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Moduli of mathematical instanton vector bundles with odd \( c_2\) on projective space. (English. Russian original) Zbl 1262.14053

Izv. Math. 76, No. 5, 991-1073 (2012); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 76, No. 5, 143-224 (2012).
Let \(E\) be a rank 2 vector bundle on the projective space \({\mathbb P}^3\) (over an algebraically closed field \(k\) of characteristic 0), with Chern classes \(c_1(E) = 0\) and \(c_2(E) = n\). Since \(\text{det}\, E \simeq {\mathcal O}_{{\mathbb P}^3}\), there exists a skew-symmetric isomorphism \(E \overset\sim\rightarrow E^\vee\). By Serre duality, \(\text{h}^0(E(-2)) = \text{h}^3(E(-2))\) and \(\text{h}^1(E(-2)) = \text{h}^2(E(-2))\) hence \(\chi(E(-2)) = 0\). \(E\) is called a (mathematical) \(n\)-instanton if \(\text{H}^i(E(-2)) = 0\), \(\forall \, i\). In this case \(n \geq -1\). If \(n = -1\) then \(E \simeq {\mathcal O}_{{\mathbb P}^3}(1) \oplus {\mathcal O}_{{\mathbb P}^3}(-1)\) and if \(n = 0\) then \(E \simeq {\mathcal O}_{{\mathbb P}^3}^2\). If \(n \geq 1\) then \(E\) is, actually, stable, i.e., \(\text{H}^0(E) = 0\), and it can be realised as the middle cohomology of an anti-selfdual monad: \[ 0 \rightarrow {\mathcal O}_{{\mathbb P}^3}(-1)^n \rightarrow {\mathcal O}_{{\mathbb P}^3}^{2n+2} \rightarrow {\mathcal O}_{{\mathbb P}^3}(1)^n \rightarrow 0\, . \] For \(n \geq 1\), the set \(\text{I}_n\) of isomorphism classes of \(n\)-instantons form an open subset of the Maruyama scheme \(\text{M}_{{\mathbb P}^3}(2; 0, n, 0)\) of semistable rank 2 torsion free sheaves on \({\mathbb P}^3\) with Chern classes \(c_1 = 0\), \(c_2 = n\), \(c_3 = 0\). It is conjectured that \(\text{I}_n\) is smooth and irreducible, of dimension \(8n - 3\). This conjecture was verified, for \(n \leq 5\), in the period 1977–2003, by several authors (W. Barth, R. Hartshorne, G. Ellingsrud, S.A. Strømme, J. LePotier, P.I. Katsylo, G. Ottaviani, A.S. Tikhomirov, G. Trautmann) using a large variety of methods. Other properties of mathematical instantons have been investigated by many more other authors.
The conjecture has a simple interpretation in terms of matrices (due to W. Barth and to A.N. Tyurin). In order to recall this interpretation, we need to introduce some notation. If \(W\), \(W^\prime\) are \(k\)-vector spaces, let \(\text{L}(W^\prime, W)\) denote the space of \(k\)-linear maps \(: W^\prime \rightarrow W\), and let \(\text{L}_s(W, W^\vee)\) (resp., \(\text{L}_a(W, W^\vee)\)) denote the subspace of \(\text{L}(W, W^\vee)\) consisting of symmetric (resp., skew-symmetric) maps. \({\mathbb P}^3\) parametrizes the 1-dimensional subspaces of a 4-dimensional \(k\)-vector space \(V\). Let \(H = H_n\) be a fixed, \(n\)-dimensional, \(k\)-vector space. Put: \[ \begin{gathered} \mathbf{S}_n := \text{L}_s(H,H^\vee)\otimes \text{L}_a(V,V^\vee) \subset \text{L}_a(H\otimes V,H^\vee \otimes V^\vee)\, ,\\ \mathbf{S}_n^\vee := \text{L}_s(H^\vee ,H)\otimes \text{L}_a(V^\vee ,V) \subset \text{L}_a(H^\vee \otimes V^\vee ,H\otimes V)\, .\end{gathered} \] For \(A \in \mathbf{S}_n\), put \(W_A := \text{Coim}\, A = H\otimes V/\text{Ker}\, A\). The canonical surjection \(a : H\otimes V \rightarrow W_A\) and the map \(b : W_A \rightarrow H^\vee \otimes V^\vee\) induced by \(A\) define an anti-selfdual complex: \[ 0 \rightarrow H\otimes {\mathcal O}_{{\mathbb P}^3}(-1) \overset\alpha\longrightarrow W_A\otimes {\mathcal O}_{{\mathbb P}^3} \overset\beta\longrightarrow H^\vee \otimes {\mathcal O}_{{\mathbb P}^3}(1) \rightarrow 0\, . \] Moreover, \(\alpha\) is a locally split monomorphism iff \(\beta\) is an epimorphism iff \(A\) is nondegenerate in the sense that \(A(h\otimes v) \neq 0\), \(\forall \, h \in H\setminus \{0\}\), \(\forall \, v \in V \setminus \{0\}\). In this case, the middle cohomology of the above complex is a vector bundle \(E_A\) on \({\mathbb P}^3\), endowed with a skew-symmetric isomorphism \(E_A \overset\sim\rightarrow E_A^\vee\). It follows that \(\text{rk}\, A = 2n + \text{rk}\, E_A \geq 2n + 2\). Put: \[ \text{MI}_n := \{A\in \mathbf{S}_n\, | \, \text{rk}\, A \leq 2n + 2\;\text{and \(A\) nondegenerate}\}\, . \] \(\text{GL}(H)\) acts to the right on \(\mathbf{S}_n\) by \((A, g) \mapsto (g\otimes \text{id}_V)^\vee \circ A \circ (g\otimes \text{id}_V)\) and the map \(\text{MI}_n \rightarrow \text{I}_n\), \(A \mapsto [E_A]\), is a principal \(\text{GL}(H)/\{\pm \text{id}\}\)-bundle, hence