Geometric Fixed Parts And Gauss Manin Semisimplicity Examples

Geometric structure and the local system. The graph $\{(t,[u:v])\in {\mathbb{C}}^{\times }\times {\mathbb{P}}^1:u^r=t{v}^r\}$ is closed, and $z\mapsto (z^r,[z:1])$ identifies ${\mathbb{C}}^{\times }$ with it over ${\mathbb{C}}^{\times }$, so $f$ is projective and proper and its fibres are the $r$ distinct $r$-th roots of the base point. On coordinate rings $f$ is module-finite, the source ring being generated over the target ring by $1,z,\dots ,z^{r-1}$; and it is smooth of relative dimension zero because the derivative $r{z}^{r-1}$ is a unit on ${\mathbb{C}}^{\times }$, so the Jacobian criterion Relative Jacobian criterion with its presentation hypothesis applies and the relative dimension is $1-1=0$, which is exactly étaleness by the definition Étale morphism of schemes. Thus $f$ is finite étale. Therefore $R^0{f}_{\ast }\mathbb{Q}$ is a finite-rank local system of rank $r$ in the sense of Local systems and pullback: over each base point the fibre is $H^0$ of $r$ reduced points, with the sheet-indicator functions as a rational basis, and these bases are transported by the deck identifications of the fibres. Every indicator function is locally constant of Hodge type $(0,0)$, so the local system is pure of weight zero. The standard positive definite dot product in the indicator basis is rational, flat because transport permutes the basis vectors and hence preserves the form, and positive on every fibre; it is therefore the polarization required for a geometric polarizable weight-zero variation, whose algebraic realization is $f$ itself on the open $U={\mathbb{C}}^{\times }$.

Monodromy and semisimplicity. The homotopy $z\mapsto ((1-u)+u\mathrm{\mid }{t}_0\mathrm{\mid }\mathrm{/}\mathrm{\mid }z\mathrm{\mid })z$ gives a deformation retraction of ${\mathbb{C}}^{\times }$ to the circle of radius $\mathrm{\mid }{t}_0\mathrm{\mid }$, fixing $t_0$ throughout. Multiplication $w\mapsto t_{0}w$ identifies that circle, based at $t_0$, with the geometric unit circle, based at $1$. Hence The trigonometric loops give ${\pi }_1(\{(x,y):x^2+y^2=1\},(1,0))\cong \mathbb{Z}$ makes $\gamma (t)=e^{2\pi it}{t}_0$ a positive generator at $t_0$. Label the roots by $z_a={\zeta }^a{z}_0$ and their indicator functions by $e_a$, where $\zeta =e^{2\pi i\mathrm{/}r}$ has order $r$ by The $n$-th roots of a complex number and the $n$ distinct roots of unity for every $n\ge 1$. The lifted loop $t\mapsto e^{2\pi it\mathrm{/}r}{z}_a$ ends at $z_{a+1}$. Covariant cohomology transport is the inverse pullback, so $T_{\gamma }{e}_a=e_{a+1}$. The left monodromy convention of Local systems and pullback uses $\rho (\gamma )=T_{\bar{\gamma }}$, hence $\rho (\gamma )e_a=e_{a-1}$. Therefore $v_j={\sum }_{a=0}^{r-1}{\zeta }^{ja}{e}_a$ satisfies $\rho (\gamma )v_j={\zeta }^j{v}_j$. These vectors form a basis: the geometric-sum identity ${\sum }_{a=0}^{r-1}{\zeta }^{ka}=0$ for $r\nmid k$ and $r$ otherwise gives the inverse Fourier matrix $r^{-1}({\zeta }^{-ja})$. Thus the complexification is a direct sum of the $r$ distinct one-dimensional characters, including the trivial character when $r=1$, and is semisimple. Finally Semisimplicity of geometric Gauss-Manin local systems applies to this smooth projective morphism and proves semisimplicity over every characteristic-zero coefficient field, including fields without the $r$-th roots of unity.