# Correspondence of Donaldson-Thomas and Gopakumar-Vafa invariants on local
Calabi-Yau 4-folds over $V_{5}$ and $V_{22}$

Kiryong Chung  Department of Mathematics Education, Kyungpook National
University, 80 Daehakro, Bukgu, Daegu 41566, Korea<EMAIL_ADDRESS>,
Sanghyeon Lee  School of Mathematics, Korea Institute for Advanced Study, 85
Hoegiro, Dongdaemun-gu, Seoul 02455, Korea<EMAIL_ADDRESS>and
Joonyeong Won  Center for Mathematical Challenges, Korea Institute for
Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Korea
<EMAIL_ADDRESS>
###### Abstract.

We compute Gromov-Witten (GW) and Donaldson-Thomas (DT) invariants (and also
descendant invariants) for local CY $4$-folds over Fano $3$-folds, $V_{5}$ and
$V_{22}$ up to degree $3$. We use torus localization for GW invariants
computation, and use classical results for Hilbert schemes on $V_{5}$ and
$V_{22}$ for DT invariants computation. From these computations, one can check
correspondence between DT and Gopakumar-Vafa (GV) invariants conjectured by
Cao-Maulik-Toda in genus $0$. Also we can compute genus $1$ GV invariants via
the conjecture of Cao-Toda, which turned out to be $0$. These fit into the
fact that there are no smooth elliptic curves in $V_{5}$ and $V_{22}$ up to
degree $3$.

###### Key words and phrases:

Gromov-Witten invariants, Gopakumar-Vafa invariants, Donaldson-Thomas
invariants, Fano varieties

###### 2010 Mathematics Subject Classification:

14N35, 14C05,14C15, 14J45

##  1\. Introduction

###  1.1. Motivation

Recently, Donaldson-Thomas (DT) invariants for Calabi-Yau 4-folds and the
correspondence among related invariants, like stable pair (PT) and Gopakumar-
Vafa invariants (GV or BPS) have been actively studied in [CL14, CMT21, CMT18,
CK20, CK18, CKM21].

Virtual cycles of $\mathrm{DT_{4}}$ moduli space has been developed by
Borisov-Joyce [BJ17] and by Oh-Thomas [OT20]. For a CY 4-fold $X$ and a curve
class $\beta\in\mathrm{H}_{2}(X,\mathbb{Z})$, we denote $M_{\beta}(X)$ the
moduli space of 1-dimensional stable sheaves $F$ on $X$, such that $[F]=\beta$
and $\chi(F)=1$. Then $\mathrm{DT}_{4}$ invariants are defined by integrations
of cohomology insertions over virtual cycles $[M_{\beta}(X)]^{\mathrm{vir}}$.
Note that the virtual cycle is defined via some suitable choice of an
orientation on the moduli space $M_{\beta}(X)$.

Insertions are defined by the following way. Consider the product space
$M\times X$ and a (normalized)universal sheaf $\mathcal{F}$ over it. For a
cohomology element $\gamma\in\mathrm{H}^{*}(X,\mathbb{Z})$, let

$\displaystyle\tau_{i}(\gamma):=(\pi_{M})_{*}(\pi_{X}^{*}\gamma\cup\mathrm{ch}_{\dim(X)+i-1}(\mathcal{F}))\in\mathrm{H}^{*+2(i-1)}(M,\mathbb{Z})$

where $M$ is an abbreviation of $M_{\beta}(X)$. $\mathrm{DT_{4}}$ invariants
and descendent invariants (with a single insertion) are defined by

$\langle\tau_{i}(\gamma)\rangle_{\beta}:=\int_{[M_{\beta}(X)]^{\mathrm{vir}}}\tau_{i}(\gamma)$

via some suitable choice of orientation on the moduli space $M_{\beta}$.
$\langle\tau_{0}(\gamma)\rangle_{\beta}$ is called _$\mathrm{DT}_{4}$
invariants_, and we denote it by $\mathrm{DT}_{4}(X)(\beta|\gamma)$. When
$i\geq 1$, we call $\tau_{i}(\gamma)$ _descendant insertions_ , and
integrations of descendant insertions over the virtual class are called
_descendant invariants_.

On the other hand, we can consider $\mathrm{DT_{3}}$ invariants on Fano
3-folds. Note that $\mathrm{DT}_{3}$ moduli space for a Fano 3-fold is
isomorphic to the $\mathrm{DT}_{4}$ moduli space of the corresponding local CY
4-fold [CL14, Theorem 6.5]. Moreover, $\mathrm{DT_{3}}$ invariants for a Fano
3-fold is equal to the $\mathrm{DT}_{4}$ invariant of the corresponding local
CY 4-fold when we choose some suitable orientation in the $\mathrm{DT}_{4}$
moduli space [Cao19a, Cao19b].

Now we review some famous conjectures between $\mathrm{DT_{4}}$ and GV
invariants. Let us denote $n_{0,\beta}(\gamma)$ (resp. $n_{1,\beta}$) be the
genus 0 (resp. genus 1) Gopakumar-Vafa invariants on a CY 4-fold $X$ defined
in [KP08].

######  Conjecture 1.1.

[CMT18, Conjecture 0.2] Via some suitable choice of orientation on the moduli
space, we have

(1)
$n_{0,\beta}(\gamma)=\mathrm{DT}_{4}(\beta|\gamma),\;\mathrm{GW}(\gamma)^{X}_{0,\beta}=\sum_{k|\beta}\frac{1}{k^{2}}\mathrm{DT}_{4}(X)(\beta/k|\gamma)$

for all $\gamma\in\mathrm{H}^{4}(X,\mathbb{Z})$.

Note that two equations in (1) are equivalent via the definition of
$n_{0,\beta}(\gamma)$ in [KP08]. In [CMT18], Conjecture 1.1 is proven for some
cases: CY 4-folds with a elliptic structure and local curves/surfaces. Also,
in [CT21], the following conjecture is proposed which relates
$\mathrm{DT}_{4}$ descendent invariants and genus one GV invariants. Let
$m_{\beta_{1},\beta_{2}}$ be the meeting invariants defined in Section 0.3 in
[KP08] (for a detail, see Section 4).

######  Conjecture 1.2.

[CT21, Conjecture 0.2] Via some suitable choice of orientation on the moduli
space, we have

$\displaystyle\langle\tau_{1}(\gamma)\rangle_{\beta}$
$\displaystyle=\frac{n_{0,\beta}(\gamma^{2})}{2(\gamma\cdot\beta)}-\sum_{\beta_{1}+\beta_{2}=\beta}\frac{(\gamma\cdot\beta_{1})(\gamma\cdot\beta_{2})}{4(\gamma\cdot\beta)}m_{\beta_{1},\beta_{2}}$
$\displaystyle-\sum_{k\geq
1,k|\beta}\frac{(\gamma\cdot\beta)}{k}n_{1,\beta/k}$

for all $\gamma\in\mathrm{H}^{2}(X,\mathbb{Z})$.

Here, the number $m_{\beta_{1},\beta_{2}}$ is a virtual count of degree
$\beta_{1}$ curves in $X$ which meets with degree $\beta_{2}$, called _meeting
invariants_. We will compute these numbers in Section 4.

In [CT21], Conjecture 1.2 is proved in several cases: CY 4-folds with an
elliptic fibration structure and local Fano $3$-fold over ${\mathbb{P}^{3}}$.
In this paper, as a continuation of the latter case, our study focus on the
Fano $3$-folds: $V_{5}$ and $V_{22}$. See Section 3.1.1 and 3.1.3 for
definitions of $V_{5}$ and $V_{22}$. These Fano $3$-folds arise as a minimal
compactification of the complex $3$-space $\mathbb{C}^{3}$. For detailed
descriptions, see the beginning parts of Section 3.1.

###  1.2. Main results

In this paper, we compute some low degree $\mathrm{DT_{3}}$ invariants for
Fano 3-folds $V_{5}$ and $V_{22}$. For the cases $\beta=d\cdot[\text{line}]$,
$1\leq d\leq 3$ where $[\text{line}]$ is a class of $[\mathbb{P}^{1}]$ in
their projective embedding via the very ample generators, we review some
classical results on description of $\mathrm{DT}_{3}$ moduli space of $V_{5}$
and $V_{22}$ and universal sheaves on them. Using this, we will compute
$\mathrm{DT}_{3}$ invariants of these Fano 3-folds(equivalently
$\mathrm{DT}_{4}$ invariants of corresponding local CY 4-fold) in Section 3.1.

On the other hand, in Section 2 we compute twisted Gromov-Witten (GW)
invariants on $V_{5}$ and $V_{22}$ for the cases $\beta=d\cdot[\text{line}]$,
$1\leq d\leq 3$, using quantum Lefschetz property [KKP03] and torus
localization [GP99]. We will introduce some recipe for computing genus zero GW
invariants on Grassmannian varieties, which is a direct analogue of the
formula in [GP99, HTK+03] for genus zero GW invariants on projective spaces.
From these calculations, we obtain the following main results of the paper.

######  Theorem 1.3 (=Theorem 3.18).

Using the choice of orientation as in [CT21, (0.7)] on $\mathrm{DT}_{4}$
moduli spaces on local CY 4-folds, the Conjecture 1.1 holds for $V_{5}$ and
$V_{22}$, $\beta=d\cdot[\text{line}]$, $1\leq d\leq 3$.

######  Theorem 1.4 (=Theorem 4.2).

Using the choice of orientation as in [CT21, (0.7)] on $\mathrm{DT}_{4}$
moduli spaces on local CY 4-folds, the Conjecture 1.2 holds for $V_{5}$ and
$V_{22}$, $\beta=d\cdot[\text{line}]$, $1\leq d\leq 3$ if and only if genus
$1$ GV invariants $n_{1,\beta}\equiv 0$.

From the fact that the degree of the defining equation of $V_{5}$ and $V_{22}$
is $\leq 2$ and the varieties does not contain plane, one can check that the
smooth elliptic curves of the degree $\leq 3$ in $V_{5}$ and $V_{22}$ does not
exist. Hence the vanishing of GV invariants $n_{1,\beta}$ is desirable.

We have posted all source codes we have used during the computation of GW
invariants and their outputs at the second author’s website:

https://sites.google.com/view/sanghyeon-lee/reference?authuser=0

### Acknowledgements

The authors gratefully acknowledge the many helpful suggestions of Yalong Cao
and Young-Hoon Kiem during the preparation of the paper. The second named
author thanks to Jeongseok Oh and Hyeonjun Park for much advice on Donaldson-
Thomas invariants.

