Hodge-Deligne and Poincaré polynomials of $\overline{\mathcal M}_{1,n}$

Introduction

The moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves of genus $g$ with $n$ marked points is one of the most important objects in algebraic geometry [10, 20]. It parametrizes connected projective curves $C$ of arithmetic genus $g$ with at worst nodal singularities together with $n$ ordered smooth marked points $p_1,\ldots,p_n$ in $C$ such that

1. $p_i$ are all distinct and

2. $\omega_C(\sum_i p_i)$ is ample.

It is a smooth proper Deligne-Mumford stack with projective coarse moduli space. It plays a fundamental role in modern enumerative geometry and serves as a model for moduli spaces of higher dimensional varieties. For over half a century, much effort has been exerted to understand the geometry and topology of $\overline{\mathcal{M}}_{g,n}$ and there has been a constant stream of new discoveries even until now [1, 2, 4, 7, 8, 9, 17].

In this paper, we focus on the cohomology of $\overline{\mathcal{M}}_{1,n}$. Getzler [12] found a formula for the $S_n$-equivariant Hodge-Deligne polynomials of $\overline{\mathcal{M}}_{1,n}$ for all $n$, by the theory of modular operads [14]. It expresses them in terms of the equivariant Hodge-Deligne polynomials of the open moduli spaces $\mathcal{M}_{1,n}$, computed in [13] by means of the Eichler-Shimura isomorphism, and of genus zero data, combined by plethystic composition. This determines the Hodge numbers of $\overline{\mathcal{M}}_{1,n}$ together with the $S_n$-action on its cohomology. Getzler listed the Hodge-Deligne polynomials of $\overline{\mathcal{M}}_{1,n}$ for $n \le15$ [12], and the $S_n$-equivariant ones were tabulated for $n \le20$ by Bergström [3].

In genus zero, on the other hand, the Poincaré polynomial of $\overline{\mathcal{M}}_{0,n}$ is given by an explicit recursive formula [21 (4.12)], which can be proved by Keel’s blowup construction [18]. Therefore it seems reasonable to ask the following.

The purpose of this paper is to provide a positive answer to this question. This gives a new proof of Getzler’s results on the Hodge numbers of $\overline{\mathcal{M}}_{1,n}$, by elementary formulas which involve only binomial coefficients, Stirling numbers and dimensions of spaces of cusp forms, and an effective way to compute the Hodge and Betti numbers of $\overline{\mathcal{M}}_{1,n}$ for large $n$. Since our recursion comes from explicit smooth blowups, it also holds for any invariant satisfying the blowup formula, such as Chow groups.

Our computation of $H^*(\overline{\mathcal{M}}_{1,n})$ is achieved in two stages. We first reduce the computation to a simpler moduli space $\overline{\mathcal{M}}_{1,(0+)^n}$, where $0^+$ denotes a positive rational number sufficiently close to $0$. We note that many of the strata in $\overline{\mathcal{M}}_{1,n}$ disappear and the computation becomes much easier if we allow all the marked points to coincide. This happens in the moduli space $\overline{\mathcal{M}}_{1,(0+)^n}$ of stable pointed curves with weight $0^+$ for each marked point [16]. By studying wall crossings in Hassett’s moduli spaces of weighted pointed curves, we prove the following.

The theorem is equivalent to [17 Theorem A]. Our proof comes from explicit blowups and gives us a Hodge conjecture comparison (Corollary 3.2).

Next we compute the cohomology of $\overline{\mathcal{M}}_{1,(0+)^n}$ piece by piece. Let

be the morphism that forgets all the marked points except the first. The inverse image $\mathcal{I}_n=\pi^{-1}(\mathcal{M}_{1,1})$ of $\mathcal{M}_{1,1}$ is the $(n-1)$-fold fiber product of the universal curve

and we have

by the relative Künneth formula, where $V=R^1q_*\mathbb{Q}$. Note that $H_c^*(\mathcal{M}_{1,1},V^{\otimes k})=0$ for $k$ odd by the action of the generic stabilizer $-I\in\operatorname{SL}_2(\mathbb{Z})$. By expanding the right hand side of (1.3) and applying the Clebsch-Gordan rule, the computation of $H_c^*(\mathcal{I}_n)$ is reduced to that of $H_c^*(\mathcal{M}_{1,1},\operatorname{Sym}^{2j}V)$ which can be obtained by the Eichler-Shimura isomorphism (cf. Lemma 4.2). Combining these, we find that the Hodge-Deligne polynomial of the interior $\mathcal{I}_n$ is (cf. Proposition 4.1)

