Holomorphic equivariant analytic torsions

Abstract

The purpose of this paper is to construct and compare two natural definitions of the equivariant holomorphic torsion. The comparison formula is shown to be compatible with the embedding formulas obtained by the first author for analytic torsion forms and equivariant analytic torsion.

A fixed point formula of Lefschetz type in Arakelov geometry I: statement and proof

“…A more general version of this corollary is a consequence of [BGo2] (see the next section). We now combine our results with the (non-equivariant) arithmetic Riemann-Roch theorem.…”

“…For a smooth differential form η it is shown in [BGo2] (see also [B1, section C,D]) that the following integrals are well-defined:…”

“…The main result of [BGo2] (announced in [BGo1]) implies that for t ∈ R \ {0}, t sufficiently small, there is a power series T t in t with T 0 = T (M, x) such that Letη(t) denote the form obtained from η(t) by multiplying the degree deg Y = j part with t −j−1 for 0 ≤ j ≤ d. By making the change of variable from u to t 2 u we get (A tK + B tK )(Td tK (T M )ch tK (x))(s) = t 2s (A K + B K )( j c qj ,K (E j ) +η(t))(s) . …”

“…We then obtain a first version of the residue formula, which arises from the fact that the left hand-side of the last equation is also computed by the (non-equivariant) arithmetic RiemannRoch. The following subsection then shows how L T (·) can be computed using the results of Bismut-Goette [BGo2]; combining the results of that subsection with the first version of the residue formula gives our final version 2.11.…”

“…Also it is shown in [BGo2] (compare [B1,Proof of theorem 7]) that s → A K ′ (η)(s) has a meromorphic extension to C which is holomorphic at s = 0 and that…”