In this paper, we develop a method of localization in equivariant cohomology based on the notion of partition of unity cohomology. We apply this method in two cases. In the first case, this method gives a refinement of the localization of Atiyah–Bott and Berline–Vergne (in the frame given by Bismut). After, we consider the Hamiltonian action of a torus, and we realise, following the idea of Witten, the localization on the critical points of the square of the moment map.

We associate to a group-like monoidal grupoid ${\mathcal C}$a principal bundle E satisfying most of the axioms defining a biextension. The obstruction to the existence of a genuine biextension structure on E is exhibited. When this obstruction vanishes, the biextension E is alternating and a trivialization of E induces a trivialization of${\mathcal C}$. The analogous theory for monoidal n-categories is also examined, as well as the appropriate generalization of these constructions in a sheaf-theoretic context. In the n-categorial situation, this produces a higher commutator calculus, in which some interesting generalizations of the notion of an alternating biextension occur. For n=2, the corresponding cocycles are constructed explicitly, by a partial symmetrization process, from the cocycle describing the n-category.

Let D≡ 7 mod 8 be a positive squarefree integer, and let h$_D$ be the ideal class number of E$_D$=Q($\sqrt{−D}$). Let d≡1 mod 4 be a squarefree integer relatively prime to D. Then for any integer k[ges ]0 there is a constant M=M(k), independent of the pair (D,D), such that if (−1)$^k$=sign (d), (2k+1,h$_D$)=1, and$\sqrt{D}$>(12/π)d$^2$(log|d+M(k)), then the central L-value L(k+1, χ$^2k+1$$_D, d$>0. Furthermore, for k[les ]1, we can take M(k)=0. Finally, if D=p is a prime, and d>0, then the associated elliptic curve A(p)$^d$ has Mordell–Weil rank 0 (over its definition field) when $\sqrt{D}$>(12/π)d$^2$ log d.