SYMPLECTIC SMALL DEFORMATIONS OF SPECIAL INSTANTON BUNDLE ON ℙ^{2n+1}

CARLA DIONISI

Abstract. Let MI_{Simp,ℙ^{2n+1}}(k) be the moduli space of stable symplectic instanton bundles on ℙ^{2n+1} with second Chern class c_{2} = k (it is a closed subscheme of the moduli space MI_{ℙ^{2n+1}}(k)) We prove that the dimension of its Zariski tangent space at a special (symplectic) instanton bundle is 2k(5n − 1) + 4n^{2} − 10n + 3, k ≥ 2.

Introduction

Symplectic instanton bundles on ℙ^{2n+1} are holomorphic bundles of rank 2n (see [1] ,[4] and [6]) that correspond to the self-dual solutions of Yang-Mills equations on ℙ^{n}(ℍ). They are given by some monads (see section 2 for precise definitions) and their only topological invariant is c_{2} = k. At present the dimension of their moduli space MI_{Simp,ℙ^{2n+1}}(k) is not known except in the cases n=1, where the dimension is 8k − 3 (see [3]), and in few other cases corresponding to small values of k. MI_{Simp,ℙ^{2n+1}}(k) is a closed subscheme of MI_{ℙ^{2n+1}}(k) and this last scheme parametrizes stable instanton bundles with structural group GL(2n). The class of special instanton bundles was introduced in [8]. Let E ∈ MI_{ℙ^{2n+1}}(k) be a special symplectic instanton bundle. The tangent dimension h^{1}(End(E)) was computed in [7] and it is equal to 4(3n − 1)k + (2n − 5)(2n − 1) . The Zariski tangent space of MI_{Simp,ℙ^{2n+1}}(k) at E is isomorphic to H^{1}(S^{2}E) and in this paper we prove that

(1)  h^{2}(S^{2}E) = ( \begin{matrix}k − 2 \\ 2 \end{matrix} ) · ( \begin{matrix}2n − 1 \\ 2 \end{matrix} )  ∀ k ≥ 2

By the Hirzebruch-Riemann-Roch formula , since h^{0}(S^{2}E) = 0 and h^{i}(S^{2}E) = 0 ∀ i ≥ 3, it follows that:

χ(S^{2}E) = h^{2}(S^{2}E) − h^{1}(S^{2}E) =

2n^{2} + n + \frac{1}{2} [k^{2} ( \begin{matrix}2n − 1 \\ 2 \end{matrix} ) − k(10n^{2} − 5n − 1)]

and

Theorem 0.1. Let E be a special symplectic instanton bundle.Then

h^{1}(S^{2}E) = 2k(5n − 1) + 4n^{2} − 10n + 3 , k ≥ 2

(for n = 1 it is well known that h^{1}(S^{2}E) = 8k−3 and, in the real case, for n = 2 the dimension 18k − 1 has been found in [5] by different techniques).

Now, since by the Kuranishi map H^{2}(S^{2}E) is the space of obstructions to the smoothness at E of MI_{Simp,ℙ^{2n+1}}(k), we obtain

Corollary 0.2. ∀k ≥ 2 the dimension of any irreducible component of MI_{Simp,ℙ^{2n+1}}(k), containing a special symplectic instanton bundle is bounded by the value 2k(5n − 1) + 4n^{2} − 10n + 3 ( linear in k )

Corollary 0.3. ∀n MI_{Simp,ℙ^{2n+1}}(3) is smooth at a special instanton bundle E,and the dimension of any irreducible component containing E is 4n^{2} + 20n − 3.

The main remark of this paper is that it is easier to compute H^{2}(S^{2}E) and

H^{2}(∧^{2} E) together as SL(2)-modules (although this second cohomology space has a geometrical meaning only for orthogonal bundles) than to compute H^{2}(S^{2}E) alone.

