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I will refer to Simpson's "Higgs bundles and local systems".

Proposition 1.4:

When $X$ is a smooth projective variety, one can build up the moduli space $\mathcal{M}(X,r)$ of rank $r$ Higgs bundles (of a certain type) over $X$ as a quasiprojective variety. Furthermore, there is a proper map from $\mathcal{M}(X,r)$ to a [finite dimensional] vector space.

The proper map this proposition talks about is the Hitchin fibration

$$ \chi \, \colon \, \mathcal{M}(X,r) \to \bigoplus_{k=1}^r H^0(X,\mathrm{Sym}^k\,\Omega^1(X)) $$

sending each isomorphism class $[(E,\phi)]$ to the coefficients of the characteristic polynomial of the Higgs field $\phi$. These are holomorphic sections of the symmetric powers of $\,T^*_X \otimes \mathbb{C}$.

There is a continuous $\mathbb{C}^*$-action on $\mathcal{M}(X,r)$ defined by rescaling the Higgs field by $z \in \mathbb{C}^*$:

$$ z \cdot [(E,\phi)] = [(E,z \phi)] $$

In the proof of Theorem 3:

Since $$ \lim_{z \to 0} \, \chi (z \cdot[(E,\phi)]) = 0 $$ and $\chi$ is a proper map, we have that the limit $$ \lim_{z \, \in \, \mathbb{C}^* , \,z \to 0} z \cdot[(E,\phi)] $$ exists and is a fixed point of the $\mathbb{C}^*$-action.

Clearly, if the limit exists, being unique, it provides a fixed point. But I do not follow Simpson's argument leading to the existence of $\lim_{z \to 0} z \cdot[(E,\phi)]$. Maybe, I'm leaving out some considerations.

Question: Why the limit $\lim_{z \to 0} z \cdot[(E,\phi)]$ does exist?

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  • $\begingroup$ Where is the question here? $\endgroup$ – Stefan Kohl Nov 27 '14 at 9:43
  • $\begingroup$ Why the limit in Theorem 3 does exist? $\endgroup$ – Ivo Nov 27 '14 at 9:48
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    $\begingroup$ Just to clarify, are you asking why $\chi$ is a proper morphism? Assuming that $\chi$ is a proper morphism, the highlighted text seems self-explanatory. $\endgroup$ – Jason Starr Nov 27 '14 at 13:14
  • $\begingroup$ @JasonStarr I am taking for granted that $\chi$ is a proper continuous mapping $\endgroup$ – Ivo Nov 27 '14 at 13:40
  • $\begingroup$ This is the valuative criterion for properness. $\endgroup$ – Daniel Litt Nov 27 '14 at 18:52

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