Prof. Dr. Bruno Klingler

Bruno Klingler ist ein weltweit renommierter Mathematiker, dessen Forschungsschwerpunkt auf der Beziehung zwischen der Topologie algebraischer Varietäten und ihren arithmetischen Strukturen liegt. Er arbeitet intensiv an der Hodge-Theorie, der algebraischen Geometrie und der Transzendenztheorie, wobei er o-minimale Geometrie als Rahmen für die Analyse von Hodge-Loci nutzt. Seine Publikationen behandeln zentrale Probleme wie die André-Oort-Vermutung, die Ax-Schanuel-Vermutung und die Zilber-Pink-Vermutung. Klingler verbindet dabei tiefe algebraische Methoden mit topologischen und analytischen Ansätzen, um die Struktur von Modulräumen und Shimura-Varietäten zu verstehen.

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• Einstein-Professur Bruno Klingler

  Quelle ↗

  Förderer: Einstein Professur Zeitraum: 07/2017 - 09/2019 Projektleitung: Prof. Dr. Bruno Klingler

• EXC 2046/1: Berlin Mathematics Research Center (MATH+)

  Quelle ↗

  Förderer: DFG Exzellenzstrategie Cluster Zeitraum: 01/2019 - 12/2025 Projektleitung: Prof. Dr. Michael Hintermüller, Prof. Dr. Andrea Walther

• EXC 2046: Berlin Mathematics Research Center (MATH+)

  Quelle ↗

  Förderer: DFG Exzellenzstrategie Cluster Zeitraum: 01/2019 - 12/2032 Projektleitung: Prof. Dr. Michael Hintermüller, Prof. Dr. Andrea Walther

• Compl�tude des vari�t�s Lorentziennes � courbure constante

  1996

  Mathematische Annalen · DOI

• The hyperbolic Ax-Lindemann-Weierstraß conjecture

  2015

  Publications mathématiques de l IHÉS · DOI

  1.1. Bi-algebraic geometry and the Ax-Lindemann-Weierstras property. — Let X and S be complex algebraic varieties and suppose π : Xan −→ San is a complex analytic, nonalgebraic, morphism between the associated complex analytic spaces. In this situation the image π(Y) of a generic algebraic subvariety Y ⊂ X is usually highly transcendental and the pairs (Y ⊂ X,V ⊂ S) of irreducible algebraic subvarieties such that π(Y) = V are rare and of particular geometric significance. We will say that an irreducible subvariety Y ⊂ X (resp. V ⊂ S) is bi-algebraic if π(Y) is an algebraic subvariety of S (resp. any analytic irreducible component of π−1(V) is an irreducible algebraic subvariety of X). Notice that V ⊂ S is bi-algebraic if and only if any analytic irreducible component of π−1(V) is bi-algebraic.

• Tame topology of arithmetic quotients and algebraicity of Hodge loci

  2020

  Journal of the American Mathematical Society · DOI

  In this paper we prove the following results: <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We show that any arithmetic quotient of a homogeneous space admits a natural real semi-algebraic structure for which its Hecke correspondences are semi-algebraic. A particularly important example is given by Hodge varieties, which parametrize pure polarized integral Hodge structures. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We prove that the period map associated to any pure polarized variation of integral Hodge structures <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper V"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {V}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a smooth complex quasi-projective variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definable with respect to an o-minimal structure on the relevant Hodge variety induced by the above semi-algebraic structure. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">3)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> As a corollary of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and of Peterzil-Starchenko’s o-minimal Chow theorem we recover that the Hodge locus of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper S comma double-struck upper V right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">V</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(S, \mathbb {V})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a countable union of algebraic subvarieties of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , a result originally due to Cattani-Deligne-Kaplan. Our approach simplifies the proof of Cattani-Deligne-Kaplan, as it does not use the full power of the difficult multivariable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S upper L 2"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">SL_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -orbit theorem of Cattani-Kaplan-Schmid.