ultimately, the
dataset shall even be curated by the end user directly, as they are the best
party to define the task through a labeled dataset. These cases motivate us to
develop the YMIR platform, which is useful for developing these applications
at scale.

##  3 Product Design

Figure 1: A typical workflow in YMIR. YMIR abstracts the data curation process
into states of model, dataset and metadata, as well as the state changing key
operations, which are training op, mining op and labeling op. YMIR uses
version control to track and mange the data evolution processes.

###  3.1 State Space and Operations

Following the typical steps in active learning, YMIR models the dataset
development process into combinations of 3 primitive operations, which are
data mining, model training and data labeling. Correspondingly, the state
space of YMIR system is abstracted as the union of datasets, models and
metadata. The operations change the states of datasets and models, where the
details of the operations are stored in metadata. The relations between these
states and operation elements are summarized in Fig.1.

Dataset. In YMIR, A dataset is a collection of raw data as assets and the
annotations of these assets. We assume most of the assets are image files.
Each dataset has an unique ID as its identifier, and may have a name as a non-
unique identifier. Different datasets may share some assets and their
annotations.

Model. From the platform’s perspective, a model is simply a binary file that
can be used to run inferences on input images. Currently, there is no
assumptions on the format of the model files, and it is the individual
operators which assume the responsibility to check its validity.

Metadata. The metadata stores the types and parameters of the operations
applied on the models and datasets which change the states of the system. To
accommodate the modularized open design, YMIR only sets very limited number of
fields as required in the metatdata, and allows implementations of individual
operations specify extra fields if necessary.

Mining Op. The mining op takes a dataset and a model as input, and outputs a
subset of the input dataset as a new dataset. The images in the new dataset
are usually the most valuable subset to develop a better model. They should be
sent to a manual labeling process and later used as training data.

Labeling Op. The labeling op takes an input dataset and outputs an updated
version of the dataset with new annotations. YMIR only send images to this op,
without any annotations from previous projects. While this design leaves the
efforts of merging different annotations to the platform itself, it has
practical advantages on the design of data storage and version control, which
will be specified in Section 4.

Training Op. The training op is expected to take a fully labeled dataset as
input and output a new model. For many projects, the scale of the training
dataset is usually well below that of COCO dataset, so YMIR trains new models
from a fixed starting point such as an imagenet pre-trained model instead of
using model checkpoints from previous iterations. This allows users to change
model architectures during the development process, which we feel out weighs
the benefits of saved computations.

It is worth mentioning that the CRUD (create, read, update and delete)
operations on datasets are implemented in the YMIR system instead of via open
APIs. This design choice helps hiding the version control concepts out of GUI,
thus lowers the barrier for entry level users.

In the training and labeling Op, different datasets may be implicitly merged.
The merge takes a primary dataset and one or more secondary datasets, combines
their images and image annotations into a new dataset. The new dataset has a
new ID, but inherits the name and other auxiliary data from the primary
dataset. In terms of conflict resolution, YMIR provides a default
implementation that detects annotation conflict using simple rules and
overwrite the old versions on conflicting parts. It is still under discussion
on whether YMIR should provide interface for custom merging rules as a
configurable part in the future versions.

###  3.2 Data and Model Version Control

Proper composition of the defined ops on defined states results in the
standard loops of an active learning process, which in theory could result in
fast creation of training data and models. YMIR borrows many code version
control concepts from git to manage the data and models from this process.
Specifically, YMIR uses git branch concept to create new projects so that
different tasks on the same set of images can run in parallel. The CRUD of
datasets as well as the the primitive operations all create commits to the
branch. Each commit stores an updated version of datasets or a new model, as
well as the metadata of the operations leading to this change. Finally, only
data changes are merged to master branch, which conceptually aggregates all
the data annotated through many projects on this platform.

###  3.3 GUI and CLI Separation

As is mentioned in the introduction, YMIR has both a web based GUI and a
command line interface (CLI).

The web interface is designed for efficiently solving typical computer vision
problems. Specifically, the GUI only exposes pre-configured primitive
operations with default parameters, and hides the concepts of metadata and
version control. This is to smooth the learning curve and invite users without
in-depth knowledge of machine learning to use our platform for computer vision
related development needs.

In contrast, YMIR exposes most of the op parameters together with full
functionalities of version control in the command line interface. While a user
needs to learn many new concepts, this user interface can unblock expert
user’s development process for highly customized development requirements.

##  4 System Architecture

Figure 2: Layered view of the system architecture.

Fig. 2 is an overview of the system architecture. YMIR puts management of
system state, _i.e_. datasets and models as the foundation, to support
efficient storage and fast retrieval of models, data and versioned data
annotations. The application layer is built on top of this foundation, to
support flexible expansions of the platform’s functions, such as the primitive
operations, as well as user and project management.

###  4.1 Dataset and Model Management as Foundations

YMIR adopted DVC[2] for version control of datasets and models, which in turn
uses git. On top of DVC, we have made several significant changes to better
support fast iterations of datasets:

1\. The image assets are moved into dedicated KV storage services, whereas the
keys (_i.e_. SHA-1 values of images) instead of images are sent to DVC for
version management. We expect data mining operation usually runs on millions
images. If the images are stored in DVC, it can easily become bottleneck when
exporting these assets to mining or training ops.

