Quot schemes and Ricci semipositivity

TL;DR

This paper investigates the geometric properties of a moduli space of torsion quotients on a Riemann surface, showing it cannot support a Kähler metric with semipositive Ricci curvature due to its anticanonical bundle.

Contribution

It proves that the anticanonical line bundle of the quot scheme ${ m extbf{Q}}_X(r,d)$ is not nef, implying the space cannot have a Ricci semipositive Kähler metric.

Findings

Anticanonical bundle of ${ m extbf{Q}}_X(r,d)$ is not nef

${ m extbf{Q}}_X(r,d)$ admits no Kähler metric with semipositive Ricci curvature

The result links geometric properties of the quot scheme to curvature conditions

Abstract

Let $X$ be a compact connected Riemann surface of genus at least two, and let ${\mathcal Q}_X(r,d)$ be the quot scheme that parametrizes all the torsion coherent quotients of ${\mathcal O}^{\oplus r}_X$ of degree $d$. This ${\mathcal Q}_X(r,d)$ is also a moduli space of vortices on $X$. Its geometric properties have been extensively studied. Here we prove that the anticanonical line bundle of ${\mathcal Q}_X(r,d)$ is not nef. Equivalently, ${\mathcal Q}_X(r,d)$ does not admit any K\"ahler metric whose Ricci curvature is semipositive.

Equations

$$\mathcal{Q}:={\mathcal{Q}}_X(r,d)$$

$$p:S^d(X)⟶{\text{Pic}}^d(X)$$

$$s_F:{\mathcal{O}}_X^{\oplus r}=({\mathcal{O}}_X^{\oplus r}{)}^{\ast }⟶F^{\ast }$$

$${\bigwedge }^r{s}_F:{\mathcal{O}}_X={\bigwedge }^r{\mathcal{O}}_X^{\oplus r}⟶{\bigwedge }^r{F}^{\ast }$$

$$\phi :\mathcal{Q}⟶S^d(X),F⟼\text{div}({\bigwedge }^r{s}_F),$$

$$f:\tilde{\mathcal{Q}}⟶\mathcal{Q}$$

$$\gamma :\tilde{\mathcal{Q}}⟶A\times C\times H\times F,$$

$$(p\circ \phi {)}_{\ast }:H_1(\mathcal{Q},\mathbb{Q})⟶H_1({\text{Pic}}^d(X),\mathbb{Q}),$$

$$f_{\ast }:H_1(\tilde{\mathcal{Q}},\mathbb{Q})⟶H_1(\mathcal{Q},\mathbb{Q})$$

$$(p\circ \phi \circ f{)}_{\ast }:H_1(\tilde{\mathcal{Q}},\mathbb{Q})⟶H_1({\text{Pic}}^d(X),\mathbb{Q})$$

$$\beta :A⟶{\text{Pic}}^d(X).$$

$$\begin{matrix}\widetilde{\mathcal{Q}}=A\times C\times H\times F&\ \stackrel{{\scriptstyle p\circ\varphi\circ f}}{{\longrightarrow}}&\text{Pic}^{d}(X)\\ q\Big{\downarrow}~{}{\ }&&~{}\,~{}\|{\rm Id}\\ A&\ \stackrel{{\scriptstyle\beta}}{{\longrightarrow}}&\text{Pic}^{d}(X)\end{matrix}$$

