Symplectic instanton bundles on ${\mathbb {P}}^3$ and ’t Hooft instantons

Abstract

We study the moduli space $I_{n,r}$ of rank-2r symplectic instanton vector bundles on ${\mathbb {P}}^3$ with $r\geqslant 2$ and second Chern class $n\geqslant r+1,\,n-r\equiv 1\,(\mathrm {mod}\,2)$. We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus $I^*_{n,r}$ of tame symplectic instantons is irreducible and has the expected dimension equal to $4n(r+1)-r(2r+1)$. The proof is inherently based on a relation between the spaces $I^*_{n,r}$ and the moduli spaces of ’t Hooft instantons.

1 Introduction

A symplectic instanton vector bundle of rank-2r and charge n on the projective 3-space ${\mathbb {P}}^3$ is an algebraic vector bundle $E=E_{2r}$ of rank-2r on ${\mathbb {P}}^3$ which is equipped with a symplectic structure , $\phi ^\vee =-\phi$ and satisfies the vanishing conditions $h^0(E)=h^1(E{\otimes } {\mathscr {O}}_{{\mathbb {P}}^3}(-2))=0$. The Chern classes $c_1(E)$ and $c_3(E)$ vanish, and we also assume $c_2(E)=n\geqslant 1$. We shall denote by $I_{n,r}$ the moduli space of symplectic (n, r)-instantons.

Rank-r symplectic instantons on ${\mathbb {P}}^3$ relate in a natural manner with “physical” $\mathbf {Sp}(r)$ instantons on the four-sphere $S^4$, i.e., connections on principal $\mathbf {Sp}(r)$-bundles on $S^4$ with self-dual curvature [1]; the moduli spaces of the former are in a sense a complexification of the moduli spaces of the latter. This relation is expressed by the so-called Atiyah–Ward correspondence [1, 3], which relies on the fact that the projective space ${\mathbb {P}}^3$ is the twistor space of the four-sphere $S^4$. The present paper and its companion [7] are the first to study the geometry of the moduli spaces $I_{n,r}$. While [7] studied the case $n \equiv r\, (\mathrm{mod}\,2)$, with $n \geqslant r$, the present paper deals with the other case, $n\equiv r+1\, (\mathrm{mod}\,2)$, with $n \geqslant r+1$. The main result of this paper is that a component $I^*_{n,r}$ of $I_{n,r}$ that is singled out by a certain open condition (which rules out some “badly behaved” monads) is irreducible.

We exploit as usual the monad method [2, 4–6, 8, 11, 12], which allows one to study instantons by means of hyperwebs of quadrics. Namely, we realize $I_{n,r}$ as the quotient space of a principal $\mathrm{GL}(H_n)/\{\pm \mathrm{id}\}$-bundle $\pi _{n,r}:MI_{n,r}\rightarrow I_{n,r}$, where $MI_{n,r}$ is a locally closed subset of the vector space ${\mathbf {S}}_n$ of hyperwebs of quadrics (precise definitions will be given later on). The tame locus $I^*_{n,r}$ being open in $I_{n,r}$, its irreducibility is equivalent to that of $MI^*_{n,r}=\pi _{n,r}^{-1}(I^*_{n,r})$. The key ingredient of our approach is the reduction of the last problem to that of certain sets $Z_{n-r+1}$ (see Sect. 3). The sets $Z_i$ as locally closed subsets of some vector spaces related to ${\mathbf {S}}_n$ were first defined in [9]. It is shown in [9, Section 9] that $Z_i$ can be interpreted essentially as open subsets of certain affine bundles over the monad spaces $M^\mathrm{tH}_{2i-1}$ of ’t Hooft rank-2 mathematical instantons of charge $2i-1$—see more details in Sect. 3.2. Thus the irreducibility of $Z_{n-r+1}$, hence that of $I^*_{n,r}$, is reduced to the irreducibility of the moduli spaces of ’t Hooft instantons of fixed charge, which is well known; see references in [9]. This nontrivial relation between the spaces $I^*_{n,r}$ and the moduli of ’t Hooft instantons is crucial for the results in this paper. Note that this process of reduction from $I^*_{n,r}$ to the moduli of ’t Hooft instantons somewhat resembles Barth’s approach in [4] to the proof of irreducibility of the moduli space $I_4$ of instantons of charge 4. In that paper, Barth reduces the problem to the irreducibility of the space $Q_n$ of commuting pairs of (good in some sense) pencils of quadrics for $n=4$. In our case the role of spaces $Q_n$ is played by the moduli spaces of ’t Hooft instantons.

Notation and conventions. Throughout this paper, we consider an algebraically closed base field $\Bbbk$ of characteristic 0. All schemes will be Noetherian. By a general point of an irreducible (but not necessarily reduced) scheme ${\mathscr {X}}$ we mean a closed point of a dense open subset of ${\mathscr {X}}$. An irreducible scheme is generically reduced if it is reduced at all general points. We follow the notation of [9]. So, we fix an integer $n\geqslant 1$, and denote by $H_n$ and V fixed vector spaces over $\Bbbk$ of dimension n and 4, respectively, and set ${\mathbb {P}}^3=P(V)$. Furthermore, ${\mathbf {S}}_n$ (the space of hyperwebs of quadrics) will denote the vector space . A hyperweb of quadrics $A\in {\mathbf {S}}_n$ is a skew-symmetric homomorphism $A:H_n{\otimes } V\rightarrow H_n^\vee {\otimes } V^\vee$, and we denote by $W_{A}$ the vector space $H_n{\otimes } V/\mathrm{ker}\, A$ and by $c_A$ the canonical epimorphism $H_n{\otimes } V\twoheadrightarrow W_{A}$. A choice of A induces a skew-symmetric isomorphism , and A is the composition

