Hodge Numbers of Generalised Borcea-Voisin Threefolds

TL;DR

This paper rederives formulas for the Hodge numbers of Borcea-Voisin type Calabi-Yau threefolds using orbifold cohomology and Euler characteristic methods.

Contribution

It provides a new proof of existing formulas for Hodge numbers of these threefolds through orbifold techniques.

Findings

Reproof of Hodge number formulas for Borcea-Voisin threefolds

Application of orbifold cohomology to Calabi-Yau geometry

Use of orbifold Euler characteristic in Hodge number calculations

Abstract

We shall reproof formulas for the Hodge numbers of Calabi-Yau threefolds of Borcea-Voisin type constructed by A. Cattaneo and A. Garbagnati, using the orbifold cohomology formula and the orbifold Euler characteristic.

Equations

$$h^{1,1}=11+5N^{\prime }-N\text{and}{h}^{2,1}=11+5N-N^{\prime },$$

$$\left(\begin{array}{cc}{\zeta }_p^{t+1}& 0\\ 0& {\zeta }_p^{p-t}\end{array}\right),\text{for }t=0,1,\dots ,p-2.$$

$$H_{\text{orb}}^{i,j}(X/G):=\underset{[g]\in \text{Conj}(G)}{⨁}{\left(\underset{U\in \Lambda (g)}{⨁}{H}^{i-\mathrm{age}(g),j-\mathrm{age}(g)}(U)\right)}^{\text{C}(g)},$$

$$H_{\text{orb}}^{i,j}(X/G)=H^{i,j}(X{)}^G\oplus \underset{U\in \Lambda (\alpha )}{⨁}\underset{k=1}{\overset{p-1}{⨁}}{H}^{i-\mathrm{age}({\alpha }^k),j-\mathrm{age}({\alpha }^k)}(U).$$

$$h^{i,j}(\widetilde{X/G})=h_{\mathrm{orb}}^{i,j}(X/G).$$

$$e_{\mathrm{orb}}(X/G):=\frac{1}{\#G}\sum _{\begin{array}{c}(g,h)\in G\times G\\ gh=hg\end{array}}e(X^g\cap X^h).$$

$$e(\widetilde{X/G})=e_{\mathrm{orb}}(X/G).$$

$$H^2(S,\mathbb{C})=H^2(S,\mathbb{C}{)}^{{\alpha }_S}\oplus H^2(S,\mathbb{C}{)}_{-1},$$

$${\stackrel{{\scriptstyle\mbox{$H^{i,j}(E,\mathbb{C})^{\alpha_{E}}$}}}{{\begin{matrix}&&&&\\ &&1&&\\ &0&&0&\\ &&1&&\\ \end{matrix}}}}\\ \quad\quad\quad\quad\quad\quad{\stackrel{{\scriptstyle\mbox{$H^{i,j}(E,\mathbb{C})_{-1}$}}}{{\begin{matrix}&&&&\\ &&0&&\\ &1&&1&\\ &&0&&\\ \end{matrix}}}}$$

$$\left(\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right)$$

$$h^{i,j}(\widetilde{S\times E/C_2})=h_{\mathrm{orb}}^{i,j}(S\times E/C_2)=h^{i,j}(S\times E{)}^{C_2}\oplus \underset{U\in \Lambda (-1)}{⨁}{h}^{i-1,j-1}(U),$$

$$h^{1,1}(\widetilde{S\times E/C_2})=r+1+4N\text{and}{h}^{2,1}(\widetilde{S\times E/C_2})=m-1+4N^{\prime }.$$

$$e(S/{\alpha }_S)=12+N-N^{\prime },$$

$$H^2(S,\mathbb{C})=H^2(S,\mathbb{C}{)}^{{\alpha }_S}\oplus H^2(S,\mathbb{C}{)}_{{\zeta }_3}\oplus H^2(S,\mathbb{C}{)}_{{\zeta }_3^2}.$$

$${\alpha }_S^{\ast }([{\omega }_S\wedge {\omega }_E])=\overline{{\zeta }_3}{\zeta }_3[{\omega }_S\wedge {\omega }_E]=[{\omega }_S\wedge {\omega }_E],$$

$${\stackrel{{\scriptstyle\mbox{$H^{i,j}(E,\mathbb{C})^{\alpha_{E}^{2}}$}}}{{\begin{matrix}&&&&\\ &&1&&\\ &0&&0&\\ &&1&&\\ \end{matrix}}}}\ \ \ \quad\quad\quad\quad\quad{\stackrel{{\scriptstyle\mbox{$H^{i,j}(E)_{\zeta_{3}}$}}}{{\begin{matrix}&&&&\\ &&0&&\\ &1&&0&\\ &&0&&\\ \end{matrix}}}}\ \ \ \quad\quad\quad\quad\quad{\stackrel{{\scriptstyle\mbox{$H^{i,j}(E)_{\zeta_{3}^{2}}$}}}{{\begin{matrix}&&&&\\ &&0&&\\ &0&&1&\\ &&0&&\\ \end{matrix}}}}$$