$${\beta }_{\ast }:H_1(A,\mathbb{Q})⟶H_1({\text{Pic}}^d(X),\mathbb{Q})$$

$$d=\dim S^d(X)\ge \dim {\text{Pic}}^d(X)=g\ge 2.$$

$${\phi }_{\ast }:{\pi }_1(\mathcal{Q})⟶{\pi }_1(S^d(X))$$

$$p_{\ast }:{\pi }_1(S^d(X))⟶{\pi }_1({\text{Pic}}^d(X))$$

$${\pi }_1(X)/[{\pi }_1(X),{\pi }_1(X)]=H_1(X,\mathbb{Z})$$

$$(p\circ \phi {)}_{\ast }:{\pi }_1(\mathcal{Q})⟶{\pi }_1({\text{Pic}}^d(X))$$

$$\mu :J⟶{\text{Pic}}^d(X)$$

$$\begin{matrix}\widetilde{\mathcal{Q}}&\ \stackrel{{\scriptstyle f}}{{\longrightarrow}}&{\mathcal{Q}}\\ {\ }~{}\Big{\downarrow}\lambda&&{\ \ \ \ \ \ }~{}~{}~{}\Big{\downarrow}p\circ\varphi\\ J&\ \stackrel{{\scriptstyle\mu}}{{\longrightarrow}}&\text{Pic}^{d}(X)\end{matrix}$$

$$f_{\ast }T\tilde{\mathcal{Q}}=f_{\ast }{f}^{\ast }T\mathcal{Q}\stackrel{p_f}{⟶}(f_{\ast }{\mathcal{O}}_{\tilde{\mathcal{Q}}})\otimes T\mathcal{Q}\stackrel{t}{⟶}{\mathcal{O}}_{\mathcal{Q}}\otimes T\mathcal{Q}=T\mathcal{Q},$$

$$\Phi :H^0(\tilde{\mathcal{Q}},T\tilde{\mathcal{Q}})=H^0(\mathcal{Q},f_{\ast }T\tilde{\mathcal{Q}})⟶H^0(\mathcal{Q},T\mathcal{Q})$$

$$H^0(\tilde{\mathcal{Q}},T\tilde{\mathcal{Q}})=H^0(A,TA)\oplus H^0(C,TC)\oplus H^0(H,TH)\oplus H^0(F,TF).$$

$$H^0(A,TA)\subset H^0(\tilde{\mathcal{Q}},T\tilde{\mathcal{Q}})$$

$$\dim H^0(A,TA)=\dim {\text{Pic}}^d(X)=g>1.$$

$$\Phi (H^0(A,TA))\subset \Phi (H^0(\tilde{\mathcal{Q}},T\tilde{\mathcal{Q}}))=H^0(\mathcal{Q},T\mathcal{Q})$$

$$\Phi {|}_{H^0(A,TA)}:H^0(A,TA)⟶H^0(\mathcal{Q},T\mathcal{Q})$$

Full text

Résumé. *Schéma quot et semi-positivité de Ricci

Soit $X$ une surface de Riemann compacte et connexe de genre au moins deux, et soit ${\mathcal{Q}}_{X}(r,d)$ le schéma quot qui paramétrise tous les quotients torsion cohérents de ${\mathcal{O}}^{\oplus r}_{X}$ de degré $d$. L’espace ${\mathcal{Q}}_{X}(r,d)$ est aussi un espace de modules de vortex sur $X$. Nous démontrons que le fibré anticanonique de $X$ n’a pas la propriété nef. De façon équivalente, ${\mathcal{Q}}_{X}(r,d)$ n’admet aucune métrique kahléienne dont la courbure de Ricci est semi-positive.

1. Introduction

Take a compact connected Riemann surface $X$. The genus of $X$, which will be denoted by $g$, is assumed to be at least two. We will not distinguish between the holomorphic vector bundles on $X$ and the torsion-free coherent analytic sheaves on $X$. For a positive integer $r$, let ${\mathcal{O}}^{\oplus r}_{X}$ be the trivial holomorphic vector bundle on $X$ of rank $r$. Fixing a positive integer $d$, let

[TABLE]

be the quot scheme that parametrizes all (torsion) coherent quotients of ${\mathcal{O}}^{\oplus r}_{X}$ of rank zero and degree $d$ [17]. Equivalently, ${\mathcal{Q}}$ parametrizes all coherent subsheaves of ${\mathcal{O}}^{\oplus r}_{X}$ of rank $r$ and degree $-d$, because these are precisely the kernels of coherent quotients of ${\mathcal{O}}^{\oplus r}_{X}$ of rank zero and degree $d$. This $\mathcal{Q}$ is a connected smooth complex projective variety of dimension $rd$. See [6], [5], [4] for properties of $\mathcal{Q}$. It should be mentioned that $\mathcal{Q}$ is also a moduli space of vortices on $X$, and it has been extensively studied from this point of view of mathematical physics; see [3], [9], [12] and references therein.