For any morphism of ${\mathscr {O}}_X$-sheaves $f:{\mathscr {F}}\rightarrow {\mathscr {F}}'$ we denote by the same letter f the induced morphism $\mathrm{id}{\otimes } f:U{\otimes }{\mathscr {F}}\rightarrow U{\otimes }{\mathscr {F}}'$, and analogously, for any homomorphism $f:U\rightarrow U'$ of $\Bbbk$-vector spaces, the induced morphism $f{\otimes } \mathrm{id}:U{\otimes }{\mathscr {F}}\rightarrow U'{\otimes }{\mathscr {F}}$. For $A\in {\mathbf {S}}_n$ we denote by $a_A$ the composition

$$\begin{aligned} H_n^\vee {\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}(-1)\xrightarrow {\,u\,\,} H_n{\otimes } V{\otimes }{\mathscr {O}}_{{\mathbb {P}}^3} \xrightarrow {\,c_A\,}{W}_{A}{\otimes } {\mathscr {O}}_{{\mathbb {P}}^3}, \end{aligned}$$

where u is the tautological subbundle morphism. By abuse of notation, we denote by the same symbol a $\Bbbk$-vector space, say U, and the associated affine space $\mathbf {V}(U^\vee )=\mathrm{Spec}(\mathrm{Sym}^*U^\vee )$.

2 Explicit construction of symplectic instantons

In this section we provide some examples and recall some facts about $MI_{n,r}$, in particular, its relation with the moduli space $I_{n,r}$ of symplectic instantons, see [7, Section 3]. Let us consider the set of (n, r)-instanton hyperwebs of quadrics

(1)

Theorem 2.1

• (i) For each $n\geqslant 1$, the space $MI_{n,r}$ of (n, r)-instanton nets of quadrics is a locally closed subscheme of the vector space ${\mathbf {S}}_n$, given locally at any point $A\in MI_{n,r}$ by

  $$\begin{aligned} \left( {\begin{array}{c}2n-2r\\ 2\end{array}}\right) =2n^2-n(4r+1)+r(2r+1) \end{aligned}$$

  (2)

  equations obtained as the rank condition (i) in (1).

• (ii) The natural morphism

  $$\begin{aligned} \pi _{n,r}:MI_{n,r}\rightarrow I_{n,r},\qquad A\mapsto [E_{2r}(A)], \end{aligned}$$

  is a principal $\mathrm{GL}(H_n)/\{\pm \mathrm{id}\}$-bundle in the étale topology. Hence $I_{n,r}$ is a quotient stack $MI_{n,r}/(\mathrm{GL}(H_n)/\{\pm \mathrm{id}\})$, and is therefore an algebraic space.

The fibre $F_{[E]}=\pi _n^{-1}([E])$ over a point $[E]\in I_{n,r}$ is a principal homogeneous space of $\mathrm{GL}(H_n)/\{\pm \mathrm{id}\}$, so that the irreducibility of $(I_{n,r})_\mathrm{red}$ amounts to the irreducibility of the scheme $(MI_{n,r})_\mathrm{red}$. Besides, (2) yields

$$\begin{aligned} \begin{aligned} \dim _AMI_{n,r}&\geqslant \dim {\mathbf {S}}_n-\bigl (2n^2-n(4r+1)+r(2r+1)\bigr )\\ {}&=n^2+4n(r+1)-r(2r+1) \end{aligned} \end{aligned}$$

(3)

at all points $A\in MI_{n,r}$. Thus, $\dim _{[E]}I_{n,r}\geqslant 4n(r+1)-r(2r+1)$ at all points $[E]\in I_{n,r}$, as $MI_{n,r}\rightarrow I_{n,r}$ is an étale principal $\mathrm{GL}(H_n)/\{\pm \mathrm{id}\}$-bundle.

2.1 Symplectic $(n{+}1,n)$-instantons

We give a construction of symplectic $(n{+}1,n)$-instantons and describe their relation to usual rank-2 instantons with second Chern class $c_2=2n$. This will be established at the level of spaces of hyperwebs of quadrics $MI_{n+1,n}$ and $MI_{2n,1}$, regarded as spaces of monads.

Denote by $\mathrm {Isom}_{n+1,n-1}$ the set of all isomorphisms

(4)

This is the principal homogeneous space of the group $\mathrm{GL}(2n)$. Moreover, for any $\zeta \in \mathrm {Isom}_{n+1,n-1}$, let $p_{\zeta }:{\mathbf {S}}_{2n}\twoheadrightarrow {\mathbf {S}}_{n+1}$ be the induced epimorphism, and, for any monomorphism $i:H_n\hookrightarrow H_{n+1}$, let $\mathrm{pr}_{(i)}:{\mathbf {S}}_{n+1}\rightarrow {\mathbf {S}}_n$ be the induced epimorphism.

Note that $MI_{2n,1}$ is irreducible [10, Theorem 1.1], and one has the following result [10, Theorem 3.1].

