A fixed point formula of Lefschetz type in Arakelov geometry II: a residue formula

对数莱夫谢茨不动点公式与共振边界指标

Logarithmic Lefschetz fixed point formulae and resonant boundary indices

该研究针对带简单正常交叉边界且有孤立不动点的紧复流形严格自映射，证明了对数莱夫谢茨不动点公式，明确了边界点处特殊化系数的来源及共振与非共振边界项的不同表现。

Localized intersection of currents and the Lefschetz coincidence point theorem

We introduce the notion of a Thom class of a current and define the localized intersection of currents. In particular we consider the situation where we have a smooth map of manifolds and study localized intersections of the source manifold and currents on the target manifold. We then obtain a residue theorem on the source manifold and give explicit formulas for the residues in some cases. These are applied to the problem of coincidence points of two maps. We define the global and local coincidence homology classes and indices. A representation of the Thom class of the graph as a Cech-de~Rham …

Quasi-projective and formal-analytic arithmetic surfaces

This memoir is devoted to the study of formal-analytic arithmetic surfaces. These are arithmetic counterparts, in the context of Arakelov geometry, of germs of smooth complex-analytic surfaces along a projective complex curve. Formal-analytic surfaces provide a natural framework for arithmetic algebraization theorems, old and new. Formal-analytic arithmetic surfaces admit a rich geometry which parallels the geometry of complex analytic surfaces. Notably the dichotomy between pseudoconvexity and pseudoconcavity plays a central role in their geometry. Our study of formal-analytic arithmetic surf…

On localization and Riemann-Roch numbers for symplectic quotients

Suppose $(M,ω)$ is a compact symplectic manifold acted on by a compact Lie group $K$ in a Hamiltonian fashion, with moment map $μ: M \to \Lie(K)^*$ and Marsden-Weinstein reduction $M_{red} = μ^{-1}(0)/K$. In this paper, we assume that $M$ has a $K$-invariant Kähler structure. In an earlier paper, we proved a formula (the residue formula) for $η_0 e^{ω_0}[M_{red}]$ for any $η_0 \in H^*(M_{red})$, where $ω_0$ is the induced symplectic form on $M_{red}$. Here we apply the residue formula in the special case $η_0 = Td(M_{red})$; when $K$ acts freely on $μ^{-1}(0)$ this yields a formula for the Rie…

Archimedean cohomology revisited

Archimedean cohomology provides a cohomological interpretation for the calculation of the local L-factors at archimedean places as zeta regularized determinant of a log of Frobenius. In this paper we investigate further the properties of the Lefschetz and log of monodromy operators on this cohomology. We use the Connes-Kreimer formalism of renormalization to obtain a fuchsian connection whose residue is the log of the monodromy. We also present a dictionary of analogies between the geometry of a tubular neighborhood of the ``fiber at arithmetic infinity'' of an arithmetic variety and the compl…