Bökstedt and Romão proved some interesting differential geometric properties of $\mathcal{Q}$ (see [12]). In [10] and [11] we proved that $\mathcal{Q}$ does not admit Kähler metrics with semipositive or seminegative holomorphic bisectional curvature. In this note, we continue the study the question of existence of metrics on $\mathcal{Q}$ whose curvature has a sign. Our aim here is to prove the following:

Theorem 1.1**.**

The quot scheme ${\mathcal{Q}}$ in (1.1) does not admit any Kähler metric such that the anticanonical line bundle $K^{-1}_{\mathcal{Q}}$ is hermitian semipositive.

Since semipositive holomorphic bisectional curvature implies semipositive Ricci curvature for a Kähler metric, Theorem 1.1 generalizes the main result of [11].

Recall that a holomorphic line bundle $L$ on a compact complex manifold $M$ is said to be hermitian semipositive if $L$ admits a smooth hermitian structure such that the corresponding hermitian connection has the property that its curvature form is semipositive. The anticanonical line bundle on $M$ will be denoted by $K^{-1}_{M}$. Note that if $M$ admits a Kähler metric such that the corresponding Ricci curvature is semipositive, then $K^{-1}_{M}$ is hermitian semipositive. Indeed, in that case the hermitian connection on $K^{-1}_{M}$ for the hermitian structure induced by such a Kähler metric has semipositive curvature. The converse statement, that hermitian semipositivity of $K_{M}^{-1}$ implies the existence of Kähler metrics with semipositive Ricci curvature, is also true by Yau’s solution of the Calabi’s conjecture [1], [2], [20].

The proof of Theorem 1.1 is based on a recent work of Demailly, Campana and Peternell on the classification of compact Kähler manifolds $M$ with semipositive $K_{M}^{-1}$ [15], [14]. This classification implies that if $K_{M}^{-1}$ is semipositive, then there is a nontrivial abelian ideal in the Lie algebra of holomorphic vector fields on $M$, provided $b_{1}(M)\,>\,0$. On the other hand, for $M\,=\,{\mathcal{Q}}$, this Lie algebra is isomorphic to $\mathfrak{sl}(r,{\mathbb{C}})$, which does not have any nontrivial abelian ideal.

2. Proof of Theorem 1.1

2.1. Semipositive Ricci curvature

Let $J^{d}(X)\,=\,\text{Pic}^{d}(X)$ be the connected component of the Picard group of $X$ that parametrizes the isomorphism classes of holomorphic line bundles on $X$ of degree $d$. Let $S^{d}(X)$ denote the space of all effective divisors on $X$ of degree $d$, so $S^{d}(X)\,=\,X^{d}/P_{d}$ is the symmetric product with $P_{d}$ being the group of permutations of $\{1\,,\cdots\,,d\}$. Let

[TABLE]

be the natural morphism that sends a divisor on $X$ to the holomorphic line bundle on $X$ defined by it.

Take any coherent subsheaf $F\,\subset\,{\mathcal{O}}^{\oplus r}_{X}$ of rank $r$ and degree $-d$. Let

[TABLE]

be the dual of the inclusion of $F$ in ${\mathcal{O}}^{\oplus r}_{X}$. Its exterior product

[TABLE]

is a holomorphic section of the holomorphic line bundle $\bigwedge^{r}F^{*}$ of degree $d$. Therefore, the divisor $\text{div}(\bigwedge^{r}s_{F})$ is an element of $S^{d}(X)$. Consequently, we have a morphism

[TABLE]

where $\mathcal{Q}$ is defined in (1.1). We note that when $r\,=\,1$, then $\varphi$ is an isomorphism.

