Prof. Dr. Bruno Klingler

Prof. Klingler erforscht die Hodge-Theorie komplexer algebraischer Varietäten, insbesondere die Frage, wann und wo spezielle algebraische Strukturen (sogenannte Hodge-Loci) in geometrischen Objekten auftreten. Sein aktueller Fokus liegt auf der Anwendung o-minimaler Geometrie, um die Transzendenz in der Hodge-Theorie zu begrenzen und klassische Vermutungen wie die Zilber-Pink-Vermutung zu beweisen. Diese Arbeiten verbinden algebraische und transzendente Aspekte der Mathematik und ermöglichen es, tiefe Fragen über die arithmetische Natur geometrischer Strukturen zu beantworten. Die Ergebnisse sind grundlegend für die Zahlentheorie und algebraische Geometrie und haben Anwendungen in der Theorie rationaler Punkte auf algebraischen Varietäten.

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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.