Prof. Dr. Bruno Klingler
Profil
Forschungsthemen6
2025-BMS-5 „Familie lokaler Systeme und Mapping-Klassengruppen”
Quelle ↗Förderer: DFG Exzellenzstrategie Cluster Zeitraum: 03/2025 - 02/2028 Projektleitung: Prof. Dr. Bruno Klingler
Einstein-Professur Bruno Klingler
Quelle ↗Förderer: Einstein Professur Zeitraum: 07/2017 - 09/2019 Projektleitung: Prof. Dr. Bruno Klingler
GRK 2965/1: „Von Geometrie zu Zahlen: Moduli, Hodge Theorie, rationale Punkte“
Quelle ↗Förderer: DFG Graduiertenkolleg Zeitraum: 10/2024 - 09/2029 Projektleitung: Prof. Dr. Gavril Farkas
GRK 2965: Von Geometrie zu Zahlen: Moduli, Hodge Theorie, rationale Punkte
Quelle ↗Förderer: DFG Graduiertenkolleg Zeitraum: 10/2024 - 09/2029 Projektleitung: Prof. Dr. Stefan Schreieder
Math+ Distinguished Fellow
Quelle ↗Förderer: DFG Exzellenzstrategie Cluster Zeitraum: 01/2023 - 12/2025 Projektleitung: Prof. Dr. Bruno Klingler
Tame geometry and transcendence in Hodge theory (TameHodge)
Quelle ↗Förderer: Horizon 2020: ERC Advanced Grant Zeitraum: 10/2021 - 09/2027 Projektleitung: Prof. Dr. Bruno Klingler
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Publikationen25
Top 25 nach Zitationen — Quelle: OpenAlex (BAAI/bge-m3 embedded für Matching).
Mathematische Annalen · 73 Zitationen · DOI
Publications mathématiques de l IHÉS · 59 Zitationen · 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.
Journal of the American Mathematical Society · 54 Zitationen · 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.
Annals of Mathematics · 48 Zitationen · DOI
In this paper we prove, assuming the Generalized Riemann Hypothesis, the Andr-Oort conjecture on the Zariski closure of sets of special points in a Shimura variety. In the case of sets of special points satisfying an additional assumption, we prove the conjecture without assuming the GRH.
Inventiones mathematicae · 38 Zitationen · DOI
Annales de l’institut Fourier · 36 Zitationen · DOI
Une structure complexe affine (resp. projective) sur une surface complexe est la donnée d’un atlas de cartes à valeur dans <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>ℂ</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> (resp. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="bold">P</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>ℂ</mml:mi> </mml:mrow> </mml:math> ) à changements de cartes localement constants dans le groupe affine <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>ℂ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> (resp. le groupe <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="bold">P</mml:mi> <mml:mi>G</mml:mi> <mml:mi>L</mml:mi> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mi>ℂ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> ). Dans cet article nous classifions les surfaces complexes affines et calculons, à surface complexe <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> fixée, l’espace de déformation des structures complexes affines sur <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> compatibles avec sa structure analytique. Nous montrons aussi que toute structure projective sur une surface complexe admettant une structure complexe affine est nécessairement affine.
Duke Mathematical Journal · 35 Zitationen · DOI
We show that any smooth complex projective variety whose fundamental group has a complex representation with infinite image must have a nonzero symmetric differential (a section of a symmetric power of the cotangent bundle). Along the way, we produce many symmetric differentials on the base of a variation of Hodge structures.
Proceedings of symposia in pure mathematics · 21 Zitationen · DOI
Inventiones mathematicae · 21 Zitationen · DOI
Inventiones mathematicae · 18 Zitationen · DOI
Abstract Given a polarizable ℤ-variation of Hodge structures $\mathbb{V}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> over a complex smooth quasi-projective base $S$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> , a classical result of Cattani, Deligne and Kaplan says that its Hodge locus (i.e. the locus where exceptional Hodge tensors appear) is a countable union of irreducible algebraic subvarieties of $S$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> , called the special subvarieties for $\mathbb{V}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> . Our main result in this paper is that, if the level of ${\mathbb{V}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> is at least 3, this Hodge locus is in fact a finite union of such special subvarieties (hence is algebraic), at least if we restrict ourselves to the Hodge locus factorwise of positive period dimension (Theorem 1.5). For instance the Hodge locus of positive period dimension of the universal family of degree $d$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> </mml:math> smooth hypersurfaces in $\mathbf{P}^{n+1}_{\mathbb{C}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>P</mml:mi> <mml:mi>C</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> , $n\geq 3$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:math> , $d\geq 5$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>5</mml:mn> </mml:math> and $(n,d)\neq (4,5)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>d</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≠</mml:mo> <mml:mo>(</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mn>5</mml:mn> <mml:