Poincare cohomology at the identity

Put $B=A^{\vee }$ and $R={\mathcal{O}}_{B,0}$, a regular local ring of dimension $g$. For $b\mathrm{\ne }0$ the fibre ${\mathcal{P}}_b$ is a nontrivial degree-zero bundle on $A$, hence a nontrivial homogeneous bundle; by [F2] all its cohomology vanishes, so the formation of $R^i{p}_{2,\ast }\mathcal{P}$ is supported at the closed point $0\in B$, and by [F4] $R^i{p}_{2,\ast }\mathcal{P}=0$ for $i<0$ and $i>g$.

Local acyclicity claim. Let $C$ be a bounded complex of finite free modules over the $g$-dimensional regular local ring $R$ with $C^i=0$ for $i<0$ and all $H^i(C)$ of finite length. Then $H^i(C)=0$ for $i<g$. For the proof choose a regular parameter system $x_1,\dots ,x_g$ and let $E$ be the augmented Cech complex $R\to {⨁}_{i}R[x_i^{-1}]\to \cdots \to R[(x_1\cdots x_g{)}^{-1}]$ in degrees $0,\dots ,g$. By [F5] $E$ is the filtered colimit of the Koszul complexes on $x_1^n,\dots ,x_g^n$, each with no cohomology below $g$, so $H^i(E)=0$ for $i<g$. Form the bounded double complex $C{\otimes }_{R}E$. Computing $E$ first, its $E_1$ page vanishes in degrees below $g$ because $E$ does and $C$ starts in degree $0$, so the total cohomology vanishes below $g$. Computing $C$ first, each $E^q$ is a direct sum of localizations and hence flat, so $E_2^{p,q}=H^p(C){\otimes }_R{E}^q$; since $H^p(C)$ is finite length it is killed by a power of the maximal ideal, and $x_i$ lies in the maximal ideal, so every nonempty localization of $H^p(C)$ vanishes: $E_2^{p,q}=0$ for $q\ge 1$ and $E_2^{p,0}=H^p(C)$. The second spectral sequence degenerates and identifies the total cohomology in degree $p$ with $H^p(C)$; comparing with the first computation gives $H^p(C)=0$ for $p<g$.

Since $A\times B\to B$ is proper and $\mathcal{P}$ is flat over $B$, applying [F3] over $\mathrm{Spec}R$ produces a bounded finite free complex; cancel its contractible summands over the local ring $R$ to obtain a minimal complex $K$. Its reduction has zero differentials, and fibre cohomology vanishes above $g$, so its terms vanish above $g$. Thus $K$ lies in degrees $0,\dots ,g$ with canonical isomorphisms $H^i(K{\otimes }_R{A}^{\mathrm{\prime }})\cong H^i(A_{A^{\mathrm{\prime }}},{\mathcal{P}}_{A^{\mathrm{\prime }}})$ for every $R$-algebra $A^{\mathrm{\prime }}$; in particular $H^i(K)\cong (R^i{p}_{2,\ast }\mathcal{P}{)}_0$ is a finite-length $R$-module by [F3] and step 1.1, and $H^i(K{\otimes }_{R}k)=H^i(A,{\mathcal{O}}_A)$, with degree-zero and degree-$g$ dimensions one by [F4]; after minimalization these determine the endpoint ranks.

Apply the local acyclicity claim to $K$: $H^i(K)=0$ for $i<g$, so by step 1.1 the complex $K$ has cohomology only in degree $g$, where $H^g(K)$ is a finite-length $R$-module. Form the shifted dual complex $C={\mathrm{Hom}}_R(K,R)[-g]$, so $C^i={\mathrm{Hom}}_R(K^{g-i},R)$ with the usual dual signs; it is again a bounded complex of finite free modules in degrees $0,\dots ,g$. Away from the closed point the finite-length cohomology of $K$ localizes to zero, so $K$ becomes split exact there and hence so does its dual $C$; therefore each $H^i(C)$ is supported at the closed point and has finite length. The local acyclicity claim applied to $C$ gives $H^i(C)=0$ for $i<g$.

If $g=0$, the smooth geometrically connected zero-dimensional pointed variety $A$ is $\mathrm{Spec}k$, as is its dual, so the conclusion is immediate. Suppose $g>0$. The minimal complex $K$ of step 2.1 has endpoint ranks one, since $H^0(A,{\mathcal{O}}_A)=H^g(A,{\mathcal{O}}_A)=k$. Thus $K^0\cong K^g\cong R$, and the dual complex $C$ has endpoint ranks one too. By step 3.1 its sole cohomology is the terminal cokernel $H^g(C)=R\mathrm{/}J$, where $J$ is the ideal generated by the entries of its last differential, equivalently by the entries of the first differential $d_K^0:R\to K^1$. The module $R\mathrm{/}J$ has finite length, and minimality ensures $J\subseteq {\mathfrak{m}}_R$.

Tensor $K$ with $R\mathrm{/}J$. Its first differential vanishes because all entries lie in $J$, and it has no negative terms, so $H^0(K{\otimes }_{R}R\mathrm{/}J)=R\mathrm{/}J$. The universal cohomology identification of [F3] turns the basis vector of $K^0$ into a section of $\mathcal{P}$ over $A{\times }_k\mathrm{Spec}R\mathrm{/}J$. Its reduction is a nonzero section of ${\mathcal{O}}_A$, hence nowhere zero. Since the Artinian local scheme $\mathrm{Spec}R\mathrm{/}J$ has the same underlying point as $\mathrm{Spec}k$, Nakayama makes this section a trivialization everywhere; normalize its value at the identity to give a rigidified trivialization. The universal property [F1] therefore makes the classifying map $\mathrm{Spec}R\mathrm{/}J\to B$ factor through the origin. As this map is the natural map induced by $R={\mathcal{O}}_{B,0}$, it follows that ${\mathfrak{m}}_R\subseteq J$. Hence $J={\mathfrak{m}}_R$ and $H^g(C)\cong k$.

Thus $C$ is a minimal finite free complex over $R$ concentrated in degrees $0,\dots ,g$ with $H^i(C)=0$ for $i<g$ and $H^g(C)\cong k$; it is a minimal free resolution of $k$ shifted by $g$. Comparing it with the Koszul resolution of $k$ on a regular parameter system by chain lifting [F7] shows that the two complexes are homotopy equivalent, and dualizing back by ${\mathrm{Hom}}_R(-,R)[-g]$ (an exact anti-equivalence on finite free complexes, carrying the Koszul resolution to its dual) yields that $K$ is quasi-isomorphic to $k[-g]$: the dual Koszul complex has sole cohomology $k$ in degree $g$. Therefore $H^i(K)=0$ for $i\mathrm{\ne }g$ and $H^g(K)\cong k$ as an $R$-module.

Translating step 6.1 back through the identification $H^i(K)=(R^i{p}_{2,\ast }\mathcal{P}{)}_0$ of step 2.1 gives $R^i{p}_{2,\ast }\mathcal{P}=0$ for $i\mathrm{\ne }g$ and $R^g{p}_{2,\ast }\mathcal{P}$ a coherent sheaf supported at $0$ with stalk $k$, i.e. the length-one skyscraper $k(0)$; the identification is natural in the local ring, so the global statement follows.