$$E\ni (x,y)\mapsto ({\zeta }_3^{2}x,y)\in E,$$

$$\left(\begin{array}{ccc}1& 0& 0\\ 0& {\zeta }_3& 0\\ 0& 0& {\zeta }_3^2\end{array}\right)\text{and}\left(\begin{array}{ccc}{\zeta }_3^2& 0& 0\\ 0& {\zeta }_3^2& 0\\ 0& 0& {\zeta }_3^2\end{array}\right).$$

$$\left(\begin{array}{ccc}{\zeta }_3& 0& 0\\ 0& {\zeta }_3& 0\\ 0& 0& {\zeta }_3\end{array}\right)\text{and}\left(\begin{array}{ccc}1& 0& 0\\ 0& {\zeta }_3^2& 0\\ 0& 0& {\zeta }_3\end{array}\right),$$

$$H_{\mathrm{orb}}^{i,j}(S\times E/C_3)=H^{i,j}(S\times E{)}^{C_3}\oplus \underset{U\in \Lambda ({\zeta }_3)}{⨁}\underset{\iota =1}{\overset{2}{⨁}}{H}^{i-\mathrm{age}({\alpha }^{\iota }),j-\mathrm{age}({\alpha }^{\iota })}(U)=$$

$$=H^{i,j}(S\times E{)}^{C_3}\oplus \left(\underset{U\in \mathcal{C}}{⨁}{H}^{i-1,j-1}(U)\oplus H^{i-1,j-1}(U)\right)\oplus$$

$$\oplus \left(\underset{U\in \mathcal{R}}{⨁}{H}^{i-1,j-1}(U)\oplus H^{i-1,j-1}(U)\right)\oplus \left(\underset{U\in \mathcal{P}}{⨁}{H}^{i-1,j-1}(U)\right).$$

$$h^{1,1}(\widetilde{X/C_3})=r+1+6\cdot 1+2\cdot (3k-3)\cdot 1+3n\cdot 1=r+1+3n+6k.$$

$$h^{1,2}(\widetilde{X/C_3})=m-1+2\cdot 3\cdot g(C)+(3k-3)\cdot 2\cdot 0+3n\cdot 0=m-1+6g(C).$$

$$h^{1,1}=r+1+3n+6k\text{and}{h}^{2,1}=m-1+6g(C).$$

$${\alpha }_E(x,y)=(-x,iy).$$

$$h_{\mathrm{orb}}^{1,1}(X{)}^{C_4}=r+1\text{and}{h}_{\mathrm{orb}}^{2,1}(X{)}^{C_4}=m-1.$$

$$M_g:=\underset{U\in \Lambda (g)}{⨁}{H}^{1-\mathrm{age}(g),1-\mathrm{age}(g)}(U).$$

$$E\ni (x,y)\mapsto (-x,-iy)\in E,$$

$$\left(\begin{array}{ccc}1& 0& 0\\ 0& i& 0\\ 0& 0& -i\end{array}\right),$$

$$\left(\begin{array}{ccc}-1& 0& 0\\ 0& -i& 0\\ 0& 0& -i\end{array}\right),$$

$$E\ni (x,y)\mapsto (x,-y)\in E,$$

Full text

This work was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.

1. Introduction

One of the many reasons behind the interest in non-symplectic automorphisms of $K3$ surfaces is the mirror symmetry construction of C. Borcea ([4]) and C. Voisin ([14]). They independently constructed a family of Calabi-Yau threefolds using a non-symplectic involutions of $K3$ surfaces and elliptic curves. Moreover C. Voisin gave a construction of explicit mirror maps.

1.1 Theorem** ([4] and [14]).**

Let $E$ be an elliptic curve with an involution $\alpha_{E}$ which does not preserve $\omega_{E}$. Let $S$ be a $K3$ surface with a non-symplectic involution $\alpha_{S}$. Then any crepant resolution of the variety $(E\times S)/(\alpha_{E}\times\alpha_{S})$ is a Calabi-Yau manifold with

[TABLE]

where $N$ is a number of curves in $S^{\alpha_{S}}$ and $N^{\prime}$ is a sum of their genera.

