Autodual connection in the Fourier transform of a Higgs bundle
A rank \(n\) and degree \(d\) Higgs bundle on a smooth projective complex curve \(X\), is a pair \(\mathsf{E}=(\mathcal E,\theta)\) where \(\mathcal E\) is a vector bundle on \(X\) of rank \(n\) and degree \(d\), and \(\theta:\mathcal E\to\mathcal E\otimes\Omega_X^1\). \(\mathsf{E}\) is semistable if \(\mu(\mathcal E')\leq\mu(\mathcal E)=\frac{d}{n}\), for every subbundle \(\mathcal E'\) of \(\mathcal E\) such that \(\theta(\mathcal E')\subset \mathcal E'\otimes\Omega_X^1\). \textit{N. J. Hitchin} [ Proc. Lond. Math. Soc., III. 55, 59--126 (1987; Zbl 0634.53045)] proved that the moduli space of semistable Higgs bundles over \(X\) contains the cotangent bundle of the moduli space of vector bundles on \(X\), and that it carries a natural hyper-Kähler structure, i.e. it is Kähler with respect to three complex structures \(I\), \(J\) and \(K\) satisfying \(IJK=-1\). Let \(\mathsf{E}=(\mathcal E,\theta)\) be a Higgs bundle of degree \(0\) on \(X\). Then, as proved by the author [J. Reine Angew. Math. 591, 21--48 (2006; Zbl 1094.14023)], its Fourier-Mukai transform \(\hat{\mathsf{E}}\) is an algebraic vector bundle on the cotangent bundle of the Jacobian of \(X\), \(T^*J(X)\cong J(X)\times H^0(X,\Omega_X^1)\), which is precisely the moduli space of rank \(1\) and degree \(0\) Higgs bundles on \(X\). The main result of this paper is the construction of a natural connection on \(\hat{\mathsf{E}}\) which is autodual with respect to the hyper-Kähler structure of \(T^*J(X)\). Given a complex manifold \(M\), its twistor space is \(\mathbf{Tw}(M)=M\times\mathbb{P}_{\mathbb{C}}^1\). Using a result of \textit{D. Kaledin} and \textit{M. Verbitsky} [Sel. Math., New Ser. 4, No. 2, 279--320 (1998; Zbl 0917.53006)] the construction of this autodual connection is reduced to the construction of a suitable holomorphic vector bundle on \(\mathbf{Tw}(T^*J(X))\), for which there is a complex analytic description (in arbitrary rank) worked out by \textit{C. Simpson} [in: Algebraic geometry. Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, July 9--29, 1995. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 62(pt.2), 217--281 (1997; Zbl 0914.14003)] and by P. Deligne. They give a description of the twistor of the moduli space of rank \(1\) and degree \(0\) Higgs bundles on \(X\) in terms of moduli spaces of \(\lambda\)-connections, which we now define. Let \(S\) be a scheme over \(\mathbb{C}\), and fix a smooth \(S\)-scheme, \(f:X\to S\). Let \(\lambda:S\to\mathbb{A}^1\) be a morphism of schemes, and let \(\mathcal E\) be a locally free \(\mathcal O_{X\times S}\)-module. Then a \(\lambda\)-connection on \(\mathcal E\) is a morphism of sheaves \(\nabla:\mathcal E\to \mathcal E\otimes\Omega^1_{X\times S/S}\) such that \(\nabla(ae)=\lambda e\otimes da+a\nabla(e)\) and \(\nabla^2=\nabla\circ\nabla=0\). Notice that a \(1\)-connection is just a flat connection and a \(0\)-connection is a Higgs bundle. Hence, \(\lambda\)-connections are object that interpolate between Higgs bundles and flat connections. Now, Deligne's and Simpson's description, shows that \(\mathbf{Tw}(T^*J(X))\) can be obtained by gluing the moduli space of rank \(1\) and degree \(0\) \(\lambda\)-connections to its own complex conjugate via a canonical antiholomorphic involution. The author uses this description to reduce the construction of the autodual connection on \(\hat{\mathsf{E}}\) to the construction of a locally free sheaf extending \(\hat{\mathsf{E}}\) to the moduli of \(\lambda\)-connections in a way such that this extension can be glued to its conjugate. This a suitable vector bundle on \(\mathbf{Tw}(T^*J(X))\), in the conditions of D. Kaledin and M. Verbitsky's result, producing therefore the required autodual connection.
scientific article; zbMATH DE number 5851255
scientific article