##  2\. Twisted GW invariants on Grassmannians

Let $Y$ be a smooth Fano $3$-fold embedded in the Grassmannian variety
$G=\mathrm{Gr}(r,d)$, which is a zero section of a bundle over $G$. In this
subsection, we present an algorithm of the computation of GW invariants of
$|K_{Y}|$, a total space of a canonical line bundle of $Y$. For any smooth
projective variety $X$, we denote the moduli space of stable maps, with
$k$-marked points with degree $\beta\in\mathrm{H}^{2}(X,\mathbb{Z})$ by
$M_{0,k}(X,\beta)$. When $X=G$, we will denote $M_{0,k}(G,d)$ the stable map
space with degree $d\cdot[\text{line}]$. We will usually abbreviate
$M_{0,k}(X,\beta)$ by $M(X)$ in the following. In this section, we will
introduce classical methods to compute GW invariants, quantum Lefschetz
principle and torus localization when the target variety is $|K_{Y}|$. If you
are not interested in this part, you may skip the details and just see
computational results in Section 2.4.

Consider the forgetful map $p:M=M_{0,k}(G,d)\to\mathfrak{M}_{0,k}$ where
$\mathfrak{M}_{0,k}$ is the moduli space of prestable genus $0$ curves with
$k$ marked points. We usually will abbreviate $\mathfrak{M}_{0,k}$ by
$\mathfrak{M}$. There is a usual (relative) perfect obstruction theory [Beh96]
$E_{M(X)/\mathfrak{M}}\to L_{M(X)/\mathfrak{M}}$ of $M(X)\to\mathfrak{M}$,
where

(2) $\displaystyle E_{M/\mathfrak{M}}:=[R\pi_{*}f^{*}T_{G}]^{\vee}\in
D^{b}(M(X)).$

Also, as in [BF97] the virtual cycle correspond to this perfect obstruction
theory is given by:

(3)
$\displaystyle[M(X)]^{\mathrm{vir}}:=0^{!}_{h^{1}/h^{0}(E_{M/\mathfrak{M}})}[\mathcal{C}_{M(X)/\mathfrak{M}}]\in
A_{\mathrm{vdim}}(M(X))$

where $h^{1}/h^{0}(E_{M/\mathfrak{M}})$ is a vector bundle stack correspond to
$E_{M/\mathfrak{M}}$, $\mathcal{C}_{M(X)/\mathfrak{M}}$ is the (relative)
intrinsic normal cone defined in [BF97], and $\mathrm{vdim}$ is given by

$\mathrm{vdim}=(1-g)(\dim(X)-3)+k-\int_{\beta}c_{1}(K_{X})$

which is called virtual dimension.

###  2.1. Quantum Lefschetz principle

We will briefly review some aspect of famous quantum Lefschetz principle
[KKP03] in this section. We consider a negative vector bundle $E$ on $G$, so
that $\mathrm{H}^{0}(G,f^{*}E)=0$ for any non-constant morphism
$f:\mathbb{P}^{1}\to G$. Let $|E|$ be the total space of $E$ and $p:|E|\to G$
be the projection. Consider a stable map $f:C\to|E|$. Then $p\circ f:C\to G$
is a stable map and $f$ induces an element of $\mathrm{H}^{0}(C,(p\circ
f)^{*}E)$. But since $E$ is a direct sum of negative degree line bundles,
$\mathrm{H}^{0}(C,(p\circ f)^{*}E)=0$. Hence we have a natural isomorphism of
moduli spaces of stable maps:

$M_{0,k}(|E|,d)\cong M_{0,k}(G,d).$

Note that $M_{0,k}(G,d)$ is well known to be smooth. Also, we have the short
exact sequence:

$0\to p^{*}T_{G}\to T_{|E|}\to p^{*}E\to 0.$

From the condition that $E$ is negative and the fact that
$\mathrm{R}^{1}\pi_{*}(f^{*}p^{*}T_{G})=0$, we obtain the following by taking
the higher direct image functor $R\pi_{*}f^{*}$ to the above short exact
sequence.

$E_{M(|E|)/\mathfrak{M}}^{\vee}=\mathrm{R}\pi_{*}T_{|E|}=\left[\pi_{*}(f^{*}p^{*}T_{G})\stackrel{{\scriptstyle
0}}{{\longrightarrow}}R^{1}\pi_{*}(f^{*}p^{*}E)\right]$

Note that $\pi_{*}(f^{*}p^{*}T_{G})\cong T_{M(G)}$. We can easily check that
the intrinsic normal cone $\mathcal{C}_{M(G)/\mathfrak{M}}$ is
$[M(G)/T_{M(G)/\mathfrak{M}}]$ and
$h^{1}/h^{0}(E^{\vee})=[R^{1}\pi_{*}(f^{*}p^{*}E)/T_{M(G)/\mathfrak{M}}]$.
From the definition of the virtual cycle in (3), we have

$\displaystyle[M(G)]^{\mathrm{vir}}$
$\displaystyle=0^{!}_{[R^{1}\pi_{*}(f^{*}p^{*}E)/T_{M(G)/\mathfrak{M}}]}[M(G)/T_{M(G)/\mathfrak{M}}]$
$\displaystyle=0^{!}_{R^{1}\pi_{*}(f^{*}p^{*}E)}[M(G)]$
$\displaystyle=e(\mathrm{R}^{1}\pi_{*}(f^{*}p^{*}E))\cap[M(G)]$

where the second identity comes from properties of Gysin pull-backs via bundle
stacks in [Kre99]. We call this phenomenon B-twist. Note that this phenomenon
also arises when we replace $G$ by any projective variety. Using this, we
define a _twisted GW invariants_ of a Fano variety $X$ by:

######  Definition 2.1 (Twisted GW invariant).

The twisted Gromov-Witten invariant a Fano variety $X$ is defined by the
integration

$\mathrm{GW}^{\text{twist}}_{0,\beta}(X)(\gamma):=\int_{[M_{0,1}(X,\beta)]^{\mathrm{vir}}}e(\mathrm{R}^{1}\pi_{*}f^{*}K_{X})\cup\mathrm{ev}^{*}(\gamma)\in\mathbb{Q}$

where $\mathrm{ev}:M_{0,1}(X,\beta)\rightarrow X$ be the evaluation map.

Note that twisted GW invariants are usually considered as a definition of GW
invariants of the total space $|K_{X}|$, because we can not define it directly
since $|K_{X}|$ is not compact.

Next we consider a complete intersection in $G=G(r,n)$. Let $E$ be the vector
bundle on $G$ and let $Y\subset G$ be a zero section of the generic section
$s:\mathcal{O}_{G}\to E$. Assume that $E$ is convex, so that
$\mathrm{H}^{1}(G,\varphi^{*}E)=0$ for any non-constant morphism
$\varphi:\mathbb{P}^{1}\to G$. For example, the direct sum
$\bigoplus_{i}\mathcal{O}(a_{i})$, $a_{i}>0$ of line bundles on $G$ is a
convex vector bundle. From the convexity of $E$, $\pi_{*}f^{*}E$ is locally
free, where $\pi:\mathcal{C}\to M$ is the universal curve and
$f:\mathcal{C}\to G$ is the universal morphism. Let $M(Y)$ denote the stable
map space $M_{0,k}(Y,d)$. Then we have

$E_{M(Y)/\mathfrak{M}}^{\vee}=\mathrm{R}\pi_{*}f^{*}T_{Y}=[\pi_{*}f^{*}T_{G}\to\pi_{*}f^{*}E]=[T_{M}\to\pi_{*}f^{*}E].$

Consider the section $s:\mathcal{O}_{G}\to E$ and the induced section
$\widetilde{s}:M\to\pi_{*}f^{*}E$ defined by
$\widetilde{s}([(C,f)])=(C,f^{*}s)\in H^{0}(C,f^{*}E)$. Then we have
$Z(\widetilde{s})=M(Y)\subset M$. We can easily check that
$\mathcal{C}_{M/\mathfrak{M}}=[C_{M(Y)/M}/T_{M/\mathfrak{M}}]$ and
$h^{1}/h^{0}(E^{\vee})=[\pi_{*}f^{*}E/T_{M/\mathfrak{M}}]$. Then by the
definition of the virtual cycle, we have

$[M(Y)]^{\mathrm{vir}}=0^{!}_{[\pi_{*}f^{*}E/T_{M}]}[C_{M(Y)/M}/T_{M/\mathfrak{M}}]=0^{!}_{\pi_{*}f^{*}E}[C_{M(Y)/M}]\in
A_{*}(M(Y)).$

Also we have

$\iota_{*}[M(Y)]^{\mathrm{vir}}=e(\pi_{*}f^{*}E)\cap[M]\in A_{*}(M)$

where $\iota:M(Y)\hookrightarrow M$ is the inclusion. See [KKP03] for the
proof of the general case. We call this phenomenon A-twist.

###  2.2. Torus localization

We will briefly review some aspect of torus localization [GP99] in this
section. Again we consider the Grassmannian variety $G=\mathrm{Gr}(r,d)$ and
the stable map space $M=M_{0,k}(G,d)$. We give a $\mathbb{C}^{*}$-action on
$\mathbb{C}^{n}$ with weights $-\alpha_{1},\dots,-\alpha_{n}$, which induces
the $\mathbb{C}^{*}$-action on $\mathrm{Gr}(r,n)$ and $M$. Let $M^{F}\subset
M$ be the fixed locus of the action and let $M^{F}=\bigcup_{i}M^{F}_{i}$ be
the irreducible decomposition. Note that $E_{M/\mathfrak{M}}^{\vee}$ has an
induced $\mathbb{C}^{*}$-action and we have a decomposition

$E_{M/\mathfrak{M}}^{\vee}|_{M^{F}_{i}}\cong N^{\mathrm{fix}}\oplus
N^{\mathrm{vir}}$

where $N^{\mathrm{fix}}$ has weight $0$ under the $\mathbb{C}^{*}$-action and
$N^{\mathrm{vir}}$ is a direct sum of vector bundles with non-zero weights.