In [6] A. Cattaneo and A. Garbagnati generalised Borcea-Voisin construction using purely non-symplectic automorphisms of order 3, 4 and 6. They obtained the following theorem

1.2 Theorem** ([6]).**

Let $S$ be a $K3$ surface admitting a purely non-symplectic automorphism $\alpha_{S}$ of order $n.$ Let $E$ be an elliptic curve admitting an automorphism $\alpha_{E}$ such that $\alpha_{E}(\omega_{E})=\zeta_{n}\omega_{E}.$ Then $n\in\{2,3,4,6\}$ and $({S\times E})/(\alpha_{S}\times\alpha_{E}^{n-1})$ is a singular variety which admits a crepant resolution which is a Calabi-Yau manifold.

Any crepant resolution of $({S\times E})/(\alpha_{S}\times\alpha_{E}^{n-1})$ will be called a Calabi-Yau 3-fold of Borcea-Voisin type. For all possible orders they computed the Hodge numbers of this varieties and constructed an elliptic fibrations on them. Their computations are more technical and rely on a detailed study of a crepant resolutions of threefolds.

In this paper we give shorter computations of the Hodge numbers using orbifold cohomology introducted by W. Chen and Y. Ruan in [7] and orbifold Euler characteristic (cf. [12]). The main advantage of our approach is that the computations are carried out on $S\times E.$

2. Preliminaries

A Calabi-Yau manifold $X$ is a complex, smooth, projective $d$-fold $X$ satisfying

1. (1)

$K_{X}=\mathcal{O}_{X},$ 2. (2)

$H^{i}\mathcal{O}_{X}=0$ for $0<i<d.$

A non-trivial generator $\omega_{X}$ of $H^{d,0}(X)\simeq\mathbb{C}$ is called a period of $X.$ For any automorphism $\alpha_{X}\in\textup{Aut}(X),$ the induced mapping $\alpha_{X}^{*}$ acts on $H^{d,0}(X)$ and $\alpha_{X}^{*}(\omega_{X})=\lambda_{\alpha}\omega_{X},$ for some $\lambda_{\alpha}\in\mathbb{C}^{*}.$

In the case of a $K3$ surface $S$ automorphism $\alpha_{S}\in\textup{Aut}(X)$ which preserves a period is called symplectic. If $\alpha_{S}$ does not preserve a period then it is called non-symplectic. If additionally $\alpha_{S}$ is of finite order $n$ and $\alpha_{S}^{*}(\omega_{S})=\zeta_{n}\omega_{S},$ where $\zeta_{n}$ is a primitive $n$-th root of unity, then it is called purely non-symplectic.

We have the following characterisation of orders of purely non-symplectic automorphisms of elliptic curves and $K3$ surfaces.

2.1 Theorem** ([13]).**

Let $E$ be an elliptic curve. If $\alpha_{E}$ is an automorphism of $E$ which does not preserve the period, then $\alpha_{E}(\omega_{E})=\zeta_{n}\omega_{E}$ where $n=2,3,4,6.$

2.2 Theorem** ([11]).**

Let $S$ be a $K3$ surface and $\alpha_{S}$ be a purely non-symplectic automorphism of order $n.$ Then $n\leq 66$ and if $n=p$ is a prime number, then $p\leq 19.$

Let $\sigma$ be a purely non-symplectic automorphism of order $p$ of a $K3$ surface $S.$ We will denote by $S^{\sigma}$ the set of fixed points of $\sigma.$ The action of $\sigma$ may be locally linearized and diagonalized at $p\in S^{\sigma}$ (cf. Cartan [5]), so the possible local actions are

[TABLE]

Clearly, if $t=0,$ then $p$ belongs to a smooth curve fixed by $\sigma,$ otherwise $p$ is an isolated point. We have the following description of the fixed locus of $\sigma.$

2.3 Theorem** ([3], Lemma 2.2, p. 5).**

Let $S$ be a $K3$ surface and let $\alpha_{S}$ be a non-symplectic automorphism of S of order $n$. Then there are three possibilities

• •

$S^{\alpha_{S}}=\emptyset;$* in this case $n=2,$*

• •

$S^{\alpha_{S}}=E_{1}\cup E_{2},$* where $E_{1},$ $E_{2}$ are disjoint smooth elliptic curves; in this case $n=2,$*

• •

$S^{\alpha_{S}}=C\cup R_{1}\cup R_{2}\cup\ldots R_{k-1}\cup\{p_{1},p_{2},\ldots,p_{h}\},$* where $p_{i}$ are isolated fixed points, $R_{i}$ are smooth rational curves and $C$ is the curve with highest genus $g(C).$*

We refer to [3] for a proof and more precise description of the fixed locus for particular values of $n$.

3. Orbifold’s cohomology

In [7] W. Chen and Y. Ruan introduced a new cohomology theory for orbifolds. We consider varieties $X/G,$ where $X$ is a projective variety and $G$ is a finite group acting on $X$ viewed as orbifold.