Theorem 2.2

There exists a dense open subset $MI_{2n,1}^*$ of $MI_{2n,1}$ such that for any hyperweb $A\in MI_{2n,1}^*$ and a general $\zeta \in \mathrm {Isom}_{n+1,n-1}$ the rank of the homomorphism $B=p_{\zeta }(A):H_{n+1}{\otimes } V\rightarrow H_{n+1}^\vee {\otimes } V^\vee$ coincides with the rank of $A:H_{2n}{\otimes } V\rightarrow H_{2n}^\vee {\otimes } V^\vee$:

$$\begin{aligned} \mathrm {rk}\,B=\mathrm {rk}\,A=4n+2. \end{aligned}$$

(5)

Set $W_{4n+2}=H_{2n}{\otimes } V/\mathrm{ker}\, A$ and define the skew-symmetric isomorphism and the morphism of sheaves $a_A:H_{2n}{\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}(-1)\rightarrow W_{4n+2}{\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}$ with $H_{2n}$ and $W_{4n+2}$ taken instead of $H_n$ and $W_{A}$, respectively. The morphism $a_A$ and its transpose ${}^ta_A=a_A^\vee {\circ } q_A:W_{4n+2}{\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}\rightarrow H_{2n}^\vee {\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}(1)$ yield a monad

$$\begin{aligned} \mathscr {M}_{A}:\ 0\,\rightarrow H_{2n}{\otimes }\mathscr {O}_{\mathbb {P}^3}(-1)\,\xrightarrow {a_A}\, W_{4n+2}{\otimes }\mathscr {O}_{\mathbb {P}^3}\,\xrightarrow {{}^ta_A}\, H_{2n}^\vee {\otimes }\mathscr {O}_{\mathbb {P}^3}(1)\,\rightarrow \,0 \end{aligned}$$

with the cohomology sheaf $E_{2}(A),\,[E_{2}(A)]\in I_{2n,1}$, see Theorem 2.1.

Let $i_\zeta :H_{n+1}\hookrightarrow H_{2n}$ be the monomorphism defined by the isomorphism (4). The composition

and its transpose ${}^ta_B=a_B^\vee {\circ } q_A$ yield a monad

$$\begin{aligned} \mathscr {M}_{B}:\ 0\,\rightarrow H_{n+1}{\otimes } \mathscr {O}_{\mathbb {P}^3}(-1) \,\xrightarrow {a_B}\,W_{4n+2}{\otimes } \mathscr {O}_{\mathbb {P}^3} \,\xrightarrow {{}^ta_B}\,H_{n+1}^\vee {\otimes } \mathscr {O}_{\mathbb {P}^3}(1)\,\rightarrow \,0 \end{aligned}$$

with the cohomology sheaf

$$\begin{aligned} E_{2n}(B)=\ker {}^ta_B/\mathrm{im}\,a_B,\quad c_2(E_{2n}(B))=n+1. \end{aligned}$$

The symplectic isomorphism induces a symplectic structure on $E_{2n}(B)$,

(6)

Moreover, (5) implies an isomorphism $H_{n+1}{\otimes } V/\mathrm{ker}\, B\simeq W_{4n+2}$, hence a monomorphism of spaces of sections

$$\begin{aligned} h^0({}^ta_B):\, W_{4n+2}{\otimes }\mathscr {O}_{\mathbb {P}^3} \xrightarrow {{}^ta_B\,} H_{n+1}^\vee {\otimes }V^\vee \end{aligned}$$

in the monad ${\mathscr {M}}_{B}$. Hence for this monad one has $h^0(E_{2n}(B))=0$. This together with (6) means that $E_{2n}(B)$ is a symplectic instanton

$$\begin{aligned}{}[E_{2n}(B)]\in I_{n+1,n}. \end{aligned}$$

Note that, by construction, the monads ${\mathscr {M}}_A$ and ${\mathscr {M}}_{B}$ fit into the commutative diagram

(7)

In view of (6) and the canonical isomorphism $H_{2n}/i_\zeta (H_{n+1})\simeq H_{n-1}$, this diagram yields the quotient monad

$$\begin{aligned} {\mathscr {M}}_{A,B}:\ 0\,\rightarrow H_{n-1}{\otimes } {\mathscr {O}}_{{\mathbb {P}}^3}(-1)&\,\xrightarrow {a_{A,B}} E_{2n}(B)\, \xrightarrow [\simeq ]{\phi _B}E_{2n}(B)^\vee \\ {}&\, \xrightarrow {a_{A,B}^\vee } H_{n-1}^\vee {\otimes } {\mathscr {O}}_{{\mathbb {P}}^3}(1)\,\rightarrow \,0 \end{aligned}$$

whose cohomology sheaf is $E_2(A)=\ker \bigl (a_{A,B}^\vee {\circ } \phi _B\bigr )/\mathrm{im}\, a_A$.

2.2 A special family of symplectic $(2n{-} r{+}1,r)$-instantons

For any integer $r,\,2\leqslant r\leqslant n$, with $n\geqslant 2$, consider a monomorphism

$$\begin{aligned} \tau :H_{2n-r+1}\hookrightarrow H_{2n} \end{aligned}$$

such that

$$\begin{aligned} \tau (H_{2n-r+1})\supset i_\zeta (H_{n+1}). \end{aligned}$$

(8)

The image of $A\in MI_{2n,1}$ under the projection ${\mathbf {S}}_{2n}\twoheadrightarrow {\mathbf {S}}_{2n-r+1}$ induced by $\tau$ produces a hyperweb of quadrics $A_\tau \in {\mathbf {S}}_{2n-r+1}$. This corresponds to a monad

$$\begin{aligned} {\mathscr {M}}_\tau :\ 0\,\rightarrow H_{2n-r+1}{\otimes } {\mathscr {O}}_{{\mathbb {P}}^3}(-1)&\,\xrightarrow {a_\tau } W_{4n+2} {\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}\\ {}&\,\xrightarrow {a_\tau ^\vee \circ q_A\,} H_{2n-r+1}^\vee {\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}(1)\,\rightarrow \,0, \end{aligned}$$

whose cohomology is the rank-2r bundle

$$\begin{aligned} E_{2r}(A_\tau )=\ker \bigl (a_\tau ^\vee {\circ } q_A\bigr )/\mathrm{im}\, a_\tau , \end{aligned}$$