Let $(A^{T})_{*}(M)$ (resp. $(A^{T})^{*}(M)$) be the equivariant Chow group
(resp. equivariant Chow cohomology group) of $M$ and $e^{T}(E)$ be the
equivariant Euler class of a locally free sheaf $E$. If $E_{M}^{\vee}$ has
locally free resolution $[E_{0}\to E_{1}]$, we define the (equivariant) Euler
class by

$e^{T}(N^{\mathrm{vir}})=\frac{e^{T}(E_{0}^{m})}{e^{T}(E_{1}^{m})}\in(A^{T})^{*}(M)\otimes\mathbb{Q}[t,1/t].$

By the virtual localization theorem in [GP99], we have

$[M]^{\mathrm{vir}}=\sum_{i}\frac{[M^{F}_{i}]^{\mathrm{vir}}}{e^{T}(N^{\mathrm{vir}})}\in
A^{T}_{*}(M)\cong(A^{T})_{*}(M)\otimes\mathbb{Q}[t,1/t].$

###  2.3. Computation of the virtual normal bundle

For $M=M_{0,k}(G,d)$, we can do more specific computation. In a similar manner
as in [GP99] and [HTK+03], which dealt with the case $G=\mathbb{P}^{N}$, fixed
loci $M^{F}_{i}$ are indexed by decorated graphs $\Gamma$. We denote
$M^{F}_{i}$ by $M_{\Gamma}$ for the corresponding decorated graph $\Gamma$.
Note that $M_{\Gamma}$ is smooth and thus
$[M_{\Gamma}]^{\mathrm{vir}}=[M_{\Gamma}]$. Also $N^{\mathrm{vir}}$ denotes
the virtual normal bundle defined in [BF97].

For a stable map $[(C,x_{1},\dots,x_{k},f)]\in M_{\Gamma}$, the fiber of the
(K-theoretic) virtual normal bundle is given by the moving part of
$\mathrm{Ext}^{1}(\Omega_{C}(x_{1}+\dots,x_{k}),\mathcal{O}_{C})-\mathrm{Ext}^{0}(\Omega_{C}(x_{1}+\dots,x_{k}),\mathcal{O}_{C})+(\mathrm{H}^{0}-\mathrm{H}^{1})(f^{*}T_{G})$.
By [GP99] and [HTK+03], we have

$e^{T}(\mathrm{Ext}^{0}(\Omega_{C}(x_{1}+\dots,x_{k}),\mathcal{O}_{C}))=\prod_{\begin{subarray}{c}v\in\mathrm{Vertices}\\\
\text{val}(v)=1\end{subarray}}\omega_{F_{v}}$

and

$e^{T}(\mathrm{Ext}^{1}(\Omega_{C}(x_{1}+\dots,x_{k}),\mathcal{O}_{C}))=\prod_{\begin{subarray}{c}F\in\mathrm{Flags}\\\
\mathrm{valency}(i(F))\geq 3\end{subarray}}(\omega_{F}-e_{F}),$

where $\omega_{F}:=\frac{\alpha_{i(F)}-\alpha_{j(F)}}{d_{e}}$ and $e_{F}$ is
the $\psi$-class correspond to the flag $F$. Also, by [GP99] and [HTK+03], we
have the following by using the normalization sequence of nodal curves.

(4) $\displaystyle(\mathrm{H}^{0}-\mathrm{H}^{1})(f^{*}T_{G})$
$\displaystyle=\bigoplus_{v\in\mathrm{Vertices}}T_{p_{v}}G+\bigoplus_{e\in\mathrm{Edges}}\mathrm{H}^{0}(C_{e},f^{*}T_{G})$
$\displaystyle-\bigoplus_{F\in\mathrm{Flags}}T_{p_{i(F)}}G-\bigoplus_{v\in\mathrm{Vertices}}\mathrm{H}^{1}(C_{v},f^{*}T_{G})$

Then the equivariant Euler classes of their moving part are given by the
followings. For the first term of (4), if $p_{v}=x_{u_{1},\dots,u_{r}}=\langle
e_{u_{1}},\dots,e_{u_{r}}\rangle\in\mathrm{Gr}(r,n)$, then

$e^{T}(T_{p_{i(v)}}G)=\prod_{1\leq j\leq
r}\prod_{k\in[n]\setminus\\{u_{1},\dots,u_{r}\\}}(\alpha_{u_{j}}-\alpha_{k}).$

For the third term of (4), if $p_{i(F)}=x_{u_{1},\dots,u_{r}}=\langle
e_{u_{1}},\dots,e_{u_{r}}\rangle\in\mathrm{Gr}(r,n)$, then we have the same
formula as above :

$T_{p_{i(F)}}=\prod_{1\leq j\leq
r}\prod_{k\in[n]\setminus\\{u_{1},\dots,u_{r}\\}}(\alpha_{u_{j}}-\alpha_{k}).$

For the fourth term of (4), by [HTK+03], we have the following. If
valency($v$)$=2$ and there is no marking on $v$, then

$e^{T}(\mathrm{H}^{1}(C_{v},f^{*}T_{G}))=\omega_{F_{v,1}}+\omega_{F_{v,2}}.$

Otherwise, we have

$e^{T}(\mathrm{H}^{1}(C_{v},f^{*}T_{G}))=1.$

For the second term of (4), consider the Euler sequence:

(5) $\displaystyle 0\to S^{\vee}\otimes S\to
S^{\vee}\otimes\mathcal{O}_{G}^{\oplus n}\to S^{\vee}\otimes Q=T_{G}\to 0$

where $S,Q$ are tautological bundle and universal quotient bundle on $G$. Take
pull-back via the map $f:C_{e}\to G$. Let $e=\\{v_{1},v_{2}\\}$,
$p_{v_{1}}=x_{u_{1},\dots,u_{r-1},u_{a}}$ and
$p_{v_{2}}=x_{u_{1},\dots,u_{r-1},u_{b}}$. Note that $u_{a}\neq u_{b}$. Then
we have
$f^{*}(S^{\vee})\cong\mathcal{O}(\alpha_{u_{1}})\oplus...\oplus\mathcal{O}(\alpha_{u_{1}})\oplus\mathcal{O}(d_{e})$
where $\mathcal{O}(\alpha_{i})$ is an equivariant trivial bundle where
$(\mathbb{C}^{*})^{n}$ acts on it with a weight $\alpha_{i}$. Note that
$\mathrm{H}^{0}(C_{e},\mathcal{O}(d_{e}))$ has weights
$\frac{c_{1}\alpha_{u_{a}}+c_{2}\alpha_{u_{b}}}{d_{e}}$ for
$c_{1}+c_{2}=d_{e}$. Let $f|_{C_{e}}=:f_{e}$. By taking $f_{e}^{*}$ and the
cohomology in (5), we have the following exact sequence:

$0\to\mathrm{H}^{0}(f_{e}^{*}(S^{\vee}\otimes
S))\to\mathrm{H}^{0}((f_{e}^{*}S^{\vee}\otimes\mathcal{O}_{G}^{\oplus
n}))\to\mathrm{H}^{0}(f_{e}^{*}T_{G})\to\mathrm{H}^{1}((f_{e}^{*}S^{\vee}\otimes
S))\to 0.$

By a direct calculation, we have

$\displaystyle e^{T}((\mathrm{H}^{1}-\mathrm{H}^{0})(f_{e}^{*}(S^{\vee}\otimes
S)))$ $\displaystyle=\frac{(-1)^{d_{e}-1}}{\prod_{1\leq i\leq
r-1}(\alpha_{u_{a}}-\alpha_{u_{i}})(\alpha_{u_{b}}-\alpha_{i})\prod_{1\leq
i<j\leq r-1}(\alpha_{u_{i}}-\alpha_{u_{j}})^{2}}.$

Also we have,

$\displaystyle
e^{T}((\mathrm{H}^{1}-\mathrm{H}^{0})(f_{e}^{*}S^{\vee}\otimes\mathcal{O}^{\oplus
n}_{G}))$ $\displaystyle=\prod_{1\leq j\leq
r}\prod_{k\in[n]\setminus\\{u_{1},\dots,u_{r}\\}}(\alpha_{u_{j}}-\alpha_{k})$
$\displaystyle\times\prod_{k\in[n]\setminus\\{u_{1},\dots,u_{r}\\}}\prod_{\begin{subarray}{c}c_{1},c_{2}\neq
0\\\ c_{1}+c_{2}=d_{e}\\\
(c_{1},k)\neq(0,u_{b}),\,(d_{e},u_{a})\end{subarray}}(\frac{c_{1}\alpha_{u_{a}}+c_{2}\alpha_{u_{b}}}{d_{e}}-\alpha_{k}).$

###  2.4. Computation of GW invariants on Fano $3$-folds

Combining above arguments, we have expression of
$e^{T}(N^{\mathrm{vir}}|_{M_{\Gamma}})$. Next we represent A-twist and B-twist
in section 2.1 in equivariant cohomology. Let
$F_{1}:=\pi_{*}f^{*}\left(\bigoplus_{i}\mathcal{O}(a_{i})\right)$ and
$F_{2}:=R^{1}\pi_{*}f^{*}\mathcal{O}(-b)$ where $b$ is the Fano index of $X$,
$\pi:\mathcal{C}\to M(X)$ is the universal curve and $f:\mathcal{C}\to X$ is
the universal morphism. We have

(6)
$\displaystyle\iota_{*}[M(X)]^{\mathrm{vir}}=\sum_{\Gamma}\frac{1}{|A_{\Gamma}|}\frac{e^{T}(F_{1}|_{M_{\Gamma}})\cup
e^{T}(F_{2}|_{M_{\Gamma}})}{e^{T}(N^{\mathrm{vir}})}[M_{\Gamma}]\in(A^{T})_{*}(M)\otimes\mathbb{Q}[t,1/t]$

where $A_{\Gamma}$ is the order of the automorphism group of a generic element
in $M_{\Gamma}$. We have
$|A_{\Gamma}|=|\mathrm{Aut}(\Gamma)|\cdot\prod_{e\in\mathrm{Edges}}d_{e}$
where $\mathrm{Aut}(\Gamma)$ is the automorphism group of the decorated graph
$\Gamma$. One can check more detail on the group $A_{\Gamma}$ in [GP99]. Note
that we can find specific expressions of $e^{T}(F_{1}|_{M_{\Gamma}})$ and
$e^{T}(F_{2}|_{M_{\Gamma}})$ using the normalization sequence:

(7) $\displaystyle
0\to\mathcal{O}_{C}\to\bigoplus_{v\in\mathrm{Verteces}}\mathcal{O}_{C_{v}}\oplus\bigoplus_{e\in\mathrm{Edges}}\mathcal{O}_{C_{e}}\to\bigoplus_{\begin{subarray}{c}F\in\mathrm{Flags}\end{subarray}}\mathbb{C}_{x_{F}}\to
0.$

Combining arguments in section 2.2, 2.3, 2.1 and Hodge integrals computed in
[FP00], we can express the right hand side terms of (6) by formal weights of
$\mathbb{C}^{*}$-action. Therefore, we can compute genus 0 Gromov-Witten
invariants of the total space of the canonical line bundle over the Fano
3-fold, which is a zero section of an equivariant vector bundle over a
Grassmannian variety.