3.1 Definition**.**

For $G\in GL_{n}(\mathbb{C})$ of order $m,$ let $e^{2\pi ia_{1}},e^{2\pi ia_{2}},\ldots,e^{2\pi ia_{n}}$ be eigenvalues of $G$ for some $a_{1},a_{2},\ldots,a_{n}\in[0,1)\cap\mathbb{Q}$. The value of the sum $a_{1}+a_{2}+\ldots+a_{n}$ is called the age of $G$ and is denoted by $\operatorname{age}(G).$

The age of $G$ is an integer if and only if $\det G=1$ i.e. $G\in SL_{n}(\mathbb{C})$.

3.2 Definition**.**

For a variety $X/G$ define the Chen-Ruan cohomology by

[TABLE]

where $\textrm{Conj}(G)$ is the set of conjugacy classes of $G$ (we choose a representative $g$ of each conjugacy class), $\textup{C}(g)$ is the centralizer of $g$, $\Lambda(g)$ denotes the set of irreducible connected components of the set fixed by $g\in G$ and $\operatorname{age}(g)$ is the age of the matrix of linearized action of $g$ near a point of $U.$

The dimension of $H_{\textrm{orb}}^{i,j}(X/G)$ will be denoted by $h_{\text{orb}}^{i,j}(X/G).$

3.3 Remark*.*

If the group $G$ is cyclic of a prime order $p$, then we can pick a generator $\alpha$ and the above formula simplifies to

[TABLE]

We have the following theorem.

3.4 Theorem** ([15], Theorem 1.1, p. 2).**

Let $G$ be a finite group acting on an algebraic smooth variety $X$. If there exists a crepant resolution $\widetilde{X/G}$ of variety $X/G,$ then the following equality holds

[TABLE]

4. Orbifold Euler characteristic

Let $G$ be a finite group acting on an algebraic variety $X.$ In a similar manner as in the case of Hodge numbers, we can use an orbifold formula to compute the Euler characteristic of a crepant resolution of $X/G$ (for details see [12]).

4.1 Definition**.**

The orbifold Euler characteristic of $X/G$ is defined as

[TABLE]

4.2 Theorem** ([12], Theorem 2, p. 534).**

For any finite abelian group $G$ acting on smooth algebraic variety $X$. If there exists a crepant resolution $\widetilde{X/G}$ of variety $X/G,$ then the following equality holds

[TABLE]

5. Computations of Hodge numbers

5.1. Order 2

Let $(S,\alpha_{S})$ be a $K3$ surface admitting a non-symplectic involution $\alpha_{S}$. Consider an elliptic curve $E$ with non-symplectic involution $\alpha_{E}$ (any elliptic curve $E$ admits such an automorphism). Let us denote by $H^{2}(S,\mathbb{C})^{\alpha_{S}}$ the invariant part of cohomology $H^{2}(S,\mathbb{C})$ under $\alpha_{S}$ and by $r$ the dimension $r=\dim H^{2}(S,\mathbb{C})^{\alpha_{S}}.$ We also denote the eigenspace for $-1$ of the induced action $\alpha_{S}^{*}$ on $H^{2}(S,\mathbb{C})$ by $H^{2}(S,\mathbb{C})_{-1}$ and by $m$ the dimension $m=\dim H^{2}(S,\mathbb{C})_{-1}.$

We see that

[TABLE]

hence the Hodge diamonds of the respective eigenspaces have the following forms

The Hodge diamonds of eigenspaces of the induced action of $\alpha_{E}^{*}$ on $H^{1}(E,\mathbb{C})$ have forms

[TABLE]

By Künneth’s formula the Hodge diamond of $H^{3}(S\times E,\mathbb{C})^{C_{2}}$ is given by

The local action of $-1$ on curve may be linearized to matrix

[TABLE]

with age equal to 1. Thus by 4.2

[TABLE]

which gives formulas

[TABLE]

Since the quotient $S/\alpha_{S}$ is a smooth surface with Euler characteristic

[TABLE]

we recover formulas from Thm. 1.1 of [6].

