Hitchin map on even very stable upward flows (Q6573206)

The main object of study are \textit{even very stable Higgs bundles}. The notion of a very stable Higgs bundle, motivated by mirror symmetry, is that of a couple \((E,\Phi)\) of a vector bundle \(E\) (on a smooth projective curve) together with a Higgs field \(\Phi\). A fixed point, under the \(\mathbb{C}^\times\) action given by scaling the Higgs field, is called very stable if the corresponding upward flow is closed.\N\NVery stable Higgs bundles of type \((1,\ldots,1)\) had been previously classified by T. Hausel and N. J. Hitchin. In this paper, such a classification is recalled and is moreover reformulated in terms of minuscule dominant weights of \(\mathrm{GL}_n\).\N\NThe authors give the definition of \emph{even} upward flows as being those fixed by the involution \((E, \Phi) \mapsto (E, -\Phi)\). This allows them to define even very stable Higgs bundles.\N\NThe main classification result is the following (Theorem 1.2): let \((E,\Phi)\) be an even very stable Higgs bundle of type \((1,\ldots,1)\). The very stable condition is equivalent to \(E\) being the direct sum of \(n\) line bundles \(L_0,\ldots, L_{n-1}\). Let \(b_i\) denote the restriction of the Higgs field \(\Phi\) to \(L_i\). Then \((E,\Phi)\) is even very stable if and only if the divisors\N\[\N\mathrm{div}(b_2) + \cdots \mathrm{div}(b_{n-2}), \quad \mathrm{div}(b_i)+ \mathrm{div}(b_{i+2k+1}) \quad \text{for } 1\leq i \leq i+2k+1\leq n-1\N\]\Nare all reduced. To make this easier to understand, this statement is reformulated (Theorem 1.3) in terms of even minuscule dominant weights.\N\NA later part of the paper (Theorem 4.1) deals with the problem of modeling the Hitchin map on some specific (cominuscule) even stable flows. This map turns out to be described by the cohomology of homogeneous spaces: quaternionic Grassmannians for \(\mathrm{GL}_{2n}\), the \(4n\)-sphere for \(\mathrm{SO}_{4n+2}\) and the real Cayley plane for \(E_6\).