In this paper, we consider two cases: (a) $Y=V_{5}$ and (b) $Y=V_{22}$. In
case (a), $F_{1}=\pi_{*}f^{*}\mathcal{O}(1)^{\oplus 3}$ and
$F_{2}=\mathrm{R}^{1}\pi_{*}f^{*}\mathcal{O}(-2)$. In case (b),
$F_{1}=\pi_{*}f^{*}(\wedge^{2}S^{\vee})$ and
$F_{2}=\mathrm{R}^{1}\pi_{*}f^{*}\mathcal{O}(-1)$. The actual computation has
been done by a computer program. Firstly we make a dataset of all possible
decorated graphs $\Gamma$ and their information. Secondly, using this dataset,
we make a code computing the right hand side terms in (6) for each
localization graph $\Gamma$ and adding up them. As a result, we obtain the
following table are twisted GW invariants.

######  Proposition 2.2 (Twisted GW invariants).

The twisted GW invariants for $Y=V_{5}$ and $V_{22}$ are given by the numbers
of the following table.

$d$ |  $\mathrm{GW}^{\text{twist}}_{0,d}(V_{5})(h_{2})$ |  $\mathrm{GW}^{\text{twist}}_{0,d}(V_{22})(h_{2})$
---|---|---
$1$ |  $-5$ |  $2$
$2$ |  $-36\frac{1}{4}$ |  $-6\frac{1}{2}$
$3$ |  $-490\frac{5}{9}$ |  $28\frac{2}{9}$

Here $h_{2}$ is the generator of $\mathrm{H}^{4}(Y,\mathbb{Z})\cong\mathbb{Z}$
.

######  Remark 2.3.

The degree $4$ twisted GW invariant on $V_{5}$ is given by
$\mathrm{GW}^{\text{twist}}_{0,4}(V_{5})(\gamma_{2})=-8829\frac{1}{16}$. In
principle we can compute GW invariants for higher degrees, but the time taken
for the calculation super-exponentially increases.

##  3\. Donaldson-Thomas type invariants

In this section, we compute DT invariants and descendant invariants for some
local Fano 3-folds $|K_{V_{5}}|$ and $|K_{V_{22}}|$ for degree $1\leq
d\cdot[\textrm{line}]\leq 3$. We will abbreviate $d\cdot[\textrm{line}]$ by
$d$. In these cases, $M_{d}$ naturally isomorphic to moduli space of stable
sheaves on $|K_{Y}|$.

######  Definition 3.1 (Twisted DT invariant).

Let $\mathbf{M}_{\beta}=\mathbf{M}_{\beta}(Y)$ be the moduli space of stable
sheaves $F$ on $Y$ with $[F]=\beta\in H_{2}(Y,\mathbb{Z})$ and $\chi(F)=1$.
Let

$\tau_{0}:\mathrm{H}^{4}(Y,\mathbb{Z})\rightarrow\mathrm{H}^{2}(\mathrm{M}_{\beta},\mathbb{Z}),\;\tau_{0}(\gamma)=\pi_{M*}(\pi_{X}^{*}\gamma\cup\mathrm{ch}_{2}(\mathcal{F}))$

be the _primary insertion_ of $\gamma\in\mathrm{H}^{4}(Y,\mathbb{Z})$. Here
$\mathcal{F}$ is the universal sheaf and the maps $\pi_{M_{\beta}}$, $\pi_{Y}$
are the canonical projection maps. The _twisted_ genus zero DT invariant is
defined by

$\mathrm{DT}_{3}^{\text{twist}}(Y)(\beta|\gamma):=(-1)^{c_{1}(Y)\cdot\beta-1}\int_{[\mathbf{M}_{\beta}(Y)]^{\mathrm{vir}}}\tau_{0}(\gamma)\in\mathbb{Z}.$

where $[\mathbf{M}_{\beta}(Y)]^{\mathrm{vir}}$ is the virtual class defined in
[Tho00, Corollary 3.39].

Since $\mathrm{DT}_{4}$ invariant of $|K_{Y}|$ is equal to $\mathrm{DT}_{3}$
invariant of $Y$, it is enough to compute twisted $\mathrm{DT}_{3}$ invariant.

Note that if the moduli space $\mathbf{M}_{\beta}(Y)$ is smooth, the virtual
cycle is the Poincaré dual of the top Chern class of the obstruction bundle.
Combining computation of GW invariants in Section 2 and DT invariant
computation in this section, we will check Conjecture 1.1 for $1\leq d\leq 3$,
which can be rewritten by

######  Conjecture 3.2.

For the cohomology class $\gamma\in\mathrm{H}^{4}(Y,\mathbb{Z})$,

$n_{0,\beta}(\gamma)=\mathrm{DT}_{4}(|K_{Y}|)(d|\gamma)$

and

$\mathrm{GW}_{0,\beta}(Y)^{\text{twist}}(\gamma)=\sum_{k|\beta}\frac{1}{k^{2}}\cdot\mathrm{DT}_{4}(|K_{Y}|)(d/k|\gamma).$

On the other hand, Cao and Toda suggest genus one GV-type invariant on CY
$4$-fold $X$ by using the descendent insertion in [CT21].

######  Definition 3.3 (Descendent insertion).

For an integral class $\gamma\in\mathrm{H}^{2}(X,\mathbb{Z})$, let us define
the _descendent insertion_ as

$\tau_{1}:\mathrm{H}^{2}(X,\mathbb{Z})\rightarrow\mathrm{H}^{2}(\mathrm{M}_{\beta},\mathbb{Z}),\;\tau_{1}(\gamma)=\pi_{M*}(\pi_{X}^{*}\gamma\cup\mathrm{ch}_{4}(\mathcal{F}_{\text{norm}})),$

where
$\mathcal{F}_{\text{norm}}:=\mathcal{F}\otimes\pi_{M}^{*}(\pi_{M*}\det(\mathcal{F}))^{-1}$
is the normalized universal sheaf of the universal sheaf
$\mathcal{F}\in\text{Coh}(\mathrm{M}_{\beta}\times X)$ and the maps $\pi_{M}$
and $\pi_{X}$ are the canonical projection maps from $M_{\beta}\times X$ into
$M_{\beta}$ and $X$ respectively.

Descendant invariants from the _descendent insertion_ s are defined by

$\langle\tau_{1}(\gamma)\rangle_{\beta}:=\int_{[\mathbf{M}_{\beta}]^{\mathrm{vir}}}\tau_{1}(\gamma).$

######  Remark 3.4.

For $\gamma\in\mathrm{H}^{4-2i}(X,\mathbb{Z})$, the insertion becomes

$\tau_{i}(\gamma)=\pi_{M*}(\pi_{Y}^{*}\gamma\cup\\{\mathrm{ch}(\mathcal{F}_{\text{norm}})\cdot\text{td}(K_{Y})^{-1}\\}_{2+i})$

by Grothendieck-Riemann-Roch theorem.

By computing descendant invariants, we can obtain genus 1 GV invariants
$n_{1,\beta}$ via Conjecture 1.2.

###  3.1. Computations on Fano $3$-folds

It is very well-known that the following list of smooth Fano $3$-folds have
the same Betti numbers of that of $\mathbb{P}^{3}$:

$\mathbb{P}^{3},\;Q_{2}\subset\mathbb{P}^{4},\;V_{5}\subset\mathbb{P}^{6},\;V_{22}\subset\mathbb{P}^{13}.$

In special, the odd cohomology of these varieties vanish. All of varieties are
rigid except $V_{22}$. The moduli of $V_{22}$’s is six-dimensional. The first
two varieties in the list are homogeneous and the others are not. In this
section, we compute the primary and descent invariants for non-homogeneous
cases.

####  3.1.1. The case $V_{5}$

The Fano threefold $V_{5}$ is defined by the linear intersection
$V_{5}=\mathrm{Gr}(2,\mathbb{C}^{5})\cap H_{1}\cap H_{2}\cap H_{3}$ where
$H_{i}$ are the general hyperplane in
$\mathbb{P}(\wedge^{2}\mathbb{C}^{5})=\mathbb{P}^{9}$.

  * •

$\mathrm{Pic}_{\mathbb{Z}}(V_{5})\cong\mathbb{Z}\langle H\rangle$,
$\mathrm{deg}(V_{5})=5$, $K_{V_{5}}=-2H$.

  * •

The cohomology ring of $Y$ over $\mathbb{Z}$ is isomorphic to

$\mathrm{H}^{*}(Y_{5},\mathbb{Z})\cong\mathbb{Z}[h_{1},h_{2},h_{3}]/\langle
h_{1}^{2}-5h_{2},h_{1}^{3}-5h_{3},h_{1}h_{2}-h_{3},h_{2}^{2}\rangle$

where $\deg(h_{i})=2i,1\leq i\leq 3$. Moreover,
$h_{1}=c_{1}(\mathcal{O}_{Y}(1))$ and $h_{i}$ is the Poincaré dual of the
linear space of dimension $3-i$ for $i=2,3$.

  * •

Let $\mathcal{S}_{Y}$ and $\mathcal{Q}_{Y}$ be the restriction of the
universal bundles $\mathcal{S}$ and $\mathcal{Q}$ on $\mathrm{Gr}(2,5)$. The
Chern classes are

    1. (1)

$c(\mathcal{S}_{V_{5}})=1-h_{1}+2h_{2}$,

    2. (2)

$c(\mathcal{Q}_{Y})=1+h_{1}+3h_{2}+h_{3}$,

    3. (3)

$c(Y)=1+2h_{1}+12h_{2}+4h_{3}$.

Unless otherwise stated, we omit the subscription $V_{5}$ in the universal
bundles.

Let us denote by $\mathbf{M}_{d}:=\mathbf{M}_{\beta}(V_{5})$ for
$\beta=d[\text{line}]\in\mathrm{H}_{2}(V_{5},\mathbb{Z})$. For $1\leq d\leq
3$, one can easily see that $\mathbf{M}_{d}(Y)$ is isomorphic to the Hilbert
scheme $\mathbf{H}_{d}(V_{5})$ of curves with Hilbert polynomial $dm+1$ (cf.
[Chu19, Proposition 3.1]). Thus one can borrow the description of Hilbert
scheme of rational curves in $V_{5}$.

######  Proposition 3.5 ([Fae05, FN89, Ili94, San14]).

$\mathbf{M}_{1}\cong\mathbb{P}^{2}$, $\mathbf{M}_{2}\cong\mathbb{P}^{4}$,
$\mathbf{M}_{3}\cong\mathrm{Gr}(2,5)$.

######  Remark 3.6.

By taking an explicit choice of the hyperplanes $H_{i}$, the results for $d=1$
and $2$ of Proposition 3.5 has been reproved by the birational-geometric
method (For the detail, see [CHL18, Section 7]).