5.2. Order 3

Let $(S,\alpha_{S})$ be a $K3$ surface admitting a purely non-symplectic automorphism $\alpha_{S}$ of order $3.$ Eigenvalues of induced mapping $\alpha_{S}^{*}$ on $H^{2}(S,\mathbb{C})$ belong to $\{1,\zeta_{3},\zeta_{3}^{2}\}.$ Let us denote by $H^{2}(S,\mathbb{C})_{\zeta_{3}^{i}}$ the eigenspace of the eigenvalue $\zeta_{3}^{i}.$ For $i=1,2$ the dimension of $H^{2}(S,\mathbb{C})_{\zeta_{3}^{i}}$ does not depend on $i$ and will be denoted by $m.$ Moreover let $r$ be a dimension of $H^{2}(S,\mathbb{C})^{\alpha_{S}}$ — invariant part of $H^{2}(S,\mathbb{C})$ under $\alpha_{S}.$

Consider an elliptic curve $E$ with the Weierstrass equation $y^{2}=x^{3}+1$ together with a non-symplectic automorphism $\alpha_{E}$ of order 3 such that $\alpha_{E}(x,y)=(\zeta_{3}x,y).$

We see that

[TABLE]

Because \alpha_{S}^{*}\bigr{|}_{H^{2,0}(S,\mathbb{C})}([\omega])=\zeta_{3}[\omega] for any $[\omega]\in H^{2,0}(S),$ we get $H^{2,0}(S)\subset H(S,\mathbb{C})_{\zeta_{3}}.$

The complex conjugation yields $H^{0,2}(S)\subset H^{2}(S,\mathbb{C})_{\zeta_{3}^{2}}.$ Finally

[TABLE]

hence the Hodge diamonds of the respective eigenspaces have the following forms

Similar analysis gives the Hodge diamonds of eigenspaces of action on the Hodge groups.

[TABLE]

By Künneth’s formula the Hodge diamond of the invariant part of $H^{3}(S\times E,\mathbb{C})$ has the same form as in the case of order 2.

We denote the automorphism $\alpha_{S}\times\alpha_{E}^{2}$ by $\alpha$. Let us now consider possible actions of elements of $\langle\alpha\rangle\simeq C_{3}$ on $S$ and $E.$

The action of $\alpha$. The action of the automorphism $\alpha$ on $E$ is given by

[TABLE]

hence it has three fixed points. Locally the action of $\alpha$ on components of the fixed locus can be diagonalised to

[TABLE]

It follows that ages are equal to 1 and 2 respectively.

The action of $\alpha^{2}$. Analogously we gets possible diagonalised matrices

[TABLE]

with ages 1 and 1.

Thus decomposing $(S\times E)^{\alpha}=\mathcal{C}\cup\mathcal{R}\cup\mathcal{P}$, where

1. \mathcal{C}:=\{\textup{3 curves with highest genus g(C)}\},
2. \mathcal{R}:=\{\textup{3k-3 rational curves}\},
3. \mathcal{P}:=\{\textup{3n isolated points}\},

the orbifold formula implies that

[TABLE]

Therefore by 3.4

[TABLE]

Hence we proved the following theorem:

5.1 Theorem**.**

If $S^{\alpha_{S}}$ consists of $k$ curves together with a curve with highest genus $g(C)$ and $n$ isolated points, then for any crepant resolution of the variety $(S\times E)/(\alpha_{S}\times\alpha_{E}^{2})$ the following holds

[TABLE]

5.3. Order 4

Let $(S,\alpha_{S})$ be a $K3$ surface with purely non-symplectic automorphism $\alpha_{S}$ of order $4.$ Consider an elliptic curve $E$ with the Weierstrass equation $y^{2}=x^{3}+x$ together with a non-symplectic automorphism $\alpha_{E}$ of order 4 such that

[TABLE]

Additionally, suppose that $S^{\alpha_{S}^{2}}$ is not a union of two elliptic curves.

We shall keep the notation of [6].

1. $X=S\times E,$
2. $P$ – the infinity point of $E,$
3. $r=\dim H^{2}(S,\mathbb{C})^{\alpha_{S}},$
4. $m=\dim H^{2}(S,\mathbb{C})_{\zeta_{6}^{i}}$ for $i\in\{1,2,\ldots,5\},$
5. $N$ – number of curves which are fixed by $\alpha_{S}^{2},$
6. $k$ – number of curves which are fixed by $\alpha_{S}$ (curves of the first type).
7. $b$ – number of curves which are fixed by $\alpha_{S}^{2}$ and are invariant by $\alpha_{S}$ (curves of the second type),
8. $a$ – number of pairs $(A,A^{\prime})$ of curves which are fixed by $\alpha_{S}^{2}$ and $\alpha_{S}(A)=A^{\prime}$ (curves of the third type),
9. $D$ – the curve of the highest genus in $S^{\alpha_{S}^{2}},$
10. $n_{1}$ – number of curves which are fixed by $\alpha_{S}$ not laying on the curve $D,$
11. $n_{2}$ – number of curves which are fixed by $\alpha_{S}$ laying on the curve $D.$

For the same reasons as in the previous cases

[TABLE]

We denote an automorphism $\alpha_{S}\times\alpha_{E}^{3}$ by $\alpha$. Furthermore for any $g\in C_{4}$ let

[TABLE]

We will consider all possible cases.