1. Preliminaries

Throughout this paper 𝕂 denotes an algebraically closed field of characteristic zero. U denotes a 2-dimensional 𝕂 vector space (U =< s, t >), S_{n} = S^{n}U its n-th symmetric power (S_{n} =< s^{n}, s^{n−1}t, …, t^{n} >) , V_{n} = U ⊗ S_{n} (V_{n} =< s ⊗ s^{n}, s ⊗ s^{n−1}t, … s ⊗ t, … t ⊗ t^{n} >) and ℙ^{2n+1} = ℙ(V_{n}).

Definition 1.1. A vector bundle E on ℙ^{2n+1} of rank 2n is called an instanton bundle of quantum number k if:

• E has Chern polinomial c_{t}(E) = (1 − t^{2})^{−k};

• E(q) has natural cohomology in the range −(2n+1) ≤ q ≤ 0, that is H^{i}(E(q)) ≠ 0 for at most one i = i(q).

By [6],[2],the Definition 1.1 is equivalent to : i)E is the cohomology bundle of a monad:

0 → O(−1)^{k} → Ω^{1}(1)^{k} → O^{2n(k−1)} → 0

or ii) E is the cohomology bundle of a monad:

0 → O(−1)^{k} ⟶^{A} O^{2n+2k} ⟶^{B^{t}} O(1)^{k} → 0

(where, after we have fixed a coordinate system, A and B can be identified with matrices in the space Mat(k, 2n + 2k, S_{1}))

Definition 1.2. An instanton bundle E is called symplectic if there is an isomorphism φ : E → E^{∨} satisfying φ = −φ^{∨}.

Definition 1.3. An instanton bundle is called special if it arises from a monad where the morfism B^{t} is defined in some system of homogeneous coordinates x_{0}, ⋯ x_{n}, y_{0} ⋯ y_{n} on ℙ^{2n+1} by the trasposed of the matrix:

B = ( \begin{matrix}x_{0} & ⋯ & x_{n} & 0 & ⋯ & 0 & y_{0} & ⋯ & y_{n} & 0 & ⋯ & 0 \\ 0 & x_{0} & ⋯ & x_{n} & 0 & ⋯ & 0 & y_{0} & ⋯ & y_{n} & 0 & ⋯ \\ ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ & ⋯ \\ ⋯ & 0 & x_{0} & ⋯ & x_{n} & 0 & ⋯ & 0 & y_{0} & ⋯ & y_{n} & 0 \\ 0 & ⋯ & 0 & x_{0} & ⋯ & x_{n} & 0 & ⋯ & 0 & y_{0} & ⋯ & y_{n} \end{matrix} )

The following lemma is well known (and easy to prove)

Lemma 1.4.

H^{0}(O(1)) ≅ V^{∨}

H^{0}(Ω^{1}(2)) ≅∧^{2} V^{∨}

H^{i}(ℙ^{n}, S^{2}Ω^{1}(1)) = \{ \begin{matrix}0 & se i ≠ 1 \\ ∧^{2} V^{∨} & se i = 1 \end{matrix}

2. Existence of a special symplectic instanton bundle

There is a natural exact sequence of GL(U)-equivariant maps for any k, n ≥ 1 (Clebsch-Gordan sequence):

(2)  0 →∧^{2} U ⊗ S_{k−1} ⊗ V_{n−1} ⟶^{β} S_{k} ⊗ V_{n} ⟶^{μ} V_{k+n} → 0

where μ is the multiplication map and β is defined by (s ∧ t) ⊗ f ⊗ g → (sf ⊗ tg − tf ⊗ sg)

We can define (see [7]) the morphism

b̃: S_{k−1}^{∨} ⊗ Ω^{1}(1) →∧^{2} U^{∨} ⊗ S_{k−2}^{∨} ⊗ V_{n−1}^{∨} ⊗ O

and it is induced the complex

(3)  A ⊗ O(−1) ⟶^{ã} S_{k−1}^{∨} ⊗ Ω^{1}(1) ⟶^{b̃}∧^{2} U^{∨} ⊗ S_{k−2}^{∨} ⊗ V_{n−1}^{∨} ⊗ O

where A is a k-dimensional subspace of S_{2n+k−1}^{∨}⊗ ∧^{2} U^{∨} such that (3) is a monad and the cohomology bundle E is a special symplectic instanton bundle. It was proved in [7] that

