samples/texts/1693838/page_8.md

Taking wedge products of (1.2) we have the exact sequence

$$\begin{array}{r}{\displaystyle 0\to \mathcal{O}(t+1-2n-2k{)}^{{\alpha }_0}\to \cdots \to \mathcal{O}(t-k-1{)}^{{\alpha }_{2n+k-2}}}\\ {\displaystyle \to \mathcal{O}(t-k{)}^{{\alpha }_{2n+k-1}}\to \stackrel{2n+k-1}{\bigwedge }S(t-k)\to 0}\end{array}\backslash begin\{align*\}\; 0\; \backslash to\; \backslash mathcal\{O\}(t+1-2n-2k)^\{\backslash alpha\_0\}\; \backslash to\; \backslash dots\; \backslash to\; \backslash mathcal\{O\}(t-k-1)^\{\backslash alpha\_\{2n+k-2\}\}\; \backslash \backslash \; \backslash qquad\; \backslash to\; \backslash mathcal\{O\}(t-k)^\{\backslash alpha\_\{2n+k-1\}\}\; \backslash to\; \backslash bigwedge^\{2n+k-1\}\; S(t-k)\; \backslash to\; 0\; \backslash end\{align*\}$$

for suitable $\alpha_i \in \mathbb{N}$ and from this sequence we can conclude.

Ellia proves Theorem 3.5 in the case of $\mathbb{P}^3$ ([E], Prop. IV.1). He also remarks that the given bound is sharp. This holds on $\mathbb{P}^{2n+1}$ as it is shown by the following theorem, which points out that the special symplectic instanton bundles are the “furthest” from having natural cohomology. $\square$

Theorem 3.6. Let $E$ be a special symplectic instanton bundle on $\mathbb{P}^{2n+1}$ with $c_2 = k$. Then

$$h^1(E(t))\mathrm{\ne }0\text{ for }-1\le t\le k-2.h^1(E(t))\; \backslash neq\; 0\; \backslash text\{\; for\; \}\; -1\; \backslash le\; t\; \backslash le\; k-2.$$

Proof. For $n=1$ the thesis is immediate from the exact sequence

$$0\to \mathcal{O}(t-1)\to E(t)\to {\mathcal{J}}_C(t+1)\to 00\; \backslash rightarrow\; \backslash mathcal\{O\}(t-1)\; \backslash rightarrow\; E(t)\; \backslash rightarrow\; \backslash mathcal\{J\}\_C(t+1)\; \backslash rightarrow\; 0$$

where $C$ is the union of $k+1$ disjoint lines in a smooth quadric surface. Then the result follows by induction on $n$ by considering the sequence

$$0\to E(t-2)\to E(t-1{)}^2\to E(t)\to E(t){\mathrm{\mid }}_{{\mathbb{P}}^{2n-1}}\to 00\; \backslash rightarrow\; E(t-2)\; \backslash rightarrow\; E(t-1)^2\; \backslash rightarrow\; E(t)\; \backslash rightarrow\; E(t)|\_\{\backslash mathbb\{P\}^\{2n-1\}\}\; \backslash rightarrow\; 0$$

and the fact that, for a particular choice of the subspace $\mathbb{P}^{2n-1}$, the restriction $E|_{\mathbb{P}^{2n-1}}$ splits as the direct sum of a rank-2 trivial bundle and a special symplectic instanton bundle on $\mathbb{P}^{2n-1}$([ST] 5.9). $\square$

Remark 3.7. In [OT] it is proved that if $E_k$ is a special symplectic instanton bundle on $\mathbb{P}^5$ with $c_2 = k$ then $h^1(\mathrm{End},E_k) = 20k - 3$.

In the following table we summarize what we know about the component $M_0(k) \subset \mathrm{MI}_{\mathbb{P}^5}(k)$ containing $E_k$.

Table 3.10

	h1(Ek ⊗ Ek*)	h2(Ek ⊗ Ek*)	dim M0(k)	MIP5(k)
k = 1	14	0	14	open subset of P14
k = 2	37	0	37	smooth, irreduc., unirat.
k = 3	57	3	54	singular
k = 4	77	12	65	singular
k ≥ 2	20k - 3	3(k - 2)2	?	?