(9)

where $a_\tau =a_A{\circ }\tau$. The bundle $E_{2r}(A_\tau )$ has a natural symplectic structure

(10)

induced by the antiselfduality of the monad ${\mathscr {M}}_{\tau }$. Moreover, by (8), the monad ${\mathscr {M}}_\tau$ can be included into diagram (7) as a middle row, thus obtaining a three-row commutative, anti-self-dual diagram. Thus, in addition to the monad ${\mathscr {M}}_{A,B}$, we also have the monads

$$\begin{aligned} {\mathscr {M}}'_{\tau }:\ 0\,\rightarrow H_{n-r}{\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}(-1)\xrightarrow {a'_\tau } E_{2n}(B)\xrightarrow [\simeq ]{\,\phi \,}E_{2n}(B)^\vee \xrightarrow {{a'}_{\tau }^\vee } H_{n-r}^\vee {\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}(1)\,\rightarrow \,0, \end{aligned}$$

with the cohomology $E_{2r}(A_\tau )=\ker \bigl ({a'}_{\tau }^\vee {\circ } \phi \bigr )/\mathrm{im} \,a'_\tau$, and

$$\begin{aligned} \begin{aligned} {\mathscr {M}}''_{\tau }:\ 0\,\rightarrow H_{r-1}{\otimes } {\mathscr {O}}_{{\mathbb {P}}^3}(-1)&\,\xrightarrow {a''_\tau } E_{2r}(A_\tau )\,\xrightarrow [\simeq ]{\phi _\tau } E_{2r}(A_\tau )^\vee \\ {}&\,\xrightarrow {{a''}_{\tau }^\vee } H_{r-1}^\vee {\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}(1)\,\rightarrow \,0, \end{aligned} \end{aligned}$$

with the cohomology $E_2(A)=\ker \bigl ({a''}_{\tau }^\vee {\circ } \phi _\tau \bigr )/\mathrm{im}\, a''_{\tau }$.

Since $E_{2n}(B)$ is a symplectic instanton, $h^0(E_{2n}(B))=h^i(E_{2n}(B)(-2))=0$, and the monad ${\mathscr {M}}'_{\tau }$ yields

$$\begin{aligned} h^0(E_{2r}(A_\tau ))=h^i(E_{2r}(A_\tau )(-2))=0,\quad i\geqslant 0,\quad c_2(E_{2r}(A_\tau ))=2n-r+1. \end{aligned}$$

This, together with (10), means that $[E_{2r}(A_\tau )]\in I_{2n-r+1,r}$.

Remark 2.3

The maps $\tau$ lie in the set

$$\begin{aligned} N_{n,r}=\bigl \{\tau \in \mathrm{Hom}(H_{2n-r+1},H_{2n}): \tau \text { is injective and } \mathrm {im}\,\tau \supset \mathrm {im}\, i_\zeta \bigr \} \end{aligned}$$

which, for fixed $A\in MI_{2n,1}(\zeta )$, parameterizes a family of hyperwebs $A_\tau$ from $MI_{2n-r+1,r}$. Now, $N_{n,r}$ is a principal $\mathrm{GL}(H_{2n-r+1})$-bundle over an open subset of the Grassmannian $\mathrm{Gr}(n{-}r,n{-}1)$, so it is irreducible. As a result, the family of the three-row extensions of diagram (7) is parameterized by the irreducible variety $MI_{2n,1}(\zeta ){\times } N_{n,r}$. This in turn implies that the family $D_{n,r}$ of isomorphism classes of symplectic rank-2r bundles obtained from these diagrams by (9) is an irreducible, locally closed subset of $I_{2n-r+1,r}$. It is not clear a priori if the closure of $D_{n,r}$ in $I_{2n-r+1,r}$ is an irreducible component of $I_{2n-r+1,r}$.

Let $2\leqslant r\leqslant n$. For every monomorphism $i:H_n\hookrightarrow H_{2n-r+1}$, denote by B(A, i) the image of $A\in MI_{2n-r+1,r}$ under the projection ${\mathbf {S}}_{2n-r+1}\twoheadrightarrow {\mathbf {S}}_n$ induced by i. It may be regarded as a homomorphism $B(A,i):H_n{\otimes } V\rightarrow H_n^\vee {\otimes } V^\vee$.

Definition 2.4

We say that $A\in MI_{2n-r+1,r}$ satisfies property ($*$) if there exists a monomorphism $i:H_n\hookrightarrow H_{2n-r+1}$ such that B(A, i) is invertible.

This is an open condition on A. By Theorem 2.1, $\pi _{2n-r+1,r}:MI_{2n-r+1,r}\rightarrow I_{2n-r+1,r}$ is a principal bundle, so that, if an element $A\in \pi _{2n-r+1,r}^{-1}([E_{2r}])$ satisfies ($*$), then any other point $A'\in \pi _{2n-r+1,r}^{-1}([E_{2r}])$ satisfies ($*$). A symplectic instanton $E_{2r}$ from $I_{2n-r+1,r}$ is said to be tame if some (hence all) $A\in \pi _{2n-r+1,r}^{-1}([E_{2r}])$ satisfies property ($*$). This is an open condition on $[E_{2r}]\in I_{2n-r+1,r}$.