H^{2}(EndE) ≅ Ker(Φ^{∨})^{∨}

where Φ^{∨} : S_{k−2}^{⊗2} ⊗ V_{n−1}^{⊗2} → S_{k−1}^{⊗2}⊗ ∧^{2} V_{n}

and there is an isomorphism of SL(2)-representations

ε : S_{k−3}^{∨} ⊗ S_{k−3}^{∨} ⊗ S^{2}V_{n−2}^{∨} → Ker(Φ^{∨})
3. How to identify H^{2}(S^{2}E) and H^{2}(∧^{2} E)

Proposition 3.1. Let E be special symplectic instanton bundle ,cohomology of monad (3) and N = Kerb̃.Then

(i): H^{2}(S^{2}E) ≅ H^{2}(S^{2}N)

(ii): H^{2}(∧^{2} E) ≅ H^{2}(∧^{2} N)

Proof. We denote B := S_{k−1}^{∨}  and  C :=∧^{2} U^{∨} ⊗ S_{k−2}^{∨} ⊗ V_{n−1}^{∨}

The result follows from the two exact sequences given by monad (3) :

(4)  0 → N → B ⊗ Ω^{1}(1) → C ⊗ O → 0

(5)  0 → A ⊗ O(−1) → N → E → 0

In fact, by performing the second symetric and alternating power of sequence (4), we have

0 → S^{2}N →Ã→ B ⊗ C ⊗ Ω^{1}(1) →∧^{2} C ⊗ O → 0

↘ ↗

M^{1}

↗ ↘ 0 0

(6)

where Ã := S^{2}(B ⊗ Ω^{1}(1)) = (S^{2}B ⊗ S^{2}(Ω^{1}(1))) ⊕ (∧^{2} B ⊗ Ω^{2}(2)) and

0 →∧^{2} N →\overline{A} → B ⊗ C ⊗ Ω^{1}(1) → O ⊗ S^{2}C → 0

↘ ↗

M

↗ ↘ 0 0

(7)

where \overline{A} :=∧^{2} (B ⊗ Ω^{1}(1)) = (∧^{2} B ⊗ S^{2}(Ω^{1}(1))) ⊕ (S^{2}B ⊗ Ω^{2}(2))

□

3.1. Identifying H^{2}(S^{2}N) and H^{2}(∧^{2} N). i) Diagram (6) gives the following two exact sequences:

(8) O → H^{0}(M^{1}) → H^{1}(S^{2}N) → H^{1}(Ã) → H^{1}(M^{1}) → H^{2}(S^{2}(N)) → H^{2}(Ã) → ⋯

O → H^{0}(M^{1}) → B ⊗ C ⊗ H^{0}(Ω^{1}(1)) →∧^{2} C → H^{1}(M^{1}) → B ⊗ C ⊗ H^{1}(Ω^{1}(1)) → ⋯

∥ ∥ 0 0

(9) Sequence (9) implies:

H^{0}(M^{1}) = 0 and H^{1}(M^{1}) ≅∧^{2} C

Then, by using the two formulas:

H^{1}(Ã) = (S^{2}B ⊗ H^{1}(S^{2}Ω^{1}(1))) ⊕ (∧^{2} B ⊗ H^{1}(Ω^{2}(2)) = S^{2}B⊗ ∧^{2} V^{∨}

and:

H^{2}(Ã) = (S^{2}B ⊗ H^{2}(S^{2}Ω^{1}(1))) ⊕ (∧^{2} B ⊗ H^{2}(Ω^{2}(2)) = 0

sequence (8) becomes:

0 → H^{1}(S^{2}N) → H^{1}(Ã) → H^{1}(M^{1}) → H^{2}(S^{2}(N)) → 0

i.e.