where

The complement $B_n=\pi^{-1}(\infty)=\overline{\mathcal{M}}_{1,(0+)^n}-\mathcal{I}_n$ consists of pointed rational nodal curves. Let $B_{n,j}\subset\overline{\mathcal{M}}_{1,(0+)^n}$ be the locus of pointed rational curves whose dual graphs are necklaces with $j$ vertices. It is obvious that there are $\gamma_{n,j}$ dual graphs with $j$ vertices, where

and $\binom{n}{j}$ denotes the Stirling number of the second kind. For $j \ge3$, each connected component of $\mathcal{B}_{n,j}$ is isomorphic to $(\mathbb{C}^{*})^{n-j}$ and hence $H_c^*(\mathcal{B}_{n,j})$ has Hodge-Deligne polynomial $\gamma_{n,j}(uv-1)^{n-j}$. For $j=1,2$, we have to consider an additional action of $S_2$ which acts on $\mathbb{C}^{*}$ by $z \mapsto z^{-1}$. Combining all these, we find that the Hodge-Deligne polynomial of the boundary $\mathcal{B}_n$ is

By (4) and (5), we have the following.

The Hodge-Deligne polynomial $\mathbf{H}_{n|0}$ of $\overline{\mathcal{M}}_{1,n}=\overline{\mathcal{M}}_{1,n|0}$ can be computed by the recursive formula (2) together with (1) and (6).

Letting $u=v=-t$, we get the Poincaré polynomial of $\overline{\mathcal{M}}_{1,n}$. Note that only the last line in (6) has odd degree terms.

It is straightforward to implement the above theorem into a computer program and we can compute the Hodge-Deligne and Poincaré polynomials of $\overline{\mathcal{M}}_{1,n}$ for large $n$ within a relatively short period of time.

Motivated by recent discoveries on the cohomology of $\overline{\mathcal{M}}_{0,n}$ [1, 2, 4, 7, 8, 9], we raise the following.

Question 1.4. (1) The sequence of blowups in § 3 is not $S_n$-equivariant. Can the method of this paper be refined to compute the $S_n$-equivariant Hodge-Deligne polynomials of $\overline{\mathcal{M}}_{1,n}$, giving a new proof of Getzler’s formula [12, 13]?

(2) Do the coefficients of the even degree Poincaré polynomial $P^{\mathrm{even}}_{\overline{\mathcal{M}}_{1,n}}(x)=\sum_{i=0}^{n}x^i\dim H^{2i}(\overline{\mathcal{M}}_{1,n})$ of $\overline{\mathcal{M}}_{1,n}$ form a log-concave or $k$-ultra log-concave sequence for some $k$?

(3) Does the probability distribution of the coefficients of $P^{\mathrm{even}}_{\overline{\mathcal{M}}_{1,n}}(x)/P^{\mathrm{even}}_{\overline{\mathcal{M}}_{1,n}}(1)$ converge to a Gaussian distribution?

(4) Can we find effective estimates of the Betti numbers of $\overline{\mathcal{M}}_{1,n}$ for large $n$?

(5) Can we also use the method in this paper to calculate the Hodge-Deligne and Poincaré polynomials of $\overline{\mathcal{M}}_{g,n}$ for $g \ge2$? Note that Theorem 3.1 holds for all $g \ge1$ and hence it suffices to calculate the Hodge-Deligne polynomial of $\overline{\mathcal{M}}_{g,0|n}$.

In subsequent papers, we will continue our investigation to seek answers to the above questions.

Convention. All schemes and stacks in this paper are defined over the complex number field $\mathbb{C}$ and all cohomology groups have rational coefficients.

Acknowledgement. It is my pleasure to thank Gergely Bérczi, Jonas Bergström, Jinwon Choi, Shiyue Li and Han-Bom Moon for useful discussions and comments.

Moduli spaces of weighted pointed curves and Hodge-Deligne polynomials

In this section, we collect necessary facts about Hassett moduli spaces [16] and Hodge-Deligne polynomials that will be used throughout this paper.