Remark 2.5

Using (8), we see that any $[E_{2r}]\in D_{n,r}$ is tame. We define

$$\begin{aligned} I_{2n-r+1,r}^*=I_{(1)}\cup \cdots \cup I_{(k)}, \end{aligned}$$

where $I_{(1)},\ldots , I_{(k)}$ are the irreducible components of $I_{2n-r+1,r}$ whose general points are tame symplectic instantons. As $D_{n,r}\subset I_{2n-r+1,r}^*$ by definition, $I_{2n-r+1,r}^*$ is nonempty. If we define $MI_{2n-r+1,r}^*= \pi _{2n-r+1,r}^{-1}\bigl (I_{2n-r+1,r}^*\bigr )$, then the map $\pi _{2n-r+1,r}:MI_{2n-r+1,r}^*\rightarrow I_{2n-r+1,r}^*$ is a principal $\mathrm{GL}(H_{2n-r+1})/\{\pm 1\}$-bundle.

3 Irreducibility of $I^*_{2n-r+1,r}$

3.1 A dense open subset of $MI^*_{2n-r+1,r}$

We want to obtain the irreducibility of $I^*_{n,r}$ by reducing it to that of $X_{n,r}$, a dense open subset of $MI^*_{2n-r+1,r}$. The subset $X_{n,r}$ is a locally closed subset of the product of an affine space and an affine cone over a Grassmannian. Given an integer $n\geqslant 1$, we define the following dense open subset of ${\mathbf {S}}_n$:

$$\begin{aligned} {\mathbf {S}}^0_n=\bigl \{A\in {\mathbf {S}}_n: A:H_n{\otimes } V\rightarrow H_n^\vee {\otimes } V^\vee \text { an invertible map}\bigr \}. \end{aligned}$$

We need some more notation. By definition, an element $B\in {\mathbf {S}}^0_n$ is an invertible anti-self-dual map $H_n{\otimes } V\rightarrow H_n^\vee {\otimes } V^\vee$. Its inverse $B^{-1}:H_n^\vee {\otimes } V^\vee \rightarrow H_n{\otimes } V$ is also anti-self-dual. Consider the vector space . An element $C\in \varvec{{\Sigma }}_{n,r}$ can be viewed as a linear map $C:H_{n-r+1}{\otimes } V\rightarrow H_n^\vee {\otimes } V^\vee$, and its dual $C^\vee :H_n{\otimes } V\rightarrow H_{n-r+1}^\vee {\otimes } V^\vee$. As the composition $C^\vee {\circ } B^{-1}{\circ } C$ is anti-self-dual, we can consider it as an element of . Thus the condition

makes sense.

Under an arbitrary direct sum decomposition

(11)

we can represent the hyperweb $A\in {\mathbf {S}}_{2n-r+1}$, regarded as a homomorphism

$$\begin{aligned} A:\, H_n{\otimes } V{\oplus } H_{n-r+1}{\otimes } V\rightarrow H_n^\vee {\otimes } V^\vee {\oplus } H^\vee _{n-r+1}{\otimes } V^\vee \!, \end{aligned}$$

as the $(8n-4r+4){\times }(8n-4r+4)$-matrix of homomorphisms

$$\begin{aligned} A=\left( \begin{matrix} A_1(\xi ) &{} A_2(\xi ) \\ -A_2(\xi )^\vee &{} A_3(\xi ) \end{matrix}\right) \!, \end{aligned}$$

(12)

where

With this notation, decomposition (11) induces an isomorphism

(13)

Let $\mathrm {Isom}_{n,r}$ be the set of all isomorphisms $\xi$ in (11). According to Definition 2.4, there exists $\xi \in \mathrm {Isom}_{n,r}$ such that the set

is a dense open subset of $MI^*_{2n-r+1,r}$. Now take $A\in MI_{2n-r+1,r}^*(\xi )$ and consider A as a matrix of homomorphisms as in (12). By definition, the submatrix $A_1(\xi )$ is invertible. By a suitable elementary transformation we reduce the matrix A to an equivalent matrix $\widetilde{A}$ of the form

$$\begin{aligned} \widetilde{A}=\left( \begin{matrix} {\mathrm{id}}_{H_n\otimes V} &{} A_1(\xi )^{-1}{\circ } A_2(\xi ) \\ 0 &{} A_2(\xi )^\vee {\circ } A_1(\xi )^{-1}{\circ } A_2(\xi )+A_3(\xi ) \end{matrix}\right) \!. \end{aligned}$$

Since $\mathrm {rk}\,\widetilde{A}=\mathrm {rk}\, A=2(2n-r+1)+2r=4n+2$, we obtain the following relation between the matrices $A_1(\xi ), A_2(\xi )$ and $A_3(\xi )$:

$$\begin{aligned} \text {rk}\bigl (A_2(\xi )^\vee {\circ } A_1(\xi )^{-1}{\circ } A_2(\xi )+A_3(\xi )\bigr )=2. \end{aligned}$$

(14)

Consider the embedding of the Grassmannian

and let be the affine cone over G. Set $KG^*=KG{\setminus }\{0\}$. We can now rewrite (14) as

$$\begin{aligned} A_2(\xi )^\vee {\circ } A_1(\xi )^{-1}{\circ } A_2(\xi )+A_3(\xi )\in KG^*\!, \end{aligned}$$

(15)

where

(16)

Now consider the set

$$\begin{aligned} \widetilde{X}_{n,r}=\bigl \{(B,C,D)\in {\mathbf {S}}^0_n{\times } {\varvec{\Sigma }}_{n,r}{\times } KG^*: D-C^\vee {\circ } B^{-1}{\circ } C\in {\mathbf {S}}_{n-r+1}\bigr \}. \end{aligned}$$

(17)