2\. The image annotations are stored incrementally. After an image was added
to the dataset, its corresponding annotation accumulates along the whole
workflow. As the annotation is the most flexible and changeable component,
YMIR stores it incrementally as individual packages. The packed annotation is
stored in our metadata storage service, and theirs keys are put to DVC for
version control. In this case, YMIR reduces the expensive cost of DVC storing
the full annotation after each change. This change also helps speeding up
exporting the annotations, because usually the training op only use partial
annotations related to a target task, which is effectively equivalent to a
single change to the annotation. Saving it incrementally allows YMIR to
quickly find the related annotation instead of parsing it from the full
annotation file.

The metadata storage service is built on top of a standard key value service.
Apart from data annotations, this service also stores other metadata: _e.g_.
the operation context and parameters (_i.e_. the metadata in Section 3.1) of
each task ops, the attributes of datasets (_i.e_. the dimension, format and
keywords of images). YMIR uses protocol buffers to define metadata message
format. Fixed metadata (_e.g_. image SHA-hash and attributes) and mutable
metadata (_e.g_. task context) are stored separately, which allows to log
fixed data in shared packed data part that is built only once, and serialize
operation specific data into flexible but low-load storage file that do not
affect the enormous fixed metadata.

###  4.2 Application Layers

The modules in application layers can be classified to two categories: the
data related primitive operations defined in Section 3.1, and user related
services.

On the data side, the primitive operations in Section 3.1 are modularized as
individual services or processes running in docker containers. As of now, YMIR
uses Label Studio [6] as the labeling op, adopts YOLOv4 [3] in DarkNet [5] as
the training op and the data scoring methods in [1] as the core of our mining
op.

On the user side, the web interface is supported by another layer of services
that implements user and project management, op status tracking, as well as a
light weight service to retrieve and visualize the versioned datasets.

##  5 System Functions Overview

Figure 3: The YMIR homepage.

Fig.3 shows the landing page of the YMIR system. Upon login, users have an
overview of the datasets, models and ongoing tasks so that users can easily
resume working from where it was left off. In the rest of this section, we
will briefly list the main functions available in YMIR GUI, as well as the
main features inside each function.

Figure 4: UI for dataset import function.

Dataset Importing. As is shown in Fig.4, YMIR provides three ways to import a
dataset: upload dataset, copy from an existing public dataset, and import from
an URI. Before importing, the external dataset should be packaged following
the PASCAl VOC [4] dataset format. Also, the administrator of the system can
import datasets and set them as public. In this way, a user can easily
replicate a dataset by having a shallow copy of it, reducing the time and work
needed for packing and importing a dataset. If an user choose to import by
URI, the URI can be either a local path, a networked folder address or a http
address.

Figure 5: Image labeling task creation interface.

Data Annotations. After data is imported to the system, users can create
annotation tasks through YMIR. YMIR then sends the task to Label Studio,
monitors its status, and retrieves labeling results from Label Studio after
the annotation task is complete. Although not required, YMIR recommends users
to provide a task description file, giving examples of what are the right and
wrong labels. A well written document can greatly help raters improve their
annotation quality, which significantly improves the accuracy of the resulting
trained model. Besides a task description file, users can also specify data
annotators through YMIR. This additional feature is based on findings that
restricting similar labeling tasks to the same group of raters usually makes
the annotation quality much more consistent.

Figure 6: UI for model training function.

Model Training. Fig.6 shows the interface for creating a model training task.
A task can be created by specifying a task name, the used datasets, as well as
the training targets. Note that only object detection task is supported as of
the current release. More types of training tasks and more models for each
task can be added by both the YMIR developing team and the third party
contributors. The UI displays many training related hyper parameters and their
default values, which gives the user the flexibility to adjust these details
and compare results from different parameter configurations. Once a model is
trained, user can see the accuracy metrics on the task result page.

Figure 7: UI for data mining.

Data Mining. Compared with some existing AI development platforms, data mining
is a differentiating feature of YMIR system, and it is the key step in a
dataset and model iteration loop. Similar to the model training task, YMIR
currently only provides one method for data mining, but exposes many detailed
parameters so that interested users can tune the algorithm and compare its
results.

Figure 8: UI for model validation.

Model Validation. For any models trained in YMIR system, YMIR provides a model
validation feature that lets users test a trained model on the fly. Through
the interface, users can upload an image and check the detection results on
that image in seconds. Fig.8 shows an example detection result of an image on
a trained detection model.

Figure 9: Model/Dataset history tracking.

Model and Dataset History The model and dataset history page (Fig.9) shows the
dependency relations between datasets and models, as models can be used to
mine new datasets, and new datsets can then be used to train new models. It
provides a story-line view of the entire model and dataset development
process, which may provide clues on discovering the good and bad practices in
the ML development process.

##  6 Summary and Disucssions

This paper presents a new platform to support efficient development of
computer vision applications at scale. Before made open source, the platform
was adopted by pilot users, who reported significant productivity boost
compared to using standard manual process.

Although not fully utilized yet, the development histories stored in this
platform may further serve as training data to optimize the productivity of
users. If widely adopted, sharing and exchange of these records may further
help the community to find even better ways to solve data centric AI problems.

## References

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