0 → H^{1}(S^{2}N) → S^{2}B⊗ ∧^{2} V^{∨} ⟶^{Φ̃}∧^{2} C → H^{2}(S^{2}N) → 0

⟹ H^{2}(S^{2}N) ≅ Coker(Φ̃) = (Ker(Φ̃^{∨}))^{∨}

Then:

H^{2}(S^{2}N)^{∨} = Ker [∧^{2} (S_{k−2} ⊗ V_{n−1}) ⟶^{Φ̃^{∨}} S^{2}(S_{k−1})⊗ ∧^{2} V_{n}]

ii)

Diagram (7) gives the following two exact sequences:

(10)

O → H^{0}(M) → H^{1}(∧^{2} N) → H^{1}(\overline{A}) → H^{1}(M) → H^{2}(∧^{2} N) → H^{2}(\overline{A}) → ⋯

O → H^{0}(M) → B ⊗ C ⊗ H^{0}(Ω^{1}(1)) → S^{2}C⊗H^{0}(O) → H^{1}(M) → 0 → ⋯ ∥ ∥

0 S^{2}C

(11)

and, from sequence (11), we get H^{0}(M) = 0 and H^{1}(M) ≃ S^{2}C Then, since :

H^{1}(\overline{A}) = (H^{1}(S^{2}(Ω^{1}(1))⊗ ∧^{2} B) ⊕ (S^{2}B ⊗ H^{1}(Ω^{2}(2))) =∧^{2} B⊗ ∧^{2} V^{∨}

and H^{2}(\overline{A}) = 0 sequence (10) becomes :

O → H^{0}(M) → H^{1}(∧^{2} N)→ H^{1}(\overline{A}) → H^{1}(M) → H^{2}(∧^{2} N) → 0

∥ 0

i.e. 0 → H^{1}(∧^{2} N) →∧^{2} B⊗ ∧^{2} V^{∨} ⟶^{\overline{Φ}} S^{2}C → H^{2}(∧^{2} N) → 0

⟹ H^{2}(∧^{2} N) ≅ Coker(\overline{Φ}) = (Ker(\overline{Φ}^{∨}))^{∨}

Then we obtain :

(H^{2}(∧^{2} N))^{∨} = Ker [S^{2}(S_{k−2} ⊗ V_{n−1}) ⟶^{\overline{Φ}^{∨}}∧^{2} S_{k−1}⊗ ∧^{2} V_{n}]

3.2. Identifying H^{2}(S^{2}E). We have

H^{2}(S^{2}E)^{∨} ≅ Ker Φ̃^{∨}
where  Φ̃^{∨} :∧^{2} (S_{k−2} ⊗ V_{n−1}) → S^{2}S_{k−1}⊗ ∧^{2} V_{n}  is explicitly given by

Φ̃^{∨}((g ⊗ v) ∧ (g^{1} ⊗ v^{1})) =sg · sg^{1} ⊗ (tv ∧ tv^{1}) − sg · tg^{1} ⊗ (tv ∧ sv^{1})+ −tg · sg^{1} ⊗ (sv ∧ tv^{1}) + tg · tg^{1} ⊗ (sv ∧ sv^{1})

i.e.  Φ̃^{∨} = p̃ ∘ (∧^{2} β)

where β : ∧^{2} U ⊗ S_{k−2} ⊗ V_{n−1} → S_{k−1} ⊗ V_{n} is such that

(s ∧ t) ⊗ (g ⊗ v) ↦ (sg ⊗ tv) − (tg ⊗ sv)

and

p̃ : ∧^{2} (S_{k−1} ⊗ V_{n}) → S^{2}S_{k−1}⊗ ∧^{2} V_{n}

∥

(∧^{2} S_{k−1} ⊗ S^{2}V_{n}) ⊕ (S^{2}S_{k−1}⊗ ∧^{2} V_{n})

is such that

(f ⊗ u) ∧ (f′ ⊗ u^{1}) ↦ f · f′ ⊗ u ∧ u^{1}.

