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Über dieses Buch

This volume presents original research articles and extended surveys related to the mathematical interest and work of Jean-Michel Bismut. His outstanding contributions to probability theory and global analysis on manifolds have had a profound impact on several branches of mathematics in the areas of control theory, mathematical physics and arithmetic geometry.

Contributions by:

K. Behrend

N. Bergeron

S. K. Donaldson

J. Dubédat

B. Duplantier

G. Faltings

E. Getzler

G. Kings

R. Mazzeo

J. Millson

C. Moeglin

W. Müller

R. Rhodes

D. Rössler

S. Sheffield

A. Teleman

G. Tian

K-I. Yoshikawa

H. Weiss

W. Werner

The collection is a valuable resource for graduate students and researchers in these fields.

Inhaltsverzeichnis

Frontmatter

Geometric Higher Groupoids and Categories

In an enriched setting, we show that higher groupoids and higher categories form categories of fibrant objects, and that the nerve of a differential graded algebra is a higher category in the category of algebraic varieties.

Kai Behrend, Ezra Getzler

Hodge Type Theorems for Arithmetic Hyperbolic Manifolds

This note is a variation on the lecture given by the first named author at the conference celebrating the work (and sixty-fifth anniversary) of Jean-Michel Bismut. In this lecture a proof of some new cases of the Hodge conjecture for Shimura varieties uniformized by complex balls was sketched following [3]. In this note we exemplify the main ideas of the proof on real hyperbolic manifolds. The Hodge conjecture does not make sense anymore but, somewhat analogously, we prove that classes of totally geodesic submanifolds generate the cohomology groups of degree k of compact congruence n-dimensional hyperbolic manifolds “of simple type” as long as k is strictly smaller than n/3. This is a particular case of the main result of [2].

Nicolas Bergeron, John Millson, Colette Moeglin

The Ding Functional, Berndtsson Convexity and Moment Maps

The paper discusses the formal structure of the Kähler–Einstein equations over Fano manifolds, introducing a “moment map” interpretation. The key is a result of Berndtsson, which leads to a metric on the space of almost-complex structures. An application to the Kähler–Ricci flow is given.

S. K. Donaldson

Dimers and Curvature Formulae

In this expository note, we discuss different formalisms for determinants of families of elliptic operators (in the continuum) and finite-difference operators (in a discrete setting), how they relate, and consequences for the asymptotic analysis of some combinatorial models in 2d statistical mechanics.

Julien Dubédat

The Norm of the Weierstrass Section

In diophantine geometry over function fields, Weierstrass divisors are an important tool. Trying to make this tool available over numberfields we prove estimates for the archimedean and p-adic norms of the sections which define them.

Gerd Faltings

Higher Analytic Torsion, Polylogarithms and Norm Compatible Elements on Abelian Schemes

We give a simple axiomatic description of the degree 0 part of the polylogarithm on abelian schemes and show that its realisation in analytic Deligne cohomology can be described in terms of the Bismut–Köhler higher analytic torsion form of the Poincaré bundle.

Guido Kings, Damian Rössler

Teichmüller Theory for Conic Surfaces

This paper develops the local deformation theory, and some aspects of the global Teichmüller theory, of constant curvature metrics on a surface Σ with a finite number of conic singularities, with all cone angles less than 2π. We approach this using techniques of geometric analysis and the theory of elliptic operators on conic spaces.

Rafe Mazzeo, Hartmut Weiss

On the Analytic Torsion of Hyperbolic Manifolds of Finite Volume

In this paper we study the analytic torsion for a complete oriented hyperbolic manifold of finite volume. This requires the definition of a regularized trace of heat operators. We use the Selberg trace formula to study the asymptotic behavior of the regularized trace for small time. The main result of the paper is a new approach to deal with the weighted orbital integrals on the geometric side of the trace formula.

Werner Müller

Log-correlated Gaussian Fields: An Overview

We survey the properties of the log-correlated Gaussian field (LGF), which is a centered Gaussian random distribution (generalized function) h on ℝd, defined up to a global additive constant.

Bertrand Duplantier, Rémi Rhodes, Scott Sheffield, Vincent Vargas

A Variation Formula for the Determinant Line Bundle. Compact Subspaces of Moduli Spaces of Stable Bundles over Class VII Surfaces

This article deals with two topics: the first, which has a general character, is a variation formula for the determinant line bundle in non-Kählerian geometry. This formula, which is a consequence of the non-Kählerian version of the Grothendieck–Riemann–Roch theorem proved recently by Bismut [Bi], gives the variation of the determinant line bundle corresponding to a perturbation of a Fourier–Mukai kernel E on a product B×X by a unitary flat line bundle on the fiber X. When this fiber is a complex surface and E is a holomorphic 2-bundle, the result can be interpreted as a Donaldson invariant.The second topic concerns a geometric application of our variation formula, namely we will study compact complex subspaces of the moduli spaces of stable bundles considered in our program for proving existence of curves on minimal class VII surfaces [Te3]. Such a moduli space comes with a distinguished point a = [A] corresponding to the canonical extension A of X [Te2], Te3]. The compact subspaces Y ⊂ Mst containing this distinguished point play an important role in our program. We will prove a non-existence result: there exists no compact complex subspace of positive dimension Y ⊂ Mst containing a with an open neighborhood a ∈ Ya ⊂ Y such that Ya \ {a} consists only of non-filtrable bundles. In other words, within any compact complex subspace of positive dimension Y ⊂ Mst containing a, the point a can be approached by filtrable bundles. Specializing to the case b2 = 2 we obtain a new way to complete the proof of the main result of [Te3]: any minimal class VII surface with b2 = 2 has a cycle of curves. Applications to class VII surfaces with higher b2 will be be discussed in a forthcoming article.

Andrei Teleman

K-stability Implies CM-stability

In this paper, we prove in details that any polarized K-stable manifold is CM-stable. This extends what I did for Fano manifolds in my 2012 paper. Our proof is based on an asymptotic formula for the K-energy and S. Paul’s works ([Pa08], [Pa12], [Pa13]) on the K-stability in terms of stable pairs.

Gang Tian

A Simple Renormalization Flow for FK-percolation Models

We present a setup that enables to define in a concrete way a renormalization flow for the FK-percolation models from statistical physics (that are closely related to Ising and Potts models). In this setting that is applicable in any dimension of space, one can interpret perturbations of the critical (conjectural) scaling limits in terms of stationary distributions for rather simple Markov processes on spaces of abstract discrete weighted graphs.

Wendelin Werner

Analytic Torsion for Borcea–Voisin Threefolds

In their study of genus-one string amplitude F1, Bershadsky–Cecotti–Ooguri–Vafa discovered a remarkable identification between holomorphic Ray–Singer torsion and instanton numbers for Calabi–Yau threefolds. The holomorphic torsion invariant of Calabi–Yau threefolds corresponding to F1 is called BCOV invariant. In this paper, we establish an identification between the BCOV invariants of Borcea–Voisin threefolds and another holomorphic torsion invariants for K3 surfaces with involution. We also introduce BCOV invariants for abelian Calabi–Yau orbifolds. Between Borcea–Voisin orbifold and its crepant resolution, we compare their BCOV invariants.

Ken-Ichi Yoshikawa
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