On the stable principal Higgs sheaves (Q1023405)

The author proves the following theorem: If the principal Higgs \(G\)-sheaf \(\mathcal E:=(E_G,E,\psi,\varphi,\hat\varphi)\) is stable, then the injective homomorphism \[ \mathfrak z(\mathfrak g)\to\mathcal A(\mathcal E) \] is surjective, too. Let us explain notations and terminology. Here, \(G\) is a connected reductive linear algebraic group defined over the set of complex numbers, with Lie algebra \(\mathfrak g\) whose center is \(\mathfrak z(\mathfrak g)\). Let \(M\) be a compact connected Kähler manifold and \(E_G\) a holomorphic principal \(G\)-bundle over some open dense set \(U\subset M\) whose complement is a complex analytic subspace of complex codimension at least two. Let \(E\) be a torsionfree coherent analytic sheaf on \(M\) together with a map \[ \psi: E_G([\mathfrak g,\mathfrak g])\to E|_U, \] where \(E_G([\mathfrak g,\mathfrak g])\) is the vector bundle over \(U\) associated to \(E_G\) for the \(G\)-module \([\mathfrak g,\mathfrak g]\). Next, \(\varphi\) is a holomorphic section over \(U\) with values in \(\Omega^1_M\otimes\text{ad}(E_G)\), where \(\text{ad}(E_G)\) is the adjoint vector bundle and \(\hat\varphi\) is a holomorphic homomorphism of coherent analytic sheaves \(\hat\varphi: E\to\Omega^1_M\otimes E\) satisfying certain conditions. Define \(\mathcal A(\mathcal E)\) to be the complex subspace of \(H^0(U,\text{ad}(E_G))\) of all sections \(\tau\) such that \([\varphi,\tau]=0\) (using the Lie algebra structure of the fibers of \(\text{ad}(E_G)\), we have a Lie algebra strucutre on \(\mathcal A(\mathcal E)\) and this algebra is identified with the Lie algebra of automorphisms of the principal Higgs \(G\)-sheaf \(\mathcal E\)). Since the adjoint action of \(G\) on \(\mathfrak g\) fixes \(\mathfrak z(\mathfrak g)\) pointwise, each element of \(\mathfrak z(\mathfrak g)\) defines a holomorphic section of \(\text{ad}(E_G)\) over \(U\), from which, we have the injective morphism \(\mathfrak z(\mathfrak g)\hookrightarrow\mathcal A(\mathcal E)\).