Now, we consider the SL(2)-equivariant morphism:

ε̃^{1} : ∧^{2} (S_{k−3} ⊗ V_{n−2}) →∧^{2} (S_{k−2} ⊗ V_{n−1})

where, up to the order of factors, the map ε̃^{1} := β^{1} ∧ β^{1} and β^{1} : S_{k−3} ⊗ V_{n−2} → S_{k−2} ⊗ V_{n−1} is defined as β. Hence, ε̃^{1} is injective.

Finally, we define

ε̃ : ∧^{2} S_{k−3} ⊗ S^{2}V_{n−2} →∧^{2} (S_{k−2} ⊗ V_{n−1})

as ε̃ = ε̃^{1} ∘ ĩ, where

ĩ : ∧^{2} S_{k−3} ⊗ S^{2}V_{n−2} →∧^{2} (S_{k−3} ⊗ V_{n−2}) such that

f ∧ f′ ⊗ u · u^{1} ⟼ (f ⊗ u) ∧ (f′ ⊗ u^{1}) + (f ⊗ u^{1}) ∧ (f′ ⊗ u)

is an injective map. Then, also ε̃ is injective.

Lemma 3.2. Im ε̃ ⊂ Ker Φ̃^{∨}

Proof. Straightforward computation. □

3.3. Identifying H^{2}(∧^{2} E). We have

H^{2}(∧^{2} E)^{∨} ≅ Ker \overline{Φ}^{∨}

where  \overline{Φ}^{∨} : S^{2}(S_{k−2} ⊗ V_{n−1}) →∧^{2} S_{k−1}⊗ ∧^{2} V_{n}  is explicity given

by \overline{Φ}^{∨}((g ⊗ v) · (g^{1} ⊗ v^{1})) =sg ∧ sg^{1} ⊗ (tv ∧ tv^{1}) − sg ∧ tg^{1} ⊗ (tv ∧ sv^{1})

−tg ∧ sg^{1} ⊗ (sv ∧ tv^{1}) + (tg ∧ tg^{1}) ⊗ (sv ∧ sv^{1})

i.e.  \overline{Φ}^{∨} = \overline{p} ∘ (S^{2}β)  where

\overline{p} : S^{2}(S_{k−1} ⊗ V_{n}) →∧^{2} S_{k−1}⊗ ∧^{2} V_{n}

∥

(∧^{2} S_{k−1}⊗ ∧^{2} V_{n}) ⊕ (S^{2}S_{k−1} ⊗ S^{2}V_{n})

is such that

\overline{p}((f ⊗ u) · (f′ ⊗ u^{1})) = f ∧ f′ ⊗ u ∧ u^{1}

We consider the SL(2)-equivariant morphism:

\overline{ε}^{1} : S^{2}(S_{k−3} ⊗ V_{n−2}) → S^{2}(S_{k−2} ⊗ V_{n−1})

such that: \overline{ε}^{1}((f ⊗ u) · (f′ ⊗ u^{1})) =(sf ⊗ tu) · (sf′ ⊗ tu^{1}) − (sf ⊗ su) · (tf′ ⊗ tu^{1})+

−(tf ⊗ tu) · (sf′ ⊗ su^{1}) + (sf ⊗ tu) · (sf′ ⊗ tu^{1})

(\overline{ε}^{1} = S^{2}β^{1} hence \overline{ε}^{1} is injective). Finally, we define

\overline{ε} : S^{2}S_{k−3} ⊗ S^{2}V_{n−2} → S^{2}(S_{k−2} ⊗ V_{n−1})

as \overline{ε} = \overline{ε}^{1} ∘ \overline{i} where \overline{i} : S^{2}S_{k−3} ⊗ S^{2}V_{n−2} → S^{2}(S_{k−3} ⊗ V_{n−2}) such that

f · f′ ⊗ uu^{1} ↦ (f ⊗ u)(f′ ⊗ u^{1}) + (f ⊗ u^{1})(f′ ⊗ u) is an injective map. Then, also \overline{ε} is injective Lemma 3.3. Im \overline{ε} ⊂ Ker \overline{Φ}^{∨}