Since for an arbitrary point $y=(B,C,D)\in \widetilde{X}_n$ the point $\widetilde{\xi }^{-1}(B,C,D-C^\vee {\circ } B^{-1}{\circ } C)$ lies in ${\mathbf {S}}_{2n-r+1}$, it may be considered as a homomorphism $A_y:H_{2n-r+1}{\otimes } V\rightarrow H_{2n-r+1}^\vee {\otimes } V^\vee$ of rank $4n+2$, and we have a well-defined $(4n{+}2)$-dimensional vector space $W_{4n+2}(y)=H_{2n-r+1}{\otimes } V/\mathrm{ker}\, A_y$ together with a canonical epimorphism $c_y:H_{2n-r+1}{\otimes } V\twoheadrightarrow W_{4n+2}(y)$ and an induced skew-symmetric isomorphism such that $A_y=c_y^\vee {\circ } q_y{\circ } c_y$. Now, similarly to the morphism $a_A:H_{2n-r+1}{\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}(-1)\rightarrow W_{4n+2}{\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}$ (see Sect. 2.1), a morphism of sheaves

$$\begin{aligned} a_y=c_y{\circ } u:\, H_{2n-r+1}{\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}(-1) \rightarrow W_{4n+2}(y){\otimes }{\mathscr {O}}_{{\mathbb {P}}^3} \end{aligned}$$

is defined, together with its transpose ${}^ta_y=a_y^\vee {\circ } q_y:W_{4n+2}(y){\otimes }{\mathscr {O}}_{{\mathbb {P}}^3} \rightarrow H_{2n-r+1}^\vee {\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}(1)$. We now introduce the following open subset $X_{n,r}$ of the set $\widetilde{X}_{n,r}$:

(18)

Since conditions (i) and (ii) on a point $y\in \widetilde{X}_{n,r}$ in (18) are open, from (15) and (16) we obtain the following result.

Proposition 3.1

There exist a decomposition $\xi \in \mathrm {Isom}_{n,r}$, a dense open subset$MI^*_{2n-r+1,r}(\xi )$ of $MI^*_{2n-r+1,r}$ and an isomorphism of reduced schemes

The inverse isomorphism is given by the formula

where $\widetilde{\xi }$ is defined in (13).

The following theorem will be proved in Sect. 3.2.

Theorem 3.2

The set $X_{n,r}$ is irreducible of dimension $(2n-r+1)^2+4(2n-r+1)(r+1)-r(2r+1)$.

Proposition 3.1 and Theorem 3.2 imply that $MI_{2n-r+1,r}^*$ is irreducible of dimension $(2n-r+1)^2+4(2n-r+1)(r+1)-r(2r+1)$ for any $n\geqslant 2$ and $2\leqslant r\leqslant n$. Thus, for these values of n and r, the space $I_{2n-r+1,r}^*$ is irreducible and has dimension $4(2n-r+1)(r+1)-r(2r+1)$. Substituting $2n-r+1\mapsto n$, we obtain the main result of this paper.

Theorem 3.3

For any integer $r\geqslant 2$ and for any integer $n\geqslant r+1$ such that $n\equiv r+1\,(\mathrm{mod}\,2)$, the moduli space $I_{n,r}^*$ of tame symplectic instantons is an open subset of an irreducible component of $I_{n,r}$ of dimension $4n(r+1)-r(2r+1)$.

3.2 Proof of irreducibility of $X_{n,r}$

We prove now Theorem 3.2. Consider the set $\widetilde{X}_{n,r}$ defined in (17). Since $X_{n,r}$ is an open subset of $\widetilde{X}_{n,r}$, it is enough to prove the irreducibility of $\widetilde{X}_{n,r}$. In view of the isomorphism , we rewrite $\widetilde{X}_{n,r}$ as

$$\begin{aligned} \widetilde{X}_{n,r}=\bigl \{(B,C,D)\in ({\mathbf {S}}_n^\vee )^0{\times } {\varvec{\Sigma }}_{n,r}{\times } KG^*: D-C^\vee {\circ } B{\circ } C\in {\mathbf {S}}_{n-r+1}\bigr \}. \end{aligned}$$

If a direct sum decomposition

has been fixed, any linear map

can be represented as a homomorphism

$$\begin{aligned} C:\, H_{n-r+1}{\otimes } V\rightarrow H_{n-r+1}^\vee {\otimes } V^\vee {\oplus } H_{r-1}^\vee {\otimes } V^\vee \!, \end{aligned}$$

and also as a block matrix

$$\begin{aligned} C= \left( \begin{matrix} \phi \\ \psi \end{matrix} \right) \!, \end{aligned}$$

(19)

with

In the same way, any can be represented as

$$\begin{aligned} B=\left( \begin{matrix} B_1 &{} \lambda \\ -\lambda ^\vee &{} \mu \end{matrix} \right) \!, \end{aligned}$$

(20)

with

(21)

By (19) and (20), the composition

can be written in the form

$$\begin{aligned} C^\vee {\circ } B{\circ } C=\phi ^\vee {\circ } B_1{\circ }\phi +\phi ^\vee {\circ } \lambda {\circ }\psi -\psi ^\vee {\circ }\lambda ^\vee {\circ }\phi + \psi ^\vee {\circ }\mu {\circ }\psi . \end{aligned}$$

(22)