Assume that ${\mathcal{Q}}$ admits a Kähler metric $\omega$ such that $K^{-1}_{\mathcal{Q}}$ is hermitian semipositive. Then there is a connected finite étale Galois covering

[TABLE]

such that $(\widetilde{\mathcal{Q}},\,f^{*}\omega)$ is holomorphically isometric to a product

[TABLE]

where

• •

$A$ is an abelian variety,

• •

$C$ is a simply connected Calabi–Yau manifold (holonomy is $\text{SU}(c)$, where $c\,=\,\dim C$),

• •

$H$ is a simply connected hyper-Kähler manifold (holonomy is $\text{Sp}(h/2)$, where $h\,=\,\dim H$), and

• •

$F$ is a rationally connected smooth projective variety such that $K^{-1}_{F}$ is hermitian semipositive.

(See [15, Theorem 3.1].) Henceforth, we will identify $\widetilde{\mathcal{Q}}$ with $A\times C\times H\times F$ using $\gamma$ in (2.4). We note that $F$ is simply connected because it is rationally connected [13, p. 545, Theorem 3.5], [18, p. 362, Proposition 2.3].

2.2. A lower bound of $d$

We know that $b_{1}({\mathcal{Q}})\,=\,2g$, and the induced homomorphism

[TABLE]

where $p$ and $\varphi$ are constructed in (2.1) and (2.2) respectively, is an isomorphism [5], [6, p. 649, Remark]. Since $f$ in (2.3) is a finite étale covering, the induced homomorphism

[TABLE]

is surjective. Therefore, the homomorphism

[TABLE]

is surjective.

There is no nonconstant holomorphic map from a compact simply connected Kähler manifold to an abelian variety. In particular, there are no nonconstant holomorphic maps from $C$, $H$ and $F$ in (2.4) to $\text{Pic}^{d}(X)$. Hence the map $p\circ\varphi\circ f$ factors through a map

[TABLE]

In other words, there is a commutative diagram

[TABLE]

where $q$ is the projection of $A\times C\times H\times F$ to the first factor. Since $H_{1}(A\times C\times H\times F,\,{\mathbb{Z}})\,=\,H_{1}(A,\,{\mathbb{Z}})$ (as $C$, $H$ and $F$ are simply connected), and $(p\circ\varphi\circ f)_{*}$ in (2.5) is surjective, it follows that the homomorphism

[TABLE]

induced by $\beta$ is surjective. This immediately implies that the map $\beta$ is surjective. Since $\beta$ is surjective, from the commutativity of (2.6) we know that the map $p$ is surjective. This implies that

[TABLE]

2.3. Albanese for $\widetilde{\mathcal{Q}}$

The homomorphism of fundamental groups

[TABLE]

induced by $\varphi$ in (2.2) is an isomorphism [8, Proposition 4.1]. Since $d\,\geq\,2$ (see (2.7)), the homomorphism of fundamental groups

[TABLE]

induced by $p$ in (2.1) is an isomorphism. Indeed, $\pi_{1}(S^{d}(X))$ is the abelianization

[TABLE]

of $\pi_{1}(X)$ [16]. Combining these we conclude that the homomorphism of fundamental groups

[TABLE]

induced by $p\circ\varphi$ is an isomorphism.

