Abstract
El trabajo de Hausel prueba que la estratificación de Bialynicki-Birula del espacio moduli de fibrados de Higgs de rango dos coincide con su estratificación de Shatz. Él usa este hecho para calcular algunos grupos de homotopía del espacio moduli de k-fibrados de Higgs de rango dos. Desafortunadamente, estas dos estratificaciones no coinciden en general. Aquí, el objetivo es presentar una prueba diferente de la estabilización de los grupos de homotopía de Mk(2, d), y generalizarla a Mk(3, d), los espacios moduli de k-fibrados de Higgs de grado d, y rangos dos y tres respectivamente, sobre una superficie de Riemann compacta X, usando los resultados de los trabajos de Hausel y Thaddeus, entre otras herramientas.
Highlights
In this work, we estimate some homotopy groups of the moduli spaces of kHiggs bundles Mk(r, d) over a compact Riemann surface X of genus g > 2.This space was first introduced by Hitchin [17]; and it was worked by Hausel [14], where he estimated some of the homotopy groups working the particular case of rank two, and denoting M∞ = lim Mk as the direct limit k→∞ of the sequence.The co-prime condition GCD(r, d) = 1 implies that Mk(r, d) is smooth
We estimate some homotopy groups of the moduli spaces of kHiggs bundles Mk(r, d) over a compact Riemann surface X of genus g > 2
We shall do the estimate with Higgs bundles of fixed determinant det(E) = Λ ∈ J d, where J d is the Jacobian of degree d line bundles on X, to ensure that N (r, d) and M(r, d) are connected
Summary
We estimate some homotopy groups of the moduli spaces of kHiggs bundles Mk(r, d) over a compact Riemann surface X of genus g > 2. Hausel [14] estimates the homotopy groups πn(Mk(2, 1)) using two main tools: first the coincidence mentioned before between the Bialynicki-Birula stratification and the Shatz stratification; and second, the well-behaved embeddings Mk(2, 1) → Mk+1(2, 1). These inclusions are well-behaved in general for GCD(r, d) = 1; those two stratifications above mentioned do not coincide in general (see for instance [11]).
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