Torified varieties and their geometries over F_1

Quantum field theory over F1

In this paper we discuss some questions about geometry over the field with one element, motivated by the properties of algebraic varieties that arise in perturbative quantum field theory. We follow the approach to F1-geometry based on torified schemes. We first discuss some simple necessary conditions in terms of Euler characteristic and classes in the Grothendieck ring, then we give a blowup formula for torified varieties and we show that the wonderful compactifications of the graph configuration spaces, that arise in the computation of Feynman integrals in position space, admit an F1-structu…

Mapping F_1-land:An overview of geometries over the field with one element

This paper gives an overview of the various approaches towards F_1-geometry. In a first part, we review all known theories in literature so far, which are: Deitmar's F_1-schemes, Toën and Vaquié's F_1-schemes, Haran's F-schemes, Durov's generalized schemes, Soulé's varieties over F_1 as well as his and Connes-Consani's variations of this theory, Connes and Consani's F_1-schemes, the author's torified varieties and Borger's Lambda-schemes. In a second part, we will tie up these different theories by describing functors between the different F_1-geometries, which partly rely on the work of other…

Algebraic groups over the field with one element

Remarks in a paper by Jacques Tits from 1956 led to a philosophy how a theory of split reductive groups over $\F_1$, the so-called field with one element, should look like. Namely, every split reductive group over $\Z$ should descend to $\F_1$, and its group of $\F_1$-rational points should be its Weyl group. We connect the notion of a torified variety to the notion of $\F_1$-schemes as introduced by Connes and Consani. This yields models of toric varieties, Schubert varieties and split reductive groups as $\Fun$-schemes. We endow the class of $\F_1$-schemes with two classes of morphisms, one …

Geometrie d'Arakelov et hauteurs canoniques sur des varietes semi-abeliennes

Canonical heights and Arakelov geometry on semi-abelian varieties. In this paper, we propose a construction of the canonical heights on an extension of an abelian variety by the multiplicative group, in the framework of Arakelov geometry. These canonical heights are the sum of some height coming from the abelian variety and something we call a relative height. We finally give some complements about the points whose relative height is zero.

对数典范模型与正几何

Log Canonical Models and Positive Geometries

本文针对双有理几何中构造开簇对数典范紧化的核心问题，提出利用正几何典范形式获取对数典范模型显式坐标的方法，并将其应用于超平面排列补集等三类具体对象。

Categories derivees et geometrie birationnelle

Originally a technical tool, the derived category of coherent sheaves over an algebraic variety has become over the last twenty years an important invariant in the birational study of algebraic varieties. Problems of birational invariance and of minimization of the derived category have appeared, inspired by Kontsevich's homological mirror symmetry conjecture and Mori's minimal model program. We present the main conjectures and their proofs in dimension 3 and for particular classes of flops.