Let $\overline{\mathcal{M}}_{g,\mathcal{A}}$ denote the moduli stack of $\mathcal{A}$-stable $n$-pointed curves of genus $g$. In particular, when $a_i>1/2$ for all $i$, $\overline{\mathcal{M}}_{g,\mathcal{A}}$ coincides with the Knudsen-Deligne-Mumford moduli stack $\overline{\mathcal{M}}_{g,n}$ of stable curves of genus $g$ with $n$ marked points [10, 20].

Let us summarize useful facts about $\overline{\mathcal{M}}_{g,\mathcal{A}}$ from [16].

Proof. (1) is [16 Theorem 2.1] and (2) is [16 Theorem 4.1] while (3) is [16 Remark 4.6] and (4) is [16 Corollary 4.7]. $\square$

In this paper, we will consider invariants $\mathbf{H}$ of smooth proper Deligne-Mumford stacks which assign an element $\mathbf{H}_X$ of a ring to each $X$ over $\mathbb{C}$ satisfying the blowup formula: If $X\to Y$ is the smooth blowup along $Z$ of codimension $c$, then

For instance, $\mathbf{H}$ may be the Hodge-Deligne or the Poincaré polynomial of $X$. Our result in §3 (Theorem 3.1) relies only on the blowup formula (2.2).

For the computation in §4 (Theorem 4.4), we will let $\mathbf{H}$ be the Hodge-Deligne polynomial

where $\operatorname{gr}_F^p$ and $\operatorname{gr}_{p+q}^W$ denote the graded parts with respect to the Hodge filtration and weight filtration respectively. Since the mixed Hodge structure is natural, the Hodge-Deligne polynomial is a motivic invariant in the sense that

(1) $\mathbf{H}_X=\mathbf{H}_{X-Z}+\mathbf{H}_Z$ for any closed $Z\subset X$ and

(2) $\mathbf{H}_{X\times Y}=\mathbf{H}_X\cdot\mathbf{H}_Y$.

By letting $u=v=-t$, we obtain the virtual Poincaré polynomial

of $X$ which is also a motivic invariant.

Reduction to minimally weighted curves

In this section, we prove an explicit formula (Theorem 3.1) that calculates the invariants of $\overline{\mathcal{M}}_{g,n}=\overline{\mathcal{M}}_{g,1^n}$ from those of $\overline{\mathcal{M}}_{g,(0^+)^n}$ where $(0^+)^n\in(0,1]^n$ denotes an element sufficiently close to zero, for $g\geq1$.

Fix $g\geq1$ and $n\geq1$. Then $\overline{\mathcal{M}}_{g,A}$ is a nonempty proper smooth Deligne-Mumford stack for any $A\in(0,1]^n$. For $0\leq r<n$ and $0<s\leq n-r$, let

where $1$ is repeated $r-1$ times and $\frac{1}{n-r}$ is repeated $n-r$ times. For instance, $A_{0,s}^{(n)}=\left(\frac{1}{n},\ldots,\frac{1}{n}\right)$ for any $s$. Let

In particular, $Y_{0,n}^{(n)}=\overline{\mathcal{M}}_{g,(0^+)^n}$ and

Moreover, by Theorem 2.2 (4),

is the moduli space of stable curves of genus $g$ with $r$ heavy points and $n-r$ light points [17].

Note that if $r>r'$ or $r=r'$ and $s>s'$, then $A_{r,s}^{(n)}-A_{r',s'}^{(n)}\geq0$ and hence we have the natural morphism

by Theorem 2.2 (2).

Fix an invariant $\mathbf{H}$ satisfying the blowup formula and let

be the invariant of $\overline{\mathcal{M}}_{g,r|s}$. Then we have $\mathbf{H}_{0|n}=\mathbf{H}_{\overline{\mathcal{M}}_{g,(0^+)^n}}$ and $\mathbf{H}_{n|0}=\mathbf{H}_{\overline{\mathcal{M}}_{g,n}}$ by (3.4).