The action of $\alpha$ and $\alpha^{3}$. The action of automorphism $\alpha$ on $E$ is given by

[TABLE]

hence it has two fixed points — $P$ and $(0,0)$. The fixed locus of $\alpha$ on $S$ consists of $k$ curves and $n_{1}+n_{2}$ isolated points. Since locally the action of $\alpha$ on $X$ along the curve can be diagonalised to

[TABLE]

we infer that its age equals $1.$ Near a fixed point we have a matrix

[TABLE]

hence the age equals 2.

In case of the action of $\alpha^{3}$ we observe that the fixed locus consists of $k$ curves and $n_{1}+n_{2}$ points with ages $1.$ We see that the summand of $h^{1,1}_{\textup{orb}}$ from both actions is equal to $2k+2k+2(n_{1}+n_{2})=4k+2(n_{1}+n_{2})$.

[TABLE]

The action of $\alpha^{2}$. The automorphism $\alpha^{2}$ acts on $E$ as

[TABLE]

hence it has four fixed points — $P$, $(0,0)$, $(i,0),$ $(-i,0)$ from which only two are invariant under the action of $\alpha$ and the other two are permuted. After identifying $M_{\zeta_{4}^{2}}$ with the vector space spanned by irreducible components we will find the action of induced map $\alpha^{*}$ on it.

Because the matrix of the action of ${\alpha_{E}^{2}}^{*}$ on $M_{\zeta_{4}^{2}}$ is

[TABLE]

it follows that it has 3-dimensional eigenspace for $+1$ and 1-dimensional eigenspace for $-1.$

The fixed locus of $\alpha_{S}^{2}$ consists of $N$ curves with $a$ pairs permuted by $\alpha_{S}.$ Hence $\alpha_{S}^{*}$ on $M_{\zeta_{4}^{2}}$ has $(N-a)$-dimensional eigenspace for $+1$ and $a$-dimensional eigenspace for $-1.$ One can see that $N=k+b+2a,$ so the total effect on $h^{1,1}_{\textup{orb}}$ equals $3(N-a)+a=3k+3b+4a.$

[TABLE]

From the orbifold formula follows that

[TABLE]

In order to compute $h^{1,2}_{\textup{orb}}$ we will use the orbifold Euler characteristic. In the table below we collect all possible intersections $X^{g}\cap X^{h},$ where $(g,h)\in C_{4}^{2}.$

[TABLE]

By 4.2 we obtain the formula

[TABLE]

If $D$ is of the first type, then by the Riemann-Hurwitz formula we conclude that

[TABLE]

Since $\widetilde{X/C_{4}}$ is Calabi-Yau we gets

[TABLE]

By ([2], Thm. 1.1) and ([2], Prop. 1) we have the following relations

[TABLE]

where $\displaystyle h=\sum_{C\subseteq S^{\alpha_{S}}}(1-g(C)).$ Moreover since $D$ is of the first type $h=k-g(D),$ $n_{2}=0,$ $n_{1}=2h+4$ and $\displaystyle b=\frac{n_{1}}{2}$, thus

[TABLE]

If $D$ is of the second type, then analogously the Riemann-Hurwitz formula yields

[TABLE]

Thus

[TABLE]

Using the additional relations $h=k,$ $n_{1}+n_{2}=2h+4$ and $b=\displaystyle\frac{n_{1}}{2}+1,$ we get

[TABLE]

Hence we proved the following theorem:

5.2 Theorem** ([6], Proposition 6.3).**

If $S^{\alpha_{S}^{2}}$ is not a sum of two elliptic curves, then for any crepant resolution of variety $X/C_{4}$ the following formulas hold

• •

If $D$ is of the first type, then

[TABLE]

• •

If $D$ is of the second type, then

[TABLE]

5.4. Order 6

Let $(S,\gamma_{S})$ be a $K3$ surface with purely non-symplectic automorphism $\gamma_{S}$ of order $6.$ Consider an elliptic curve $E$ with the Weierstrass equation $y^{2}=x^{3}+1$ together with a non-symplectic automorphism $\gamma_{E}$ of order 6 such that

[TABLE]

We shall keep the notation of [6].