The universal sheaves over $\mathbf{M}_{d}$ were explicitly presented in
Proposition 2.20, Proposition 2.32, and Proposition 2.46 of [San14]. Let
$i:\mathcal{C}_{d}\hookrightarrow\mathbf{M}_{d}\times V_{5}$ be the universal
curve in $\mathbf{M}_{d}\times V_{5}$. The free resolutions of
$i_{*}\mathcal{O}_{\mathcal{C}_{d}}$ on $\mathbf{M}_{d}\times V_{5}$ are

  1. (1)

($d=1$)
$0\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(-3)\boxtimes\mathcal{S}\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(-2)\boxtimes\mathcal{Q}^{*}\rightarrow\mathcal{O}_{\mathbb{P}^{2}\times
V_{5}}\rightarrow i_{*}\mathcal{O}_{\mathcal{C}_{1}}\rightarrow 0$,

  2. (2)

($d=2$)
$0\rightarrow\mathcal{O}_{\mathbb{P}^{4}}(-2)\boxtimes\mathcal{O}_{V_{5}}(-1)\rightarrow\mathcal{O}_{\mathbb{P}^{4}}(-1)\boxtimes\mathcal{S}\rightarrow\mathcal{O}_{\mathbb{P}^{4}\times
V_{5}}\rightarrow i_{*}\mathcal{O}_{\mathcal{C}_{2}}\rightarrow 0$,

  3. (3)

($d=3$)
$0\rightarrow\mathcal{S}(-1)\boxtimes\mathcal{O}_{V_{5}}(-1)\rightarrow\mathcal{O}_{\mathrm{Gr}(2,5)}(-1)\boxtimes\mathcal{Q}(-1)\rightarrow\mathcal{O}_{\mathrm{Gr}(2,5)\times
V_{5}}\rightarrow i_{*}\mathcal{O}_{\mathcal{C}_{3}}\rightarrow 0$.

Note that each of the moduli spaces is smooth, and we can check that the
virtual class $[\mathbf{M}_{d}]^{\text{vir}}$ is the Euler class of a
K-theoretic obstruction bundle $\text{ob}_{\mathbf{M}_{d}}$. One can compute
this K-theoretic obstruction bundle by the formula

$[\text{ob}_{\mathbf{M}_{d}}]=[T_{\mathbf{M}_{d}}]+[\mathbf{R}_{\pi_{\mathbf{M}_{d}}}\mathbf{R}\mathcal{H}om_{\mathbf{M}_{d}\times
V_{5}}(i_{*}\mathcal{O}_{\mathcal{C}_{d}},i_{*}\mathcal{O}_{\mathcal{C}_{d}})[1]\
]-[\mathcal{O}]$

which appears in the proof of [CT21, Proposition 2.13].

$d$ |  $\qquad\mathbf{R}_{\pi_{\mathbf{M}_{d}}}\mathbf{R}\mathcal{H}om_{\mathbf{M}_{d}\times V_{5}}(i_{*}\mathcal{O}_{\mathcal{C}_{d}},i_{*}\mathcal{O}_{\mathcal{C}_{d}})[1]$ |  $\qquad\qquad\qquad[\text{ob}_{\mathbf{M}_{d}}]$
---|---|---
$1$ |  $-3[\mathcal{O}_{\mathbb{P}^{2}}]+3[\mathcal{O}_{\mathbb{P}^{2}}(1)]+5[\mathcal{O}_{\mathbb{P}^{2}}(2)]-5[\mathcal{O}_{\mathbb{P}^{2}}(3)]$ |  $[\mathcal{O}_{\mathbb{P}^{2}}]-5[\mathcal{O}_{\mathbb{P}^{2}}(2)]+5[\mathcal{O}_{\mathbb{P}^{2}}(3)]$
$2$ |  $-3[\mathcal{O}_{\mathbb{P}^{4}}]+10[\mathcal{O}_{\mathbb{P}^{4}}(1)]-7[\mathcal{O}_{\mathbb{P}^{4}}(2)]$ |  $[\mathcal{O}_{\mathbb{P}^{4}}]-5[\mathcal{O}_{\mathbb{P}^{4}}(1)]+7[\mathcal{O}_{\mathbb{P}^{4}}(2)]$
$3$ |  $-2[\mathcal{O}_{\mathrm{Gr}(2,5)}]+10[\mathcal{O}_{\mathrm{Gr}(2,5)}(1)]-7[\mathcal{S}^{*}(1)]+5[\mathcal{S}^{*}]-[\mathcal{S}^{*}\otimes\mathcal{S}]$ |  $[\mathcal{O}_{\mathrm{Gr}(2,5)}]-10[\mathcal{O}_{\mathrm{Gr}(2,5)}(1)]+7[\mathcal{S}^{*}(1)]-5[\mathcal{S}^{*}]+[\mathcal{S}^{*}\otimes\mathcal{S}]+[\mathcal{S}^{*}\otimes\mathcal{Q}]$.

Here, the bundles $\mathcal{S}$ and $\mathcal{Q}$ in the fourth row are the
universal sub-bundle and quotient bundle of $\mathbf{M}_{3}=\mathrm{Gr}(2,5)$.
By using the computer algebra system, Macaulay2 ([GS]), we have

######  Proposition 3.7.

The invariants $\langle\tau_{i}(h_{2-i})\rangle_{d}$ are given by the numbers
of the following table.

$d$ |  $i=0$ |  $i=1$
---|---|---
$1$ |  $5$ |  $\frac{25}{2}$
$2$ |  $35$ |  $\frac{35}{2}$
$3$ |  $490$ |  $-\frac{490}{2}$

###### Proof.

Let us present the computation of the invariants for the degree $d=3$ case.
The other cases are more simple and thus we omit it. The cohomology ring
structure of $\mathrm{Gr}(2,5)$ is very well-known as follow. Let
$m_{i}:=c_{i}(\mathcal{S})$ be the $i$-th Chern class of the universal
subbundle $\mathcal{S}$ of Grassmannian $\mathrm{Gr}(2,5)$. The cohomology
ring of $\mathrm{Gr}(2,5)$ is given by ([EH16, Theorem 5.26])

$\mathrm{H}^{*}(\mathrm{Gr}(2,5),\mathbb{Z})=\mathbb{Z}[m_{1},m_{2}]/\langle-
m_{1}^{5}+4m_{1}^{3}m_{2}-3m_{1}m_{2}^{2},\;m_{1}^{4}-3m_{1}^{2}m_{2}+m_{2}^{2}\rangle.$

Note that the dual of the point class is $[\text{point}]^{*}=m_{2}^{3}$. Then
the Chern class of the obstruction bundle $\text{ob}_{\mathbf{M}_{d}}$ is

$c(\text{ob}_{\mathbf{M}_{d}})=1-11m_{1}+(48m_{1}^{2}+7m_{2})+(-102m_{1}^{3}-56m_{1}m_{2}+451m_{1}^{2}m_{2})-78m_{2}^{2}-490m_{1}m_{2}^{2}.$

Thus the virtual class of $\mathbf{M}_{3}$ is
$[\mathbf{M}_{3}]^{\text{vir}}=-490m_{1}m_{2}^{2}$.

On the other hand, the insertion classes on
$\mathrm{H}^{*}(\mathbf{M}_{3}\times V_{5})$ are

$\begin{split}\tau_{0}(h_{2})&=m_{1}^{2}h_{2}-m_{2}h_{2}-m_{1}h_{3},\\\
\tau_{1}(h_{1})&=m_{1}^{3}h_{1}-\frac{3}{2}m_{1}m_{2}h_{1}-\frac{5}{2}m_{1}^{2}h_{2}+\frac{1}{2}m_{1}h_{3}\end{split}$

From these one, we have

$\int_{[\mathbf{M}_{3}]^{\text{vir}}}\tau_{0}(h_{2})=490m_{2}^{3}h_{3},\;\int_{[\mathbf{M}_{3}]^{\text{vir}}}\tau_{1}(h_{1})=-245m_{2}^{3}h_{3}.$

∎

######  Remark 3.8.

From the description of the universal curve $\mathcal{C}_{1}$ in [FN89, Lemma
2.1, 2.2], one can easily check that the obstruction bundle is isomorphic to
$\mathrm{ob}_{\mathbf{M}_{1}}\cong\mathcal{O}_{\mathbf{M}_{1}}(5)$ and thus
its the cohomology matches with our computation.

######  Remark 3.9.

The universal curve $\mathcal{C}_{2}$ is a regular section of the vector
bundle $\mathcal{O}_{\mathbb{P}^{4}}(1)\boxtimes\mathcal{S}^{*}$ ([San14,
Proposition 2.32]). Hence the Chern charcter is given by

$\text{ch}(i_{*}\mathcal{O}_{\mathcal{C}_{2}})=c_{2}(\mathcal{O}_{\mathbb{P}^{4}}(1)\boxtimes\mathcal{S}^{*})\cdot\text{td}(\mathcal{O}_{\mathbb{P}^{4}}(1)\boxtimes\mathcal{S}^{*})^{-1},$

and thus its cohomology class matches with our computation. In the following
subsection, we find the fundamental class of $\mathcal{C}_{3}$ by using
Porteous’ formula.

####  3.1.2. The universal cubic curves $\mathcal{C}_{3}$ via degeneracy loci

Recall that the space $\mathbf{M}_{3}$ is isomorphic to $\mathrm{Gr}(2,V)$
such that $\dim V=5$. In this subsection, we describe the universal family
$\mathcal{C}_{3}$ of cubic curves in a geometric way which confirms the
calculation of previous subsection. Let us recall the isomorphism
$\mathbf{M}_{3}\cong\mathrm{Gr}(2,V)$. Consider the Schubert variety

$\sigma_{2,0}(l):=\\{[l^{\prime}]\in\mathrm{Gr}(2,V)|l\cap
l^{\prime}\neq\emptyset\\}$

which is a degree $3$ and $4$-dimensional subvariety of $\mathrm{Gr}(2,V)$. By
taking the hyperplane sections $H_{1}\cap H_{2}\cap H_{3}$ with this
$\sigma_{2,0}(l)$, we obtain a twisted cubic curve

$C_{l}:=\sigma_{2,0}(l)\cap H_{1}\cap H_{2}\cap
H_{3}\subset\mathrm{Gr}(2,5)\cap H_{1}\cap H_{2}\cap H_{3}=V_{5},$

that is, $\pi_{V_{5}}(\pi_{\mathbf{M}_{3}}^{-1}([l]))=C_{l}$. Conversely, for
a point $p=[L]\in Y\subset\mathrm{Gr}(2,V)$, the inverse image
$\pi_{V_{5}}^{-1}(p)$ consists of the twisted cubic curves $[C_{l}]$ such that
$l\cap L\neq\emptyset$. This implies that
$\pi_{\mathbf{M}_{3}}(\pi_{V_{5}}^{-1}(p))=\sigma_{2,0}(L)\subset\mathrm{Gr}(2,V)=\mathbf{M}_{3}$.
Note that the Schubert variety $\sigma_{2,0}$ is a cone of rational normal
scroll $\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(1)^{\oplus
3})\cong\mathbb{P}^{2}\times\mathbb{P}^{1}$ in $\mathbb{P}^{6}$. Thus the
universal cubic $\mathcal{C}_{3}$ is a irreducible variety of dimension $7$.
Also, it is well-known that the Schubert variety $\sigma_{2,0}$ can be defined
by a degeneracy loci of vector bundles over Grassmannian.