Now our goal is to reach $Y_{n-2,2}^{(n)}=\overline{\mathcal{M}}_{g,n}$ starting from $Y_{0,n}^{(n)}=\overline{\mathcal{M}}_{g,(0^+)^n}$ by smooth blowups so that we can reduce the computation of $\mathbf{H}_{n|0}$ to $\mathbf{H}_{0|n}$. Let us first reach $Y_{1,n-1}^{(n)}$. The natural map

is the smooth blowup along the locus $\overline{\mathcal{M}}_{g,1|0}=\overline{\mathcal{M}}_{g,1}$ of $p_1=p_2=\cdots=p_n$ by Theorem 2.2 (3). Likewise, for $1\leq k\leq n-3$, the natural map

is the smooth blowup along the locus of $p_1=p_{i_1}=\cdots=p_{i_{n-k-1}}$ for $\{i_1,\ldots,i_{n-k-1}\}\subset\{2,\ldots,n\}$ which consists of $\binom{n-1}{k}$ copies of $\overline{\mathcal{M}}_{g,1|k}$. Note that $Y^{(n)}_{1,n-2}=Y^{(n)}_{1,n-1}$ by (12). By applying the blowup formula (8) to these morphisms, we find that

Next, let us reach $Y^{(n)}_{2,n-2}$ from $Y^{(n)}_{1,n-1}$. The natural map

is the smooth blowup along the locus $\overline{\mathcal{M}}_{g,2|0}=\overline{\mathcal{M}}_{g,2}$ of $p_2=p_3=\cdots=p_n$ by Theorem 2.2 (3). For $1\leq k\leq n-4$, the natural map

is the smooth blowup along the locus of $p_2=p_{i_1}=\cdots=p_{i_{n-k-2}}$ for $\{i_1,\ldots,i_{n-k-2}\}\subset\{3,\ldots,n\}$ which consists of $\binom{n-2}{k}$ copies of $\overline{\mathcal{M}}_{g,2|k}$. We have $Y^{(n)}_{2,n-3}=Y^{(n)}_{2,n-2}$ by (12) and the blowup formula (8) gives us

Repeating this way, we obtain the following.

The above factorization of the natural morphism

into smooth blowups in the proof of Theorem 3.1 tells us that the Chow motive of $\overline{\mathcal{M}}_{g,n}$ is a direct sum of Tate twists of the Chow motives of $\overline{\mathcal{M}}_{g,(0+)m}$ for $m\leq n$. Moreover, recall that the Hodge conjecture is about the surjectivity of the cycle class map

and that we have the blowup formulas for the Chow and ordinary cohomology groups [11, 15]. So the following are straightforward from the proof of Theorem 3.1.

Note that the Hodge conjecture holds on $\overline{\mathcal{M}}_{1,n}$ for any $n \ge1$ by (26).

Moduli spaces of minimally weighted curves

In this section, we let $g = 1$ and let $\mathbf{H}$ be the Hodge-Deligne polynomial. We find an explicit formula that calculates the Hodge-Deligne polynomials of the moduli spaces $\overline{\mathcal{M}}_{1,(0+)^n}$ of stable curves of genus $1$ with $n$ marked points whose weights are $\le1/n$. By Theorem 3.1, we can then calculate $\mathbf{H}_{\overline{\mathcal{M}}_{1,n}}$ for all $n > 0$.

Note that in $\overline{\mathcal{M}}_{1,(0+)^n}$, all the marked points are allowed to coincide as long as they stay away from the nodes. Let

be the morphism that forgets all the marked points except the first. Using the stratification

where $\infty$ parameterizes the unique singular rational curve, we have the decomposition

into the interior and the boundary.

Let $I_n = \mathbf{H}_{\mathcal{I}_n}$ and $B_n = \mathbf{H}_{\mathcal{B}_n}$ be the Hodge-Deligne polynomials of $\mathcal{I}_n$ and $\mathcal{B}_n$ respectively so that

is the Hodge-Deligne polynomial of $\overline{\mathcal{M}}_{1,(0+)^n}$.

The Hodge-Deligne Polynomial $I_n$ of $\mathcal{I}_n$

In this subsection, we prove the following.

Let $q : \mathcal{C} \to\mathcal{M}_{1,1}$ denote the universal curve. By our choice of stability condition, all the marked points are allowed to coincide and hence $\mathcal{I}_n$ is nothing but the $(n-1)$-fold fiber product

which is a smooth projective fibration with fiber $E^{n-1}$ for an elliptic curve $E \in\mathcal{M}_{1,1}$.