1. $X=S\times E,$
2. $P$ – the infinity point of $E,$
3. $r=\dim H^{2}(S,\mathbb{C})^{\gamma_{S}},$
4. $m=\dim H^{2}(S,\mathbb{C})_{\zeta_{6}^{i}}$ for $i\in\{1,2,\ldots,5\},$
5. $l$ – number of curves fixed by $\gamma_{S},$
6. $k$ – number of curves fixed by $\gamma_{S}^{2},$
7. $N$ – number of curves fixed by $\gamma_{S}^{3},$
8. $p_{(2,5)}+p_{(3,4)}$ – number of isolated points fixed by $\gamma_{S}$ of type $(2,5)$ and $(3,4)$ i.e. the action of $\gamma_{S}$ near the point linearizes to respectively

[TABLE] 9. $n$ – number of isolated points fixed by $\gamma_{S}^{2},$ 10. $2n^{\prime}$ – number of isolated points fixed by $\gamma_{S}^{2}$ and switched by $\gamma_{S},$ 11. $a$ – number of triples $(A,A^{\prime},A^{\prime\prime})$ of curves fixed by $\gamma_{S}^{3}$ such that $\gamma_{S}(A)=A^{\prime}$ and $\gamma_{S}(A^{\prime})=A^{\prime\prime},$ 12. $b$ – number of pairs $(B,B^{\prime})$ of curves fixed by $\alpha_{S}^{2}$ such that $\gamma_{S}(B)=B^{\prime},$ 13. $D$ – the curve with the highest genus in the fixed locus of $\gamma_{S},$ 14. $G$ – the curve with the highest genus in the fixed locus of $\gamma_{S}^{2},$ 15. $F_{1},F_{2}$ – the curves with the highest genus in the fixed locus of $\gamma_{S}^{3},$

5.3 Remark*.*

From ([9], Thm. 4.1) follows that $g(D)\in\{0,1\}.$ Moreover by [3] we see that if $g(F_{1})\neq 0,$ $g(F_{2})\neq 0,$ then $g(F_{1})=g(F_{2})=1.$ Clearly if $g(D)=1,$ then $D=G\in\{F_{1},F_{2}\}.$

Denote by $\gamma$ an automorphism $\gamma_{S}\times\gamma_{E}^{5}$. Clearly $\langle\gamma\rangle\simeq C_{6}.$ Similar computations as in the previous cases imply that

[TABLE]

For any $g\in C_{6}$ let

[TABLE]

The action of $\gamma$ and $\gamma^{5}$. An automorphism $\gamma$ acts on $E$ by $\gamma(x,y)=(\zeta_{6}^{4}x,-y),$ hence it has only one fixed point — $P$. The fixed locus of $\gamma$ on $S$ consists of $l$ curves and $p_{(2,5)}+p_{(3,4)}$ isolated points. Locally the action of $\gamma$ on $S\times E$ along the curve can be diagonalised to a matrix

[TABLE]

with age equal to $1.$ In the fixed point of type $(2,5)$ and $(3,4)$ we get respectively a matrices

[TABLE]

hence their ages equal 2.

In case of the action of $\gamma^{5}$ we observe that locus consists of $l$ curves and $p_{(2,5)}+p_{(3,4)}$ isolated points. Along the curve we have a matrix

[TABLE]

while in fixed points we have a matrices

[TABLE]

hence their ages are equal to 1. The above analysis shows that the effect on $h^{1,1}_{\textup{orb}}$ from both actions equals $l+l+p_{(2,5)}+p_{(3,4)}=2l+p_{(2,5)}+p_{(3,4)}$.

[TABLE]

The action of $\gamma^{2}$ and $\gamma^{4}$. Automorphisms $\gamma^{2}$ and $\gamma^{4}$ act on $E$ by $\gamma^{2}(x,y)=(\zeta_{6}^{2}x,y)$ and $\gamma^{4}(x,y)=(\zeta_{6}^{4}x,y)$, they have three fixed points — $\{P,(0,i),(0,-i)\}$ from which only $P$ is invariant under $\gamma$ and the remaining two are switched. Identifying $M_{\zeta_{6}^{2}}$ and $M_{\zeta_{6}^{4}}$ with the vector space spanning by irreducible components we will find the action of induced map $\gamma^{*}$ on it.

The matrix of the action of $\gamma^{*}$ on $M_{\zeta_{6}^{2}}$ is

[TABLE]

hence it produces a 2-dimensional eigenspace for $+1$ and 1-dimensional eigenspace for $-1.$ We have similar decomposition in the case of the action $\gamma^{*}$ on $M_{\zeta_{6}^{2}}$.

The fixed locus of $\gamma_{S}^{2}$ consists of $p_{(2,5)}$ and $n^{\prime}$ pairs of points switched by $\gamma_{S}.$ Locally, the action of $\gamma^{2}$ at point has a matrix

[TABLE]

with age equals 2. In the case of automorphism $\gamma^{4}$ we have a matrix

[TABLE]

with age equal 1.