In view of (19)–(21), we have

$$\begin{aligned} {\mathbf {S}}^\vee _n{\times }{\varvec{\Sigma }}_{n,r}= {\mathbf {S}}^\vee _{n-r+1}{\times }{\varvec{\Phi }}_{n-r+1} {\times }{\varvec{\Psi }}_{n,r}{\times }{\mathbf {L}}_{n,r}{\times } \mathbf {M}_{r-1}, \end{aligned}$$

well-defined morphisms

$$\begin{aligned} \widetilde{p}:\widetilde{X}_{n,r}\rightarrow {\mathbf {L}}_{n,r}{\times } {\mathbf {M}}_r{\times } KG,\quad (B_1,\phi ,\psi ,\lambda ,\mu ,D)\mapsto (\lambda ,\mu ,D), \end{aligned}$$

and

$$\begin{aligned} p=\widetilde{p}{|}\overline{X}_{n,r}:\,\overline{X}_{n,r}\rightarrow {\mathbf {L}}_{n,r}{\times }{\mathbf {M}}_{r-1}{\times } KG. \end{aligned}$$

Here $\overline{X}_{n,r}$ is the closure of $\widetilde{X}_{n,r}$ in $({\mathbf {S}}_n^\vee )^0{\times }{\varvec{\Sigma }}_{n,r}{\times } KG$. Moreover, we have

Proposition 3.4

Let $n\geqslant 2$. For any $B\in ({\mathbf {S}}^\vee _n)^0$ and for a general choice of the decomposition $H_n\simeq H_{n-r+1}{\oplus } H_{r-1}$, the block $B_1$ of B in (20) is nondegenerate.

Proof

By applying [9, Proposition 7.3], r times, one obtains a decomposition such that $B_1:H_{n-r+1}^\vee {\otimes } V^\vee \rightarrow H_{n-r+1}{\otimes } V$ in (20) is nondegenerate, that is, $B_1\in ({\mathbf {S}}^\vee _{n-r+1})^0$. $\square$

If ${\mathscr {X}}$ is any irreducible component of $X_{n,r}$, taken with its reduced structure, and $\overline{{\mathscr {X}}}$ is its closure in $\overline{X}_{n,r}$, we pick up a point $z=(B_1,\phi ,\psi ,\lambda ,\mu ,D)\in {\mathscr {X}}$ not lying in the components of $X_{n,r}$ different from ${\mathscr {X}}$, and such that the decomposition $H_n\simeq H_{n-r+1}{\oplus } H_{r-1}$ is general. Then, by Proposition 3.4, $B_1\in ({\mathbf {S}}^\vee _{n-r+1})^0$. Consider the morphism

$$\begin{aligned} f:{\mathbb {A}}^1\rightarrow \overline{{\mathscr {X}}},\quad t\mapsto \bigl (B_1,t^2\phi ,t\psi ,t\lambda ,t^2\mu ,t^4D\bigr ),\qquad f(1)=z. \end{aligned}$$

This is well defined as a consequence of (22). The point $f(0)=(B_1,0,0,0,0,0)$ lies in the fibre $p^{-1}(0,0,0)$, so that $p^{-1}(0,0,0)\cap \overline{{\mathscr {X}}}\ne \varnothing$. In different terms,

$$\begin{aligned} \rho ^{-1}(0,0,0)\ne \varnothing ,\quad \text {where}\quad \rho =p{|}\overline{{\mathscr {X}}}. \end{aligned}$$

(23)

By (22) and the definition of $\widetilde{X}_{n,r}$, one has

$$\begin{aligned} \begin{aligned} \widetilde{p}^{-1}(0,0,0)=\bigl \{&(B_1,\phi ,\psi )\in ({\mathbf {S}}^\vee _{n-r+1})^0{\times }{\varvec{\Phi }}_{n-r+1}{\times } {\varvec{\Psi }}_{n,r}:\\ {}& \phi ^\vee {\circ } B_1{\circ }\phi \in {\mathbf {S}}_{n-r+1}\bigr \}. \end{aligned}\quad \quad \quad \quad \end{aligned}$$

(24)

Now for each $i\geqslant 1$ consider the set $Z_i$ mentioned in the introduction. This set $Z_i$ is defined in [9, Section 7] as

$$\begin{aligned} Z_i=\bigl \{(B,\phi )\in ({\mathbf {S}}^\vee _i)^0{\times }{\varvec{\Phi }}_i: \phi ^\vee {\circ } B{\circ }\phi \in {\mathbf {S}}_i\bigr \}, \end{aligned}$$

(25)

and has a natural structure of closed subscheme of $({\mathbf {S}}^\vee _i)^0{\times }{\varvec{\Phi }}_i$. The key point in the sequel is the fact that $Z_i$ is an integral scheme of dimension $4i(i{+}2)$—see [9, Theorem 7.2]. This statement is based on the following relation between $Z_i$ for $i\geqslant 2$ and the moduli space of ’t Hooft instantons of charge $2i-1$. Fix a monomorphism $j:H_{i-1}\hookrightarrow H_i$. For an arbitrary point $z=(B,\phi )\in Z_i$, let $E_{2i}$ be a symplectic vector bundle of rank-2i defined as a cokernel of a morphism of sheaves $\widetilde{B}:H_i{\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}(-1)\rightarrow H_i^{\vee }{\otimes }{\Omega }_{{\mathbb {P}}^3}(1)$ naturally induced by B. Let $s(z):H_i\rightarrow H^0(E_{2i}(1))$ be the composition of $\phi$ understood as a homomorphism and of the evaluation map , and let $s_z$ be the composition

$$\begin{aligned} s_z:H_i{\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}(-1)\xrightarrow {s(z)} H^0(E_{2i}(1)){\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}(-1)\xrightarrow {\mathrm{ev\,}}E_{2i}, \end{aligned}$$

where $\mathrm{ev}$ is the evaluation morphism. Using the symplecticity of $E_{2i}$, one obtains an antiselfdual monad