Let $V = R^1q_*\mathbb{Q}$. Then $V$ is a local system of rank $2$ over $\mathcal{M}_{1,1}$ and

Let $\pi' = \pi|_{\mathcal{I}_n}$. By the relative Künneth theorem, we have

where $C_k = \left(\mathbb{Q} \oplus\mathbb{Q}(-1)[-2]\right)^{\otimes n-1-k}$.

Since $\mathcal{M}_{1,1} \cong SL_2(\mathbb{Z})\backslash\mathbb{H}$ where $\mathbb{H}$ denotes the upper half plane and the generic stabilizer $-I \in SL_2(\mathbb{Z})$ acts on $V^{\otimes k}$ as $(-1)^k$, we have the vanishing

Hence (19) gives us

By the Clebsch-Gordan rule and (18), we have

Therefore (20) gives us

Since $H_c^*(\mathcal{M}_{1,1},\mathbb{Q})\cong\mathbb{Q}(-1)[-2]$, Proposition 4.1 follows immediately from the following.

Proof. Let $j:\mathcal{M}_{1,1}\to\overline{\mathcal{M}}_{1,1}$ denote the inclusion map. Then we have an exact triangle

By the Eichler-Shimura isomorphism [23 Theorem 8.4], we have

where $S_{2k+2}$ has Hodge type $(2k+1,0)$. We thus find that $H_c^*(\mathcal{M}_{1,1},V_{2k})=H_c^1(\mathcal{M}_{1,1},V_{2k})$ and have an exact sequence

As the cusp $\infty$ has the stabilizer group generated by

$H^0(\infty,j_*V_{2k}|_{\infty})$ is the $\sigma$-invariant part $(\operatorname{Sym}^{2k}\mathbb{Q}^2)^\sigma\cong\mathbb{Q}$. This completes the proof. □

It is well known that $\dim S_{2k+2}=\beta_{2k+2}$ for $k>0$ by Riemann-Roch, using the notation of (18).

The Hodge-Deligne Polynomial $B_n$ of $B_n$

Let $\gamma_{n,1}=1$, $\gamma_{n,2}=2^{n-1}-1$ and for $k>2$, let

where $\binom{n}{k}$ denotes the Stirling number of the second kind, i.e. the number of ways to partition $[n]=\{1,2,\cdots,n\}$ into $k$ unordered subsets. In this subsection, we prove the following.

See Example 5.2 for the calculation of $B_n$ for $n\leq10$.

Observe that we have a decomposition

where $B_{n,j}$ denotes the locus of curves with $j$ rational components i.e. the dual graph is a necklace with $j$ vertices.

If $(C,p_1,\cdots,p_n)\in B_{n,1}$, $C$ is a rational curve with normalization $\mathbb{P}^1$. Let $0,\infty\in\mathbb{P}^1$ map to the node of $C$. Then we find that

where $-1\in S_2$ acts by $(z_1,\cdots,z_{n-1})\mapsto(z_1^{-1},\cdots,z_{n-1}^{-1})$. Therefore, we have

whose Hodge-Deligne polynomial is

If $(C,p_1,\cdots,p_n)\in B_{n,2}$, $C=C_1\cup C_2$ with $C_1,C_2\cong\mathbb{P}^1$. The locus $B_{n,2,k}$ of such pointed curves where one component has $k$ marked points while the other has $n-k$ points is

where $S_2$ interchanges the two nodes. We thus find that

For $j\geq3$, the number of ways to distribute $n$ marked points to $j$ components is $\gamma_{n,j}$ in (22). For each such distribution, we have a connected component of $B_{n,j}$ isomorphic to $(\mathbb{C}^*)^{n-j}$. Hence we find that

Adding (23), (24) and (25), we obtain Proposition 4.3.

The Hodge-Deligne polynomial $\mathrm{H}_{\overline{\mathcal{M}}_{1,(0+)^n}}$

By Propositions 4.1 and 4.3, we obtain the following.

See Example 5.3 for a computation of $\mathrm{H}_{0|n}$ for $n\leq10$.