Notice that both $\gamma_{S}^{2}$ and $\gamma_{S}^{4}$ has the same fixed points but with different ages. Thus $\gamma_{S}^{*}$ on $M_{\zeta_{6}^{2}}$ has $(p_{2,5}^{1}+n_{1})$-dimensional eigenspace for $+1$ and $n^{\prime}_{1}$-dimensional eigenspace for $-1,$ while $\alpha_{S}^{*}$ on $M_{\zeta_{6}^{4}}$ has $(p_{2,5}^{2}+n_{2})$-dimensional eigenspace for $+1$ and $n_{2}$-dimensional eigenspace for $-1,$ where $p_{2,5}^{1},$ $p_{2,5}^{2},$ $n^{\prime}_{1}$ and $n^{\prime}_{2}$ are naturals defined as $n^{\prime}_{1}+n^{\prime}_{2}=n^{\prime}$ and $p_{2,5}^{1}+p_{2,5}^{2}=p_{2,5}.$

Moreover, $a$ pairs of curves in the locus of $\gamma_{S}^{2}$ are switched by $\gamma.$ Hence $\gamma_{S}^{*}$ has $(k-b)$-dimensional eigenspace for $+1$ and $b$-dimensional eigenspace for $-1.$ The same decomposition we will obtain in case of the action of $\gamma^{4}.$

By the Künneth formula both actions add to $h^{1,1}_{\operatorname{orb}}$

[TABLE]

[TABLE]

The action of $\gamma^{3}.$ An automorphism $\gamma^{3}$ acts on $E$ by $\gamma^{3}(x,y)=(x,-y),$ hence it has four fixed points — $\{P,(1,0),(\zeta_{3},0),(\zeta_{3}^{2},0)\}$ from which only $P$ is invariant under $\gamma$ and the remaining three form a 3-cycle. Thus the action of ${\gamma_{E}^{5}}^{*}$ on $M_{\zeta_{6}^{3}}$ has the matrix

[TABLE]

so it produces 2-dimensional eigenspace for 1, 1-dimensional eigenspace for $\zeta_{3}$ and 1-dimensional eigenspace for $\zeta_{3}^{2}.$

Clearly the local action of $\gamma^{3}$ on curve has age equal to 1. The locus of $\gamma_{S}^{3}$ consists of $N$ curves with $a$ triples of curves permuted by $\gamma_{S}.$ Thus $\gamma_{S}^{*}$ on $M_{\zeta_{6}^{3}}$ has $(N-2a)$-dimensional eigenspace for +1 and two $a$-dimensional eigenspaces for $\zeta_{3}$ and $\zeta_{3}^{2}.$

The effect on $h^{1,1}_{\operatorname{orb}}$ equals $2(N-2a)+a+a=2N-2a.$

[TABLE]

Consequently, by the orbifold formula we see that

[TABLE]

By the orbifold cohomology formula we see that non-zero contribution to $h^{1,2}_{\textup{orb}}$ have only curves in $\Lambda(g)$ for any $g\in C_{6}.$

If $g(D)\geq 1,$ by 5.3 we can assume that $D=G=F_{1}.$ We see that contributions of $\gamma$ and $\gamma^{5}$ equal to $2g(D).$ The automorphisms $\gamma^{2}$ and $\gamma^{4}$ have three fixed points on $E$ with one $2$-point orbit, hence by Künneth’s formula the effect on $h^{1,2}_{\textup{orb}}$ in this case equals $4g(D).$ Since $\gamma^{3}$ has four fixed points with 3-points orbit, we find that it’s contribution is equal to $2g(D)+g(F_{2})+g(F_{2}/\gamma_{S}).$ Thus

[TABLE]

Now consider the case $g(D)=0.$ By the same argument as above, we see that contribution of $\gamma^{2}$ and $\gamma^{4}$ equals $2g(G)+2g(G/\gamma_{S}),$ while the summand from $\gamma^{3}$ equals $g(F_{21})+g(F_{1}/\gamma_{S})+g(F_{2})+g(F_{2}/\gamma_{S}),$ hence

[TABLE]

We proved the following theorem:

5.4 Theorem** ([6], Proposition 7.3).**

For any crepant resolution of variety $X/C_{6}$ the following formulas hold

[TABLE]

Now, we will compute $h^{2,1}_{\operatorname{orb}}$ using orbifold Euler characteristic. In the table below we collect all possible intersections $X^{g}\cap X^{h},$ where $(g,h)\in C_{6}^{2}.$

[TABLE]

From 4.2 we see that

[TABLE]

Thus

[TABLE]

By the Riemann-Hurwitz formula we obtain

[TABLE]

hence after simplifying

[TABLE]

Comparing both formulas we get:

5.5 Corollary**.**

With the notation above, the following relation holds

[TABLE]

Acknowledgments

This paper is a part of author’s master thesis. I am deeply grateful to my advisor Sławomir Cynk for recommending me to learn this area and his help.