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Pierre Deligne

Publications and source records attributed to Pierre Deligne.

6 recordsLinked to original sources
arXiv2024

Motives, Periods, and Functoriality

Given a pure motive $M$ over $\mathbb{Q}$ with a multilinear algebraic structure $\mathsf{s}$ on $M$, and given a representation $V$ of the group respecting $\mathsf{s}$, we describe a functorial transfer $M^V$. We formulate a criterion that guarantees when the two periods of $M^V$ are equal. This has an implication for the critical values of the $L$-function attached to $M^V.$ The criterion is explicated in a variety of examples such as: tensor product motives and Rankin-Selberg $L$-functions; orthogonal motives and the standard $L$-function for even orthogonal groups; twisted tensor motives and Asai $L$-functions.

math.NT↗
arXiv2019

On the $K(π, 1)$-problem for restrictions of complex reflection arrangements

Let $W\subset GL(V)$ be a complex reflection group, and ${\mathscr A}(W)$ the set of the mirrors of the complex reflections in $W$. It is known that the complement $X({\mathscr A}(W))$ of the reflection arrangement ${\mathscr A}(W)$ is a $K(π,1)$ space. For $Y$ an intersection of hyperplanes in $\mathscr A(W)$, let $X(\mathscr A(W)^Y)$ be the complement in $Y$ of the hyperplanes in $\mathscr A(W)$ not containing $Y$. We hope that $X(\mathscr A(W)^Y)$ is always a $K(π,1)$. We prove it in case of the monomial groups $W = G(r,p,\ell)$. Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this $K(π,1)$ property remains to be proved.

math.AT↗
arXiv2017

Arithmetic monodromy actions on pro-metabelian fundamental groups of once-punctured elliptic curves

We prove structure theorems for the moduli stack of elliptic curves equipped with $G$-structures, where $G$ is a finite 2-generated metabelian group. In particular, we show that if $G$ has exponent $e$, then there is a subgroup $H\le GL_2(\mathbb{Z}/e)$ such that $G$-structures on elliptic curves $E$ are equivalent to "congruence structures of level $H$". Our methods are almost entirely group theoretic. Let $\widehat{M}$ denote the free profinite metabelian group of rank 2, then along the way we prove a decomposition of $Out(\widehat{M})$ as an internal semi-direct product of the subgroup of "braid-like outer automorphisms" with the subgroup of "IA" outer automorphisms which induce the identity on the abelianization. We also show a surprising result that all IA-automorphisms leave every open normal subgroup stable.

math.AG↗
arXiv1999

Supersolutions

We develop classical globally supersymmetric theories. As much as possible, we treat various dimensions and various amounts of supersymmetry in a uniform manner. We discuss theories both in components and in superspace. Throughout we emphasize geometric aspects. The beginning chapters give a general discussion about supersymmetric field theories; then we move on to detailed computations of lagrangians, etc. in specific theories. An appendix details our sign conventions. This text will appear in a two-volume work "Quantum Fields and Strings: A Course for Mathematicians" to be published soon by the American Mathematical Society. Some of the cross-references may be found at http://www.math.ias.edu/~drm/QFT/

hep-th↗
arXiv1994

On the Locus of Hodge Classes

Let $f: X \rightarrow S$ be a family of non singular projective varieties parametrized by a complex algebraic variety $S$. Fix $s \in S$, an integer $p$, and a class $h \in {\rm H}^{2p}(X_s,\Z)$ of Hodge type $(p,p)$. We show that the locus, on $S$, where $h$ remains of type $(p,p)$ is algebraic. This result, which in the geometric case would follow from the rational Hodge conjecture, is obtained in the setting of variations of Hodge structures.

alg-geom↗