Projective Perverse Hard Lefschetz And Decomposition Examples

The standard characteristic maps exhibit the cellular filtration ${\mathbb{P}}^0\subset {\mathbb{P}}^1\subset \cdots \subset {\mathbb{P}}^n$ with exactly one real cell in each dimension $0,2,\dots ,2n$: the map from the closed unit ball in ${\mathbb{C}}^j$ sending $w$ to $[w:\sqrt{1-\parallel w{\parallel }^2}]$ has interior mapping homeomorphically onto the last-coordinate affine chart and boundary mapping onto ${\mathbb{P}}^{j-1}$. Adjacent cellular groups sit in degrees of opposite parity, so the cellular differential vanishes and Cellular cochains compute cohomology with local coefficients gives $H^{2j}({\mathbb{P}}^n;k)=k$ for $0\le j\le n$ and $H^m({\mathbb{P}}^n;k)=0$ for all other $m$.

On a point the perverse support and cosupport inequalities are $H^q=0$ for $q>0$ and for $q<0$, respectively, by Ordinary middle perverse t structure, recollement and intersection complexes. Thus the perverse t-structure equals the ordinary one, its heart consists of finite-dimensional $k$-vector spaces in degree zero, and ${}^p{H}^i(R{f}_{\ast }k[n])=H^{n+i}({\mathbb{P}}^n;k)$. Since ${\mathbb{P}}^n$ is smooth projective and $f$ is projective, Compact smooth projective pushforwards have semisimple perverse cohomology in characteristic zero supplies the finite splitting $R{f}_{\ast }k[n]\cong {⨁}_i{}^p{H}^i(R{f}_{\ast }k[n])[-i]$. Substituting step 1.1 gives $R{f}_{\ast }k[n]\cong {⨁}_{j=0}^{n}k[n-2j]$, with a simple copy of $k$ exactly in degrees $-n,-n+2,\dots ,n$. For $n=0$ this is the single summand $k$; an empty source is outside the example.