arXiv 1507.045682015-07-16math.AG

Higgs丛和与态射相关的表示空间

Higgs bundles and representation spaces associated to morphisms

Indranil Biswas, Carlos Florentino

摘要：

设$G$是定义在复数上的连通约化仿射代数群，$K\subset G$是极大紧子群.设$X , Y$是不可约光滑复射影簇，$f: X \rightarrow Y$是代数态射，使得$π_1(Y)$是实幂零的，同态$f_* : π_1(X) \rightarrowπ_1(Y)$是满射.定义$$ {\mathcal R }^f(π_1(X),\, G)\,=\, \{ρ\, \in\, \text{Hom}(π_1(X),\, G)\, \mid\, A\circρ\ \text{ factors through }~ f_*\}\, , $$ $$ {\mathcal R }^f(π_1(X),\, K)\,=\, \{ρ\, \in\, \text{Hom}(π_1(X),\, K)\, \mid\, A\circρ\ \text{ factors through }~ f_*\}\, , $$，其中$A: G \rightarrow \text{GL}(\text{Lie}(G))$是伴随动作。我们证明了几何不变理论商${\mathcal R }^f(π_1(X, x_0), G)/\!\!/G$允许变形收缩到${\mathcal R }^f(π_1(X, x_0),\, K)/K$ .我们还证明了$G$中的$n$几乎可换元的共轭类空间可以变形收缩到$G$中的$n$几乎可换元的共轭类空间。

英文摘要：

Let $G$ be a connected reductive affine algebraic group defined over the complex numbers, and $K\subset G$ be a maximal compact subgroup. Let $X , Y$ be irreducible smooth complex projective varieties and $f: X \rightarrow Y$ an algebraic morphism, such that $π_1(Y)$ is virtually nilpotent and the homomorphism $f_* : π_1(X) \rightarrowπ_1(Y)$ is surjective. Define $$ {\mathcal R }^f(π_1(X),\, G)\,=\, \{ρ\, \in\, \text{Hom}(π_1(X),\, G)\, \mid\, A\circρ\ \text{ factors through }~ f_*\}\, , $$ $$ {\mathcal R }^f(π_1(X),\, K)\,=\, \{ρ\, \in\, \text{Hom}(π_1(X),\, K)\, \mid\, A\circρ\ \text{ factors through }~ f_*\}\, , $$ where $A: G \rightarrow \text{GL}(\text{Lie}(G))$ is the adjoint action. We prove that the geometric invariant theoretic quotient ${\mathcal R }^f(π_1(X, x_0), G)/\!\!/G$ admits a deformation retraction to ${\mathcal R }^f(π_1(X, x_0),\, K)/K$. We also show that the space of conjugacy classes of $n$ almost commuting elements in $G$ admits a deformation retraction to the space of conjugacy classes of $n$ almost commuting elements in $K$.

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摘要：

英文摘要：

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