Since the homomorphism in (2.8) is an isomorphism, the covering $f$ in (2.3) is induced by a covering of $\text{Pic}^{d}(X)$. In other words, there is a finite étale Galois covering

[TABLE]

and a morphism $\lambda\,:\,\widetilde{\mathcal{Q}}\,\longrightarrow\,J$ such that the following diagram is commutative:

[TABLE]

where $f$ is the covering map in (2.3). The projection $q$ in (2.6) is clearly the Albanese morphism for $\widetilde{\mathcal{Q}}$, because $C$, $H$ and $F$ are all simply connected. On the other hand, $p\circ\varphi$ is the Albanese morphism for $\mathcal{Q}$ [11, Corollary 2.2]. Therefore, its pullback, namely, $\lambda$, is the Albanese morphism for $\widetilde{\mathcal{Q}}$. Consequently, we have $A\,=\,J$ with $\lambda$ coinciding with the projection $q$ in (2.6). Henceforth, we will identify $A$ and $q$ with $J$ and $\lambda$ respectively.

2.4. Vector fields

The differential $df$ of $f$ identifies $T{\widetilde{\mathcal{Q}}}$ with $f^{*}T{\mathcal{Q}}$, because $f$ is étale. Using the trace homomorphism $t\,:\,f_{*}{\mathcal{O}}_{\widetilde{\mathcal{Q}}}\,\longrightarrow\,{\mathcal{O}}_{\mathcal{Q}}$, we have

[TABLE]

where $p_{f}$ is given by the projection formula. This produces a homomorphism

[TABLE]

(the equality $H^{0}(\widetilde{\mathcal{Q}},\,T{\widetilde{\mathcal{Q}}})\,=\,H^{0}({\mathcal{Q}},\,f_{*}T{\widetilde{\mathcal{Q}}})$ follows from the fact that $f$ is a finite morphism). This homomorphism $\Phi$ is surjective. Indeed, as $f^{*}T{\mathcal{Q}}\,=\,T{\widetilde{\mathcal{Q}}}$, any section of $T{{\mathcal{Q}}}$ pulls back to a section of $T{\widetilde{\mathcal{Q}}}$.

Since $\widetilde{\mathcal{Q}}\,=\,A\times C\times H\times F$, we have

[TABLE]

Note that $H^{0}(\widetilde{\mathcal{Q}},\,T{\widetilde{\mathcal{Q}}})$ is a Lie algebra under the operation of Lie bracket of vector fields, and the subspace

[TABLE]

(see (2.12)) is an ideal in this Lie algebra. Since $A\,=\,J$ is a covering of $\text{Pic}^{d}(X)$, we have

[TABLE]

Since $H^{0}(A,\,TA)$ is an ideal in $H^{0}(\widetilde{\mathcal{Q}},\,T{\widetilde{\mathcal{Q}}})$, it follows immediate that

[TABLE]

is an ideal, where $\Phi$ is constructed in (2.11). Note that $H^{0}(A,\,TA)$ is an abelian Lie algebra, so the Lie algebra $\Phi(H^{0}(A,\,TA))$ is also abelian.

Since $\mu\,:\,J\,=\,A\,\longrightarrow\,\text{Pic}^{d}(X)$ in (2.9) is a covering map between abelian varieties, the trace map $H^{0}(A,\,TA)\,\longrightarrow\,H^{0}(\text{Pic}^{d}(X),\,T\text{Pic}^{d}(X))$ is an isomorphism. In view of this, from the commutativity of the diagram in (2.10) it follows that the restriction

[TABLE]

is injective (see (2.12) and (2.11)). But $H^{0}({\mathcal{Q}},\,T{\mathcal{Q}})\,=\,\mathfrak{sl}(r,{\mathbb{C}})$ [7, p. 1446, Theorem 1.1]. Hence the Lie algebra $H^{0}({\mathcal{Q}},\,T{\mathcal{Q}})$ does not contain any nonzero abelian ideal. This is in contradiction with the earlier result that $\Phi(H^{0}(A,\,TA))$ is a nonzero abelian ideal in $H^{0}({\mathcal{Q}},\,T{\mathcal{Q}})$ of dimension $g$ (see (2.13)). This completes the proof of Theorem 1.1.