the conjecture about \(\text{I}_n\) is equivalent to the fact that \(\text{MI}_n\) is smooth (the scheme structure of \(\text{MI}_n\) being defined by Pfaffians) and irreducible, of dimension \(n^2 + 8n - 3\). Recalling that: \[ \{A \in \text{L}_a(H\otimes V, H^\vee \otimes V^\vee)\, | \, \text{rk}\, A \leq 2n + 2\} \] is an irreducible subvariety of \(\text{L}_a(H\otimes V, H^\vee \otimes V^\vee)\) of codimension \(\binom{2n-2}{2}\), it follows that every irreducible component of: \[ \widetilde{\text{MI}}_n := \{ A \in \mathbf{S}_n \, | \, \text{rk}\, A \leq 2n + 2\} \] has dimension \(\geq n^2 + 8n - 3\). Consequently, the last part of the conjecture says that the condition “\(A\) nondegenerate” evidences an (open part of an) irreducible component of \(\widetilde{\text{MI}}_n\) of the expected dimension \(n^2 + 8n - 3\) (it is known that, for large \(n\), \(\widetilde{\text{MI}}_n\) has components of dimension \(> n^2 + 8n - 3\)).
The main result of the paper under review asserts that \(\text{MI}_n\) is irreducible, of dimension \(n^2 + 8n - 3\), for every odd integer \(n \geq 1\). Assume, from now on, that \(n\) is odd, \(n = 2m + 1\). The starting point of the proof is the following general position result: if \(E\) is an \(n\)-instanton then, for a general \(m\)-dimensional subspace \(K \subset \text{H}^1(E(-1))\), the multiplication map \(K \otimes \text{H}^0({\mathcal O}_{{\mathbb P}^3}(1)) \rightarrow \text{H}^1(E)\) is an isomorphism. Since one has identifications \(\text{H}^1(E_A(-1)) \simeq H^\vee\) and \(\text{H}^1(E_A) \simeq \text{Coker}\, A\), this general position result (appearing, also, in the paper of I. Coandă, A. Tikhomirov and G. Trautmann [Int. J. Math. 14, No. 1, 1–45 (2003; Zbl 1059.14018)]) has the following interpretation in terms of matrices: consider a fixed decomposition \(H = H_{m+1} \oplus H_m\) and the open subset \(\mathcal U\) of \(\widetilde{\text{MI}}_n\) consisting of the elements \(A\) for which the composite map (which we shall denote by \(A_1\)): \[ H_{m+1}\otimes V \hookrightarrow H\otimes V \overset{A}\longrightarrow H^\vee \otimes V^\vee \twoheadrightarrow H_{m+1}^\vee \otimes V^\vee \] is an isomorphism. Then the above general position result implies that \(\text{MI}_n \subseteq {\mathcal U}\cdot \text{GL}(H)\). Now, denoting by \(A_2\) (resp., \(A_3\)) the composite map: \[ \begin{gathered} H_m\otimes V \hookrightarrow H\otimes V \overset{A}\longrightarrow H^\vee \otimes V^\vee \twoheadrightarrow H_{m+1}^\vee \otimes V^\vee\\ (\text{resp.,}\;H_m\otimes V \hookrightarrow H\otimes V \overset{A}\longrightarrow H^\vee \otimes V^\vee \twoheadrightarrow H_m^\vee \otimes V^\vee)\end{gathered} \] it follows that, for \(A \in {\mathcal U}\), one has \(A_3 = - A_2^\vee \circ A_1^{-1} \circ A_2\) (since \(A_1\) has, already, rank \(2n + 2\)). Putting: \[ \boldsymbol{\Sigma}_{m+1} := \text{L}(H_m, H_{m+1}^\vee) \otimes \text{L}_a(V, V^\vee) \subset \text{L}(H_m \otimes V, H_{m+1}^\vee \otimes V^\vee)\, , \] one deduces that \(\mathcal U\) is isomorphic to the locally closed subscheme \(\widetilde{X}_m\) of \(\mathbf{S}_{m+1}^\vee \times \boldsymbol{\Sigma}_{m+1}\) consisting of the pairs \((D, C)\) with \(D\) an isomorphism and \(C^\vee \circ D \circ C \in \mathbf{S}_m\). For example, for \(m = 2\), representing \(D\) by a \(3\times 3\) symmetric matrix \((\delta_{ij})\) with entries in \(\text{L}_a(V^\vee , V)\) and \(C\) by a \(3\times 2\) matrix \((\gamma_{il})\) with entries in \(\text{L}_a(V,V^\vee)\), one has that \(C^\vee \circ D \circ C\) (which is represented by a \(2\times 2\) matrix with entries in \(\text{L}(V,V^\vee)\)) belongs to \(\mathbf{S}_m\) iff the \(2\times 2\) matrix representing it has entries in \(\text{L}_a(V,V^\vee)\), and this is equivalent to the fact that the following element of \(\text{L}_s(V,V^\vee)\): \[ { \sum_{i=1}^3\sum_{j=1}^3}(\gamma_{i1}\circ \delta_{ij} \circ \gamma_{j2} - \gamma_{j2} \circ \delta_{ij} \circ \gamma_{i1}) \] is 0, that is, \(\widetilde{X}_2\) is defined by ten cubic equations. Then, as much as the reviewer was able to understand, the author shows, by induction on \(m\), that \(\widetilde{X}_m\) (or, maybe, its open part \(X_m\) corresponding to \({\mathcal U}\cap \text{MI}_n\)) is irreducible.

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
14M12 Determinantal varieties

Citations:

Zbl 1059.14018
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