By Theorems 4.4 and 3.1, we can now calculate $\mathrm{H}_{\overline{\mathcal{M}}_{1,n}}(u,v)$ recursively for all $n$. By [22], the cycle class map induces an isomorphism

where $R^*(\overline{\mathcal{M}}_{1,n})$ denotes the tautological ring and hence we can also compute $\dim R^i(\overline{\mathcal{M}}_{1,n})$ as the coefficient of $(uv)^i$ in $\mathrm{H}_{\overline{\mathcal{M}}_{1,n}}(u,v)$. In particular, the Poincaré polynomial of the tautological ring is the same as the even degree Poincaré polynomial $P^{\mathrm{even}}_{\overline{\mathcal{M}}_{1,n}}(x)$.

Computations for $n\leq10$

Letting $x=uv$, by Proposition 4.1, we can evaluate $I_n$ as follows.

Example 5.1.

$I_1(x)=x$

$I_2(x)=x^2+x$

$I_3(x)=x^3+3x^2+x-1$

$I_4(x)=x^4+6x^3+6x^2-2x-3$

$I_5(x)=x^5+10x^4+20x^3+4x^2-14x-7$

$I_6(x)=x^6+15x^5+50x^4+40x^3-30x^2-49x-15$

$I_7(x)=x^7+21x^6+105x^5+160x^4-183x^2-139x-31$

$I_8(x)=x^8+28x^7+196x^6+469x^5+280x^4-427x^3-700x^2-356x-63$

$I_9(x)=x^9+36x^8+336x^7+1148x^6+1386x^5-406x^4-2436x^3-2224x^2-860x-127$

$I_{10}(x)=x^{10}+45x^9+540x^8+2484x^7+4662x^6+1764x^5-6090x^4-9804x^3-6372x^2-2003x-255$

By Proposition 4.3, we can evaluate $B_n$ as follows with $x=uv$.

By (3.8) and Theorem 4.4, we can compute $\mathbf{H}_{r|s}=\mathbf{H}_{\overline{\mathcal{M}}_{1,r|s}}$ as follows with $x=uv$.

Example 5.4.

Note that this matches perfectly with Table 1 in [17] after applying the hook-length formula to extract the Hodge-Deligne polynomials.

By (17) and Theorem 4.4, we can compute $H_{n|0}=H_{\overline{\mathcal M}_{1,n}}$ as follows with $x=uv$.

Example 5.5.

Note that this matches perfectly with [3 eM1bar.txt] and [12 page 491].

By Theorems 3.1 and 4.4, it is straightforward to write up a computer program that computes the Hodge-Deligne and Poincaré polynomials of $\overline{\mathcal M}_{1,n}$ for higher $n$, quite efficiently. For $n=50$, we get the following picture for the probability distribution

The red curve in the picture is the graph of a Gaussian function.

Image not extracted

In the subsequent paper [5], we will prove that indeed the probability distribution (27) is asymptotically Gaussian. Furthermore, we will prove a formula that gives us effective estimates of the Betti numbers for $n$ large and show that the Betti sequence is asymptotically 3-ultra log concave.

Update on Question 1.4. With the formulas of this paper, the Betti numbers of $\overline{\mathcal{M}}_{1,n}$ can be computed easily for large $n$, and one can experiment with various properties of the Betti sequence. After posting this paper in the arXiv, Gergely Bérczi informed me that the even degree Poincaré polynomial (or the Poincaré polynomial of the tautological ring) of $\overline{\mathcal{M}}_{1,n}$ is not real-rooted when $n=8706$. Still it seems reasonable to expect that the even degree Betti sequence is ultra-log-concave (ULC) based on numerical computations.

The reduction to minimally weighted curves was made $S_n$-equivariant in [19 Theorem 3.9, Remark 3.10], and together with an $S_n$-equivariant computation for $\overline{\mathcal{M}}_{1,(0+)^n}$, this gives a new proof of Getzler’s formula, answering Question 1.4(1) [6]. In [19], the strategy of this paper is applied to compute the $S_n$-equivariant Hodge-Deligne polynomials of $\overline{\mathcal{M}}_{2,n}$, which answers Question 1.4(5) for $g=2$.

On the other hand, in [5], it was proved for all $g \ge1$ that the even degree Betti sequence of $\overline{\mathcal{M}}_{g,n}$ is asymptotically 3-ULC but not 4-ULC. Interestingly, the even degree Betti sequence of $\overline{\mathcal{M}}_{g,(0+)^n}$ is also asymptotically normally distributed with slightly larger variance and asymptotically 3-ULC.

References

School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Korea

Email address: [email protected]