######  Example 3.10.

Let $i:W\subset V$ be one-dimensional subvector space of $V$. For a line
$l=\mathbb{P}(W)\subset\mathbb{P}(V)$, let

$\phi:\mathcal{U}\rightarrow(V/W)\otimes\mathcal{O}_{\mathrm{Gr}(2,V)}$

be the canonical morphism induced by the injection $i$. Then one can check
that the degeneracy locus $D_{\leq 1}(\phi)\subset\mathrm{Gr}(2,V)$ of the map
$\phi$ whose rank is $\leq 1$ has the support $\sigma_{2,0}(l)$. Also it has
the expected dimension $\dim\mathrm{Gr}(2,V)-(2-1)\cdot(3-1)=4$. Thus, by
Porteous’ formula, the fundamental class of $D_{\leq 1}(\phi)$ is given by

$[D_{\leq
1}(\phi)]=c_{2}([(V/W)\otimes\mathcal{O}_{\mathrm{Gr}(2,V)}]-\mathcal{U}).$

In our case, by relativizing over $V_{5}$, we can find the fundamental form
$[\mathcal{C}_{3}]$ over $\mathbf{M}_{3}\times V_{5}$.

######  Proposition 3.11.

Let $\mathcal{S}_{\mathbf{M}_{3}}$ be the universal subbundle of
$\mathbf{M}_{3}$ and $\mathcal{Q}_{V_{5}}$ be the restriction of the universal
quotient bundle $V_{5}\subset\mathrm{Gr}(2,V)$. Then the fundamental class of
the universal cubic curves is given by

$[\mathcal{C}_{3}]=c_{2}(\mathcal{Q}_{V_{5}}-\mathcal{S}_{\mathbf{M}_{3}})\in\mathrm{H}^{2}(\mathbf{M}_{3}\times
Y).$

###### Proof.

Since $V_{5}\subset\mathrm{Gr}(2,V)$, we have an universal sequence

(8)
$0\rightarrow{\mathcal{S}_{V_{5}}}\rightarrow{V\otimes\mathcal{O}_{V_{5}}}\rightarrow{\mathcal{Q}_{V_{5}}}\rightarrow
0.$

Let us consider the relative Grassmannian bundle
$\mathrm{Gr}(2,V\otimes\mathcal{O}_{V_{5}})\rightarrow V_{5}$ with the
structure morphism
$\pi_{V_{5}}:\mathrm{Gr}(2,V\otimes\mathcal{O}_{V_{5}})\rightarrow V_{5}$.
Here we denote the same notation with the projection map because
$\mathrm{Gr}(2,V\otimes\mathcal{O}_{V_{5}})\cong\mathrm{Gr}(2,V)\times V_{5}$.
From the universal sequence,

$0\rightarrow{\mathcal{S}_{\mathbf{G}}}\rightarrow{\pi_{Y}^{*}(V\otimes\mathcal{O}_{V_{5}})}\rightarrow{\mathcal{Q}_{\mathbf{G}}}\rightarrow
0$

over $\mathbf{G}:=\mathrm{Gr}(2,V\otimes\mathcal{O}_{V_{5}})$ and the pull-
back of the sequence (8), we obtain a bundle morphism

$\phi_{\mathbf{G}}:\mathcal{S}_{\mathbf{G}}\rightarrow\pi_{V_{5}}^{*}\mathcal{Q}_{V_{5}}$

over $\mathbf{G}$. Note that
$\mathcal{S}_{\mathbf{G}}=\pi_{\mathbf{M}_{3}}^{*}\mathcal{S}_{\mathbf{M}_{3}}$
by its definition. The space $\mathcal{C}_{3}$ is reduced because it is a
generically reduced and Cohen–Macaulay space. Thus the degeneracy locus
$D_{\leq 1}(\phi_{\mathbf{G}})$ of the map $\phi_{\mathbf{G}}$ is
$\mathcal{C}_{3}$. By Porteous’ formula,

$[D_{\leq
1}(\phi_{\mathbf{G}})]=c_{2}(\mathcal{Q}_{V_{5}}-\mathcal{S}_{\mathbf{M}_{3}}).$

∎

######  Remark 3.12.

The Poincaré dual of the fundamental class of the universal cubic curves is

$[\mathcal{C}_{3}]=m_{1}^{2}-m_{2}-m_{1}h_{1}+3h_{2}\in\mathrm{H}^{2}(\mathbf{M}_{3}\times
V_{5}),$

which matches our computation of Subsection 3.1.1.

####  3.1.3. The case $V_{22}$

Let us recall the definition of the variety $V_{22}$. Let $\mathcal{S}$ and
$\mathcal{Q}$ be the universal bundles of $\mathrm{Gr}(3,7)$. Then $V_{22}$ is
defined as a zero section of $\wedge^{2}(\mathcal{S}^{*})^{\oplus 3}$.
Alternatively, $V_{22}$ can be regraded as a subvariety of the net of quadrics
$\mathbf{N}(4;2,3)$.

  * •

$\mathrm{Pic}_{\mathbb{Z}}(V_{22})\cong\mathbb{Z}\langle H\rangle$,
$\mathrm{deg}(V_{22})=22$, $K_{V_{22}}=-H$.

  * •

The cohomology ring of $V_{22}$ over $\mathbb{Z}$ is isomorphic to

$\mathrm{H}^{*}(V_{22},\mathbb{Z})\cong\mathbb{Z}[h_{1},h_{2},h_{3}]/\langle
h_{1}^{2}-22h_{2},h_{1}^{3}-22h_{3},h_{1}h_{2}-h_{3},h_{2}^{2}\rangle$

where $\deg(h_{i})=i,1\leq i\leq 3$. Moreover,
$h_{1}=c_{1}(\mathcal{O}_{V_{22}}(1))$ and $h_{i}$ is the Poincaré dual of the
linear space of dimension $3-i$ for $i=2,3$.

  * •

The Chern classes of tautological bundles on $V_{22}$ are

    1. (1)

$c(\mathcal{S}_{V_{22}})=1-h_{1}+10h_{2}-2h_{3},$

    2. (2)

$c(\mathcal{Q}_{V_{22}})=1+h_{1}+12h_{2}+4h_{3},$

    3. (3)

$c({V_{22}})=1+h_{1}+24h_{2}+4h_{3}$

where $\mathcal{S}_{V_{22}}$ and $\mathcal{Q}_{V_{22}}$ are the restriction of
the universal bundles $\mathcal{S}$ and $\mathcal{Q}$ on $\mathrm{Gr}(3,7)$.
Unless otherwise stated, we omit the subscription $V_{22}$ of the universal
bundles.

By the same reason as in the case of $V_{5}$, the moduli space
$\mathbf{M}_{d}(V_{22})$ is isomorphic to the Hilbert scheme
$\mathbf{H}_{d}(V_{22})$ of curves with Hilbert polynomial $dm+1$ for $1\leq
d\leq 3$. The later space $\mathbf{H}_{d}(V_{22})$ has been studied by many
authors.

######  Proposition 3.13 ([KPS18, Fae14, KS04]).

$\mathbf{M}_{1}\cong Q$, $\mathbf{M}_{2}\cong\mathbb{P}^{2}$,
$\mathbf{M}_{3}\cong\mathbb{P}^{3}$, where $Q$ is a singular planar quartic
curve.

Let us compute the degree $d=1$ case by the result of Pirola ([Pir85]). By the
Chern class computation, the virtual dimension of $\mathbf{M}_{1}$ is
$\text{virt. dim}\mathbf{M}_{1}=1$ and the virtual fundamental class is given
by the following.

######  Lemma 3.14.

$[\mathbf{M}_{1}]^{\mathrm{vir}}=[\mathbf{M}_{1}]$

###### Proof.

By the deformation invariance of DT invariants, we may assume that $V_{22}$ is
not Mukai-Umemura 3-folds. Then $M_{1}\cong Q\subset\mathbb{P}^{2}$ is a
regular embedding, hence the intrinsic normal cone
$\mathcal{C}_{\mathbf{M}_{1}}$ of $\mathbf{M}_{1}$ is given by the bundle
stack

$\mathcal{C}_{\mathbf{M}_{1}}=[N_{Q/\mathbb{P}^{2}}/T_{\mathbb{P}^{2}}|_{Q}].$

Let $E\to L_{\mathbf{M}_{1}}$ be the usual perfect obstruction theory. Then by
[BF97], we have the closed embedding

$\mathcal{C}_{\mathbf{M}_{1}}\hookrightarrow h^{1}/h^{0}(E^{\vee}).$

Since they are both bundle stacks with dimension 0 (as Artin stacks), we have
$\mathcal{C}_{\mathbf{M}_{1}}=h^{1}/h^{0}(E^{\vee})$. Hence, by the definition
of the virtual cycle in [BF97], we have

$[\mathbf{M}_{1}]^{\mathrm{vir}}=0^{!}_{{h^{1}/h^{0}(E^{\vee})}}[\mathcal{C}_{\mathbf{M}_{1}}]=0^{!}_{{h^{1}/h^{0}(E^{\vee})}}[h^{1}/h^{0}(E^{\vee})]=[\mathbf{M}_{1}].$

where the last equality comes from properties of Gysin pull-back via bundle
stacks [Kre99]. ∎

######  Proposition 3.15.

The degree $d=1$ DT invariant and descendent invariant on $V_{22}$ are given
by

$\langle\tau_{0}(h_{2})\rangle_{1}=2\;\text{and}\;\langle\tau_{1}(h_{1})\rangle_{1}=22$

###### Proof.

Let $\mathcal{C}_{1}$ be the universal curve over $\mathbf{M}_{1}(=Q)$. We
compute the invariants by using the degeneracy loci method. In [AF06, Lemma
3.1], the authors describe how to obtain lines in $V_{22}$. We relativize
their construction. Let $\mathcal{K}$ be the vector bundle on $V_{22}$ with
data $\mathrm{rk}(\mathcal{K})=5$, $c_{1}(\mathcal{K})=-2$,
$c_{2}(\mathcal{K})=40$, $c_{3}(\mathcal{K})=-20$, and
$\dim\mathrm{Hom}(\mathcal{K},\mathcal{S})=3$. Let
$B^{*}=\mathrm{Hom}(\mathcal{K},\mathcal{S})$. Note that
$Q\subset\mathbb{P}(B^{*})$. The universal curve
$\mathcal{C}_{1}\subset\mathbb{P}(B^{*})\times V_{22}$ is the degeneracy loci
of the canonical homomorphism

$\Phi:\mathcal{S}^{*}\rightarrow\mathcal{K}^{*}\boxtimes\mathcal{O}_{\mathbb{P}(B^{*})}(1).$

In fact, the map $\Phi$ is the dual of the composition of the pull-back of the
evaluation map
$\text{ev}:\mathcal{K}\otimes\mathrm{Hom}(\mathcal{K},\mathcal{S})\rightarrow\mathcal{S}$
on $V_{22}$ and the tautological map
$\mathcal{O}_{\mathbb{P}(B^{*})}(-1)\rightarrow
B\otimes\mathcal{O}_{\mathbb{P}(B^{*})}$ on $\mathbb{P}(B^{*})$.