Proof. Straightforward computation. □

Theorem 3.4. For any special symplectic instanton bundle E

H^{2}(S^{2}E) ≃ ∧^{2} (S_{k−3})^{∨} ⊗ S^{2}(V_{n−2})^{∨}

Proof. By lemma 3.2 and 3.3 we have the following diagram with exact rows and columns:

0  0  0  0
↓  ↓  ↓  ↓

0 → H^{2}(∧^{2} N)^{∨} → S^{2}(S_{k−2} ⊗ V_{n−1}) ⟶^{\overline{Φ}^{∨}}∧^{2} S_{k−1}⊗ ∧^{2} V_{n} → H^{1}(∧^{2} N)^{∨} → 0

↓  ↓  ↓  ↓

0 → H^{2}(N ⊗ N)^{∨} → S_{k−2}^{⊗2} ⊗ V_{n−1}^{⊗2} ⟶^{Φ^{∨}} S_{k−1}^{⊗2}⊗ ∧^{2} V_{n} → H^{1}(N ⊗ N)^{∨} → 0

↓  ↓  ↓  ↓

0 → H^{2}(S^{2}N)^{∨} →∧^{2} (S_{k−2} ⊗ V_{n−1}) ⟶^{Φ̃^{∨}} S^{2}S_{k−1}⊗ ∧^{2} V_{n} → H^{1}(S^{2}N)^{∨} → 0

↓  ↓  ↓  ↓
0  0  0  0

It was shown in [7] that:

H^{2}(EndE) ≃ Ker Φ^{∨} = H^{2}(N ⊗ N)^{∨} ≃ S_{k−3}^{⊗2} ⊗ S^{2}V_{n−2}

We have proved that there are two injective maps:

ε̃ : ∧^{2} (S_{k−3}) ⊗ S^{2}V_{n−2} → Ker Φ̃^{∨} ≃ H^{2}(S^{2}N)^{∨} ≃ H^{2}(S^{2}E)^{∨}

\overline{ε} : S^{2}(S_{k−3}) ⊗ S^{2}V_{n−2} → Ker \overline{Φ}^{∨} ≃ H^{2}(∧^{2} N)^{∨} ≃ H^{2}(∧^{2} E)^{∨}

Then, we can consider the following diagram:

0  0  0
↓  ↓  ↓

0 → S^{2}S_{k−3} ⊗ S^{2}V_{n−2} → S_{k−3}^{⊗2} ⊗ S^{2}V_{n−2} →∧^{2} S_{k−3} ⊗ S^{2}V_{n−2} → 0

↓ \overline{ε}  ↓ ε  ↓ ε̃

0 → H^{2}(∧^{2} E)^{∨} → H^{2}(EndE)^{∨} → H^{2}(S^{2}E)^{∨} → 0

↓ 0

and by the Snake-Lemma there is the exact sequence :

0 → Ker \overline{ε} → Ker ε → Ker ε̃ → Coker \overline{ε} → Coker ε → Coker ε̃ → 0

∥  ∥  ∥  ∥
0  0  0  0

⇒ Coker \overline{ε} = 0 ⇒ \overline{ε} is an isomorphism ⇒ ε̃ is an isomorphism. Thus:

H^{2}(S^{2}E)^{∨} ≅ ∧^{2} (S_{k−3}) ⊗ S^{2}(V_{n−2})

i.e. H^{2}(S^{2}E) ≃ ∧^{2} (S_{k−3})^{∨} ⊗ S^{2}(V_{n−2})^{∨} as we wanted.

□

Remark 3.5. By this theorem formula 1 and theorem 0.1 are easily proved.

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Carla Dionisi Dipartimento di Matematica ed Applicazioni ”R.Caccioppoli” Università di Napoli Federico II via Cintia (loc. Monte S.Angelo) I-80138 Napoli, Italy E-mail address: dionisi@matna3.dma.unina.it