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Irreducibility of the moduli space of orthogonal instanton bundles on \(\mathbb{P}^n\). (English) Zbl 1473.14016

The paper studies orthogonal instanton bundles with no global sections over \(\mathbb{P}^n(n\geq 3)\). Let \(V^{\vee}:=H^0(\mathcal{O}_{\mathbb{P}^n}(1))\). The authors consider the triple \((\mathcal{E},\phi,f)\) with \(\mathcal{E}\) an orthogonal instanton bundle such that \(H^0(\mathcal{E})=0\), \(\phi:\mathcal{E}\xrightarrow{\simeq}\mathcal{E}^{\vee}\) an orthogonal structure and \(f:H_c\xrightarrow{\simeq} H^{n-1}(\mathcal{E}(-n))\). The author proved that there is a 1-1 correspondence between isomorphism classes of \((\mathcal{E},\phi,f)\) and elements \(A\in \bigwedge^2H_c^{\vee}\otimes\bigwedge^2V^{\vee}\) satisfying some conditions.
As \(\mathcal{E}\) is an instanton bundle, the Beilinson spectral sequence for it degenerates at \(E_2\) level and we have a monad \[ 0\rightarrow H^1(\mathcal{E}\otimes \Omega_{\mathbb{P}^n}^2(1))\otimes \mathcal{O}_{\mathbb{P}^n}(-1)\rightarrow H^1(\mathcal{E}\otimes \Omega_{\mathbb{P}^n}^1)\otimes \mathcal{O}_{\mathbb{P}^n}\rightarrow H^1(\mathcal{E}(-1))\otimes \mathcal{O}_{\mathbb{P}^n}(1)\rightarrow 0 \] with \(\mathcal{E}\) as its middle cohomology. Also the boundary maps \[ H^j(\mathcal{E}\otimes \Omega_{\mathbb{P}^n}^i)\xrightarrow{\partial}H^{j+1}(\mathcal{E}\otimes \Omega_{\mathbb{P}^n}^{i+1}),~~H^k(\mathcal{E}\otimes \Omega_{\mathbb{P}^n}^{l})\xrightarrow{\partial}H^{k+1}(\mathcal{E}\otimes \Omega_{\mathbb{P}^n}^{l+1}) \] are isomorphisms for \(2\leq i\leq n-1,~1\leq j\leq n-2,~1\leq l\leq n-2,~0\leq k\leq n-1\).
Therefore, applying Euler sequences one can get a map \(A\) as the composition of all the maps as follows: \[ H_c\otimes \bigwedge^n V^{\vee}\rightarrow H^{n-1}(\mathcal{E}\otimes \Omega_{\mathbb{P}^n}^{n-1})\xrightarrow{\simeq}H^1(\mathcal{E}\otimes \Omega_{\mathbb{P}^n}^1)\xrightarrow{i_1}H^1(\mathcal{E}(-1))\otimes V^{\vee}. \] As \(H^1(\mathcal{E}(-1))\cong H^{n-1}(\mathcal{E}(-n))^{\vee}\) by Serre duality and \(\bigwedge^n V^{\vee}\cong V\), we have \(A\in (H_c^{\vee})^{\otimes 2}\otimes (V^{\vee})^{\otimes 2}\). The author showed that \(A\in \bigwedge^2H_c^{\vee}\otimes\bigwedge^2V^{\vee}\) by using the monad associated to \(\mathcal{E}\).
The map \(i_1:H^1(\mathcal{E}\otimes \Omega_{\mathbb{P}^n}^1)\rightarrow H^1(\mathcal{E}(-1))\otimes V^{\vee}\) is injective as \(H^0(\mathcal{E})=0\). Hence by Riemann-Roch one has \(2c+r=h^1(\mathcal{E}\otimes \Omega_{\mathbb{P}^n}^1)\leq h^1(\mathcal{E}(-1))(n+1)\) where \(r\) and \(c\) are the rank and the charge (i.e \(c=-\chi(\mathcal{E}(-1))\)) respectively of \(\mathcal{E}\). Thus we get an inequality \(r\leq c(n-1)\).
Using the correspondence above, the author constructed by GIT the moduli space \(\mathcal{M}\) of the triples \((\mathcal{E},\phi,f)\) and showed that it is affine, smooth and irreducible. At last, the author associated to each \((\mathcal{E},\phi,f)\) a Kronecker module \(\gamma\) and showed that for any line \(L\subset \mathbb{P}^n\), \(\mathcal{E}|_{L}\) is trivial iff \(\gamma (L)\) is an isomorphism viewing \(L\) as an element in \(\bigwedge^2 V\) up to scalars.
Reviewer: Yao Yuan (Beijing)

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli

Software:

Macaulay2
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References:

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