Let
$\mathrm{H}^{\bullet}(\mathbb{P}(B^{*}),\mathbb{Z})=\mathbb{Z}[m_{1}]/\langle
m_{1}^{3}\rangle$ with $\deg(m_{1})=1$. By Proposition 3.14 in [Pir85], the
Chern character of the structure sheaf $\mathcal{O}_{\mathcal{C}_{1}}$ over
$\mathbb{P}(B^{*})\times V_{22}$ is given by

$\text{ch}(\mathcal{O}_{\mathcal{C}_{1}})=(2m_{1}^{2}h_{1}+4m_{1}h_{2})+(-8m_{1}^{2}h_{2}+2m_{1}h_{3})+(\text{the
terms of higher degree}).$

Thus, by the Grothendieck-Riemann-Roch theorem, the invariants are

$\int_{[\mathbf{M}_{1}]}\tau_{0}(h_{2})=2m_{1}^{2}h_{3}=2[pt],\;\int_{[\mathbf{M}_{1}]}\tau_{1}(h_{1})=22m_{1}^{2}h_{3}=22[pt].$

∎

######  Remark 3.16.

By Porteous’ formula, the dual class of the fundamental class
$[\mathcal{C}_{1}]$ is

$[\mathcal{C}_{1}]=2m_{1}^{2}h_{1}+4m_{1}h_{2}.$

The intersection number of $[\mathcal{C}_{1}]$ with the line class $h_{2}$ in
$V_{22}$ is $[\mathcal{C}_{1}]\cdot h_{2}=2m_{1}^{2}h_{3}$. This matches with
the fact that the degree of the surface $S$ sweeping out by lines in $V_{22}$
is $\deg(S)=2$ ([Ame98, Section 3]).

For the degree $d=2$ and $3$ cases, the universal curves over $\mathbf{M}_{d}$
have been studied in [Fae14, Lemma 4.1] and [KS04, Theorem 2.4]. Let
$i:\mathcal{C}_{d}\hookrightarrow\mathbf{M}_{d}\times V_{22}$ be the universal
curve in $\mathbf{M}_{d}\times V_{22}$. The free resolutions of
$i_{*}\mathcal{O}_{\mathcal{C}_{d}}$ on $\mathbf{M}_{d}\times V_{22}$ are

  1. (1)

($d=2$)
$0\rightarrow\mathcal{S}\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(-4)\rightarrow\mathcal{Q}^{*}\boxtimes\mathcal{O}_{\mathbb{P}^{2}}(-3)\rightarrow\mathcal{O}_{V_{22}\times\mathbb{P}^{2}}\rightarrow
i_{*}\mathcal{O}_{\mathcal{C}_{2}}\rightarrow 0$,

  2. (2)

($d=3$)
$0\rightarrow\mathcal{E}\boxtimes\mathcal{O}_{\mathbb{P}^{3}}(-3)\rightarrow\mathcal{S}\boxtimes\mathcal{O}_{\mathbb{P}^{3}}(-2)\rightarrow\mathcal{O}_{V_{22}\times\mathbb{P}^{3}}\rightarrow
i_{*}\mathcal{O}_{\mathcal{C}_{3}}\rightarrow 0$,

where $\text{rk}(\mathcal{E})=2$, $c_{1}(\mathcal{E})=-1$,
$c_{2}(\mathcal{E})=7$. By the same method for the case $V_{5}$, one can find
the (virtual) fundamental class $[\mathbf{M}_{d}]$.

$d$ |  $\qquad\mathbf{R}_{\pi_{\mathbf{M}_{d}}}\mathbf{R}\mathcal{H}om_{\mathbf{M}_{d}\times V_{22}}(i_{*}\mathcal{O}_{\mathcal{C}_{d}},i_{*}\mathcal{O}_{\mathcal{C}_{d}})[1]$ |  $\qquad\qquad\qquad[\text{ob}_{\mathbf{M}_{d}}]$
---|---|---
$2$ |  $-3[\mathcal{O}_{\mathbb{P}^{2}}]+7[\mathcal{O}_{\mathbb{P}^{2}}(3)]-7[\mathcal{O}_{\mathbb{P}^{2}}(4)]+3[\mathcal{O}_{\mathbb{P}^{2}}(1)]$ |  $[\mathcal{O}_{\mathbb{P}^{2}}]-7[\mathcal{O}_{\mathbb{P}^{2}}(3)]+7[\mathcal{O}_{\mathbb{P}^{2}}(4)]$
$3$ |  $-3[\mathcal{O}_{\mathbb{P}^{3}}]+7[\mathcal{O}_{\mathbb{P}^{3}}(2)]-8[\mathcal{O}_{\mathbb{P}^{3}}(3)]+4[\mathcal{O}_{\mathbb{P}^{3}}(1)]$ |  $[\mathcal{O}_{\mathbb{P}^{3}}]-7[\mathcal{O}_{\mathbb{P}^{2}}(2)]+8[\mathcal{O}_{\mathbb{P}^{2}}(3)]$

Therefore we have

######  Proposition 3.17.

The invariants $\langle\tau_{i}(h_{2-i})\rangle_{d}$ are given by the numbers
of the following table.

$d$ |  $i=0$ |  $i=1$
---|---|---
$1$ |  $2$ |  $22$
$2$ |  $7$ |  $28$
$3$ |  $28$ |  $28$

By combining Proposition 2.2, Proposition 3.7 and Proposition 3.17, we have

######  Theorem 3.18.

Conjecture 1.1 (which is equivalent to Conjecture 3.2) is true when $X=V_{5}$
and $V_{22}$ up to the degree $3$.

##  4\. Proof of Conjecture 1.2

Let us recall the definition of meeting invariants
$m_{\beta_{1},\beta_{2}}\in\mathbb{Z}$ for
$\beta_{1},\beta_{2}\in\mathrm{H}_{2}(X,\mathbb{Z})$ ([KP08, Section 0.3]). It
is given by the following rules:

  1. (1)

$m_{\beta_{1},\beta_{2}}=m_{\beta_{2},\beta_{1}}$.

  2. (2)

If either $\deg(\beta_{1})\leq 0$ or $\deg(\beta_{2})\leq 0$, then
$m_{\beta_{1},\beta_{2}}=0$.

Let $\\{S_{1},\cdots,S_{k}\\}$ be the basis of the torsion free part of
$\mathrm{H}^{4}(X,\mathbb{Z})$. Let $(g^{ij})$ be the inverse matrix of the
intersection matrix $(g_{ij})$, $g_{ij}=\langle S_{i},S_{j}\rangle$.

  3. (3)

If $\beta_{1}\neq\beta_{2}$,

$m_{\beta_{1},\beta_{2}}=\sum_{i,j}n_{0,\beta_{1}}(S_{i})g^{ij}n_{0,\beta_{2}}(S_{j})+m_{\beta_{1},\beta_{2}-\beta_{1}}+m_{\beta_{1}-\beta_{2},\beta_{2}}.$

  4. (4)

If $\beta_{1}=\beta_{2}=\beta$, then

$m_{\beta,\beta}=n_{0,\beta}(c_{2}(X))+\sum_{i,j}n_{0,\beta}(S_{i})g^{ij}n_{0,\beta}(S_{j})-\sum_{\beta_{1}+\beta_{2}=\beta}m_{\beta_{1},\beta_{2}}.$

We recall Conjecture 1.2 here. For each
$\gamma\in\mathrm{H}^{2}(X,\mathbb{Z})$,

$\langle\tau_{1}(\gamma)\rangle_{\beta}=\frac{n_{0,\beta}(\gamma^{2})}{2(\gamma\cdot\beta)}-\sum_{\beta_{1}+\beta_{2}=\beta}\frac{(\gamma\cdot\beta_{1})(\gamma\cdot\beta_{2})}{4(\gamma\cdot\beta)}m_{\beta_{1},\beta_{2}}-\sum_{k\geq
1,k|\beta}\frac{(\gamma\cdot\beta)}{k}n_{1,\beta/k}$

######  Remark 4.1.

It is believed that the invariants $n_{1,d}$ come from a space of elliptic
curves in $Y$ (cf. [KP08, Section 3 and 5]). The defining equations of $V_{5}$
and $V_{22}$ can be generated by quadric equations and they does not contain
planes. Thus the space of elliptic curves of degree $d\leq 3$ should be empty.
This implies that the invariants are $n_{1,d}=0$.

In this section, we will prove the following.

######  Theorem 4.2.

Conjecture 1.2 holds for $Y=V_{5}$ and $V_{22}$ when we assume that
$n_{1,d}=0$ for $1\leq d\leq 3$.

The remaining two subsections is devoted to the proof of Theorem 4.2.