$$\begin{aligned} M(z):\ 0\,\rightarrow H_{i-1}{\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}(-1)\xrightarrow {s_z\circ j\,}E_{2i}\xrightarrow {{}^t(s_z\circ j)}H_{i-1}^{\vee }{\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}(1)\,\rightarrow \,0 \end{aligned}$$

with a rank-2 cohomology vector bundle $E_2(z)$ with $c_1=0$ and $c_2=2i-1$. A standard diagram chase yields a monomorphism $H_i/j(H_{i-1}){\otimes }{\mathscr {O}}_{{\mathbb {P}}^3}(-1)\rightarrow E_2(z)$ showing that $h^0(E_2(z)(1))\ne 0$, i.e., that $E_2(z)$ is a ’t Hooft instanton vector bundle. Thus the association $z\rightsquigarrow M(z)$ yields a morphism of $Z_i$ to the space $M^\mathrm{tH}_{2i-1}$ of the ’t Hooft monads, which is irreducible since the moduli space of ’t Hooft instantons of charge $2i-1$ is known to be irreducible. It is shown in [9, Section 9] that this morphism $Z_i\rightarrow M^\mathrm{tH}_{2i-1}$ is a composition of a dense open embedding and the structure map of an affine bundle over $M^\mathrm{tH}_{2i-1}$. This implies the irreducibility of $Z_i$.

Now, comparing (25) for $i=n-r+1$ with (24), we obtain scheme-theoretic inclusions

$$\begin{aligned} \rho ^{-1}(0,0,0)\subset p^{-1}(0,0,0)\subset \widetilde{p}^{-1}(0,0,0)=Z_{n-r+1} {\times }{\varvec{\Psi }}_{n,r}. \end{aligned}$$

(26)

By the above, $Z_{n-r+1}$ is an integral scheme of dimension $4(n-r+1)(n-r+3)$. This together with (26) implies that

$$\begin{aligned} \begin{aligned} \dim \rho ^{-1}(0,0,0)&\leqslant \dim p^{-1}(0,0,0)\leqslant \dim Z_{n-r+1}+ \dim {\varvec{\Psi }}_{n,r}\\ {}&=4(n-r+1)(n-r+3) +6(r-1)(n-r+1)\\ {}&=(n-r+1)(4n+2r+6). \end{aligned} \end{aligned}$$

(27)

Hence, in view of (23),

$$\begin{aligned} {\begin{matrix} \dim \overline{\mathscr {X}}&{}\leqslant \dim \rho ^{-1}(0,0,0)+ \dim \mathbf {L}_{n,r}+\dim \mathbf {M}_{r-1}+\dim KG \\ {} &{} \leqslant (n-r+1)(4n+2r+6) +6(r-1)(n-r+1)\\ {} &{}+3(r-1)r+(8n-8r+5) \\ {} &{} =(2n-r+1)^2+4(2n-r+1)(r+1) -r(2r+1). \end{matrix}} \end{aligned}$$

(28)

On the other hand, formula (3)—with n replaced by $2n-r+1$—and Proposition 3.1 show that, for any point $x\in {\mathscr {X}}$ such that $A=f_{n,r}^{-1}(x)\in MI_{2n-r+1,r}^*(\xi )$,

$$\begin{aligned} \begin{aligned} (2n-r+1)^2+4(2n-r+1)(r+1)-r(2r+1)&\leqslant \dim _A MI_{2n-r+1,r}^*(\xi )\\ {}&= \dim \overline{{\mathscr {X}}}. \end{aligned} \end{aligned}$$

(29)

Comparing (28) with (29), we see that all inequalities in (27)–(29) are equalities. In particular,

$$\begin{aligned} \dim \rho ^{-1}(0,0)=\dim (Z_{n-r+1}{\times }{\varvec{\Psi }}_{n,r})= \dim \overline{{\mathscr {X}}}-\dim ({\mathbf {L}}_{n,r} {\times }{\mathbf {M}}_{r-1}{\times } KG). \end{aligned}$$

(30)

Since, by [9, Theorem 7.2], the scheme $Z_{n-r+1}$ is integral and so $Z_{n-r+1}{\times }{\varvec{\Psi }}_{n,r}$ is integral as well, (26) and (30) yield the coincidence of the integral schemes

$$\begin{aligned} \rho ^{-1}(0,0,0)=p^{-1}(0,0,0)= \widetilde{p}^{-1}(0,0,0)=Z_{n-r+1}{\times }{\varvec{\Psi }}_{n,r}. \end{aligned}$$

(31)

We need now the following easy lemma, which is a slight generalization of [9, Lemma 7.4].

Lemma 3.5

Let $f:X\rightarrow Y$ be a morphism of reduced schemes, with Y an integral scheme. Assume that there exists a closed point $y\in Y$ such that, for any irreducible component $X'$ of X,

• (a) $\dim f^{-1}(y)=\dim X'-\dim Y$,

• (b) the scheme-theoretic inclusion of fibres $(f{|}_{X'})^{-1}(y)\subset f^{-1}(y)$ is an isomorphism of integral schemes.

Then

• (i) there exists an open subset U of Y containing y such that the morphism $f{|}_{f^{-1}(U)}:f^{-1}(U)\rightarrow U$ is flat, and

• (ii) X is integral.

By applying this lemma to $X=X_{n,r},\,X'={\mathscr {X}},\,Y={\mathbf {L}}_{n,r}{\times } {\mathbf {M}}_{r-1}{\times } KG,\,y=(0,0),\,f=p$, also in view of (30) and (31), one obtains that $X_{n,r}$ is integral and is of dimension

$$\begin{aligned} (2n-r+1)^2+4(2n-r+1)(r+1)-r(2r+1). \end{aligned}$$

Theorem 3.2 is thus proved.