###  4.1. The case $V_{5}$

Let $X=|K_{Y}|$, $\bar{X}=\mathbb{P}(K_{Y}\oplus\mathcal{O}_{Y})$ and
$\pi:\bar{X}\to Y$ be the canonical projection map. By construction, we have

$\mathrm{H}^{4}(\bar{X},\mathbb{Z})=\langle T_{1},T_{2}\rangle,$

where $T_{1}=\text{PD}(H\cap Y)$, $T_{2}=\pi^{*}(\text{PD}(L_{1}))$ for linear
spaces $L_{i}$ of dimension $i$. Since the normal bundle of $H\cap Y$ in $X$
is $\mathcal{N}_{H\cap
Y,X}\cong\pi^{*}(\mathcal{O}_{Y}(1)\oplus\mathcal{O}_{Y}(-2))$, $T_{1}\cdot
T_{1}=-2H^{3}\cap Y=-10$. Hence the intersection matrix is given by

$(g_{ij})=$ $T_{1}$ $T_{2}$ $T_{1}$ $-10$ $1$ $T_{2}$ $1$ $0$ ,
$g^{ij}=(g_{ij})^{-1}=\left(\begin{matrix}0&1\\\ 1&10\\\ \end{matrix}\right).$

From $\mathcal{T}_{X}=\pi^{*}(\mathcal{T}_{Y}\oplus\mathcal{O}_{Y}(-2))$, we
have $c(X)=\pi^{*}(c(Y)\cdot(1-2h_{1}))$. Hence $c_{2}(X)=-8T_{2}$. Also we
have $T_{1}|_{Y}=(T_{1}\cdot T_{1})T_{2}|_{Y}=-10T_{2}|_{Y}$. Using the
formula of meeting invariants, we have

$\displaystyle m_{1,1}$
$\displaystyle=n_{0,1}(-8T_{2})+2n_{0,1}(T_{1})n_{0,1}(T_{2})+10n_{0,1}(T_{2})^{2}$
$\displaystyle=40-20\times 5^{2}+10\times 5^{2}=-210,$ $\displaystyle m_{1,2}$
$\displaystyle=n_{0,1}(T_{1})n_{0,2}(T_{2})+n_{0,1}(T_{2})n_{0,2}(T_{1})+10n_{0,1}(T_{2})n_{0,2}(T_{2})+m_{1,1}$
$\displaystyle=-10n_{0,1}(T_{2})n_{0,2}(T_{2})-210=-1960.$

Motivated from the fact that the spaces of genus one curves on $Y$ with degree
$d=1,2,3$ are empty sets, let us assume that $n_{1,1}=n_{1,2}=n_{1,3}=0$. Then
Conjecture 1.2 has been written as

$\langle\tau_{1}(H)\rangle_{d}=\frac{n_{0,d}(H^{2})}{2d}-\sum_{d_{1}+d_{2}=d}\frac{d_{1}\cdot
d_{2}}{4d}m_{d_{1},d_{2}}.$

Note that we need to choose some suitable orientation of the moduli space.
Here we use the orientation in [CT21, (0.7)]. By a direction calculation, one
can check the identity as follows.

$\begin{split}-\langle\tau_{1}(H)\rangle_{1}&=\frac{5n_{0,1}(T_{2})}{2},\;-\langle\tau_{1}(H)\rangle_{2}=\frac{5n_{0,2}(T_{2})}{4}-\frac{1}{8}m_{1,1},\\\
-\langle\tau_{1}(H)\rangle_{3}&=\frac{5n_{0,3}(T_{2})}{6}-\frac{1}{3}m_{1,2}\end{split}$

Therefore the suitable choice of sign for the conjecture should be $-1$.

###  4.2. The case $V_{22}$

Let $Y=V_{22}$, $X=|K_{V_{22}}|$, and
$\bar{X}=\mathbb{P}(K_{Y}\oplus\mathcal{O}_{Y})$ and $\pi:\bar{X}\to Y$ be the
canonical projection map. Let $T_{1}$ and $T_{2}$ be generators of the
cohomology group $\mathrm{H}^{2}(X,\mathbb{Z})$, defined by

$T_{1}:=PD(H\cap Y),\ T_{2}:=\pi^{*}(PD(L_{1}))$

where $L_{1}$ is a class of line, equal to $(H^{2}/22)\cap Y$. Then the
intersection matrix is computed by:

$(g_{ij})=$ $T_{1}$ $T_{2}$ $T_{1}$ $-22$ $1$ $T_{2}$ $1$ $0$ ,
$g^{ij}=(g_{ij})^{-1}=\left(\begin{matrix}0&1\\\ 1&22\\\ \end{matrix}\right)$

Note that we can check $c_{2}(X)=2T_{2}$ by direct calculation. Also, by the
same manner as did in the case $Y=Y_{5}$, the meeting invariants are

$m_{1,1}=-84,\ m_{1,2}=224.$

Under the assumption $n_{1,1}=n_{1,2}=n_{1,3}=0$, we have the identities:

$\begin{split}\langle\tau_{1}(H)\rangle_{1}&=\frac{22n_{0,1}(T_{2})}{2},\;-\langle\tau_{1}(H)\rangle_{2}=\frac{22n_{0,2}(T_{2})}{4}-\frac{1}{8}m_{1,1},\\\
\langle\tau_{1}(H)\rangle_{3}&=\frac{22n_{0,3}(T_{2})}{6}-\frac{1}{3}m_{1,2},\end{split}$

which confirms Theorem 4.2.

## References

  * [AF06] Enrique Arrondo and Daniele Faenzi.  Vector bundles with no intermediate cohomology on Fano threefolds of type $V_{22}$.  Pacific Journal of Mathematics, 225(2):201–220, 2006.
  * [Ame98] E. Amerik.  On a problem of Noether Lefschetz type.  Compositio Mathematica, 112:255–271, 1998.
  * [Beh96] K. Behrend.  Gromov-Witten invariants in algebraic geometry.  arXiv preprint arXiv:alg-geom/9601011, 1996.
  * [BF97] K. Behrend and B. Fantechi.  The intrinsic normal cone.  Invent. Math., 128(1):45–88, 1997.
  * [BJ17] Dennis Borisov and Dominic Joyce.  Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds.  Geometry & Topology, 21(6):3231–3311, 2017.
  * [Cao19a] Yalong Cao.  Counting conics on sextic 4-folds.  Mathematical Research Letters, 26(5):1343–1357, 2019.
  * [Cao19b] Yalong Cao.  Genus zero-gopakumar-vafa type invariants for calabi-yau $4$-folds ii: Fano 3-folds.  Commun. Contemp. Math., 1950060:1–25, 2019.
  * [CHL18] Kiryong Chung, Jaehyun Hong, and SangHyeon Lee.  Geometry of moduli spaces of rational curves in linear sections of Grassmannian $\text{Gr}(2,5)$.  Journal of Pure and Applied Algebra, 222(4):868 – 888, 2018.
  * [Chu19] Kiryong Chung.  A desingularization of Kontsevich’s compactification of twisted cubics in $V_{5}$.  arXiv:1902.01658, 2019.
  * [CK18] Yalong Cao and Martijn Kool.  Zero-dimensional Donaldson-Thomas invariants of Calabi–Yau 4-folds.  Advances in Mathematics, 338:601–648, 2018.
  * [CK20] Yalong Cao and Martijn Kool.  Curve counting and DT/PT correspondence for Calabi-Yau 4-folds.  Advances in Mathematics, 375:107371, 2020.
  * [CKM21] Yalong Cao, Martijn Kool, and Sergej Monavari.  Stable pair invariants of local Calabi–Yau 4-folds.  International Mathematics Research Notices, 04 2021.  rnab061.
  * [CL14] Yalong Cao and N. C. Leung.  Donaldson-Thomas theory for Calabi-Yau 4-folds.  arXiv:1407.7659, 2014.
  * [CMT18] Yalong Cao, Davesh Maulik, and Yukinobu Toda.  Genus zero Gopakumar-Vafa type invariants for Calabi-Yau 4-folds.  Advances in Mathematics, 338:41–92, 2018.
  * [CMT21] Yalong Cao, Davesh Maulik, and Yukinobu Toda.  Stable pairs and Gopakumar-Vafa type invariants for Calabi-Yau 4-folds.  Journal of the European Mathematical Society, 2021.
  * [CT21] Yalong Cao and Yukinobu Toda.  Gopakumar-Vafa type invariants on Calabi-Yau 4-folds via descendent insertions.  Communications in Mathematical Physics, 383(1):281–310, 2021.
  * [EH16] David Eisenbud and Joseph Harris.  3264 and All That: A Second Course in Algebraic Geometry.  Cambridge: Cambridge University Press, 2016.
  * [Fae05] Daniele Faenzi.  Bundles on the Fano threefold $V_{5}$.  Commun. Algebra, 33(9):3061–3080, 2005.
  * [Fae14] Daniele Faenzi.  Even and odd instanton bundles on Fano threefolds of Picard number one.  manuscripta math, 144:199–239, 2014.
  * [FN89] Mikio Furushima and Noboru Nakayama.  The family of lines on the Fano threefold $V_{5}$.  Nagoya Mathematical Journal, 116:111–122, 1989.
  * [FP00] Carel Faber and Rahul Pandharipande.  Hodge integrals and Gromov-Witten theory.  Inventiones mathematicae, 139(1):173–199, 2000.
  * [GP99] T. Graber and R. Pandharipande.  Localization of virtual classes.  Invent. Math., 135(2):487–518, 1999.
  * [GS] Daniel R. Grayson and Michael E. Stillman.  Macaulay2, a software system for research in algebraic geometry.  Available at http://www.math.uiuc.edu/Macaulay2/.
  * [HTK+03] Kentaro Hori, Richard Thomas, Sheldon Katz, Cumrun Vafa, Rahul Pandharipande, Albrecht Klemm, Ravi Vakil, and Eric Zaslow.  Mirror symmetry, volume 1.  American Mathematical Soc., 2003.
  * [Ili94] Atanas Iliev.  The fano surface of the Gushel threefold.  Compositio Mathematica, 94(1):81–107, 1994.
  * [KKP03] Bumsig Kim, Andrew Kresch, and Tony Pantev.  Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee.  Journal of Pure and Applied Algebra, 179(1-2):127–136, 2003.
  * [KP08] A. Klemm and R. Pandharipande.  Enumerative geometry of Calabi-Yau 4-folds.  Commun. Math. Phys., 281:621–653, 2008.
  * [KPS18] Alexander G. Kuznetsov, Yuri G. Prokhorov, and Constantin A. Shramov.  Hilbert schemes of lines and conics and automorphism groups of Fano threefolds.  Japan. J. Math., pages 685–789, 2018.
  * [Kre99] Andrew Kresch.  Cycle groups for Artin stacks.  Invent. Math., 138(3):495–536, 1999.
  * [KS04] J. Koll´ar and F.-O. Schreyer., editors.  Real Fano 3-folds of type $V_{22}$, volume 62. Dipartimento di Matematica dell’Universit‘a di Torino, 2004.
  * [OT20] Jeongseok Oh and Richard P Thomas.  Counting sheaves on Calabi-Yau 4-folds, i.  arXiv preprint arXiv:2009.05542, 2020.
  * [Pir85] Gian Pietro Pirola.  Chern character of degeneracy loci and curves of special divisors.  Annali di Matematica Pura ed Applicata, 142(1):77–90, 1985.
  * [San14] Giangiacomo Sanna.  Rational curves and instantons on the Fano threefold $Y_{5}$.  arXiv:1411.7994, 2014.
  * [Tho00] R. P. Thomas.  A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations.  Journal of Differential Geometry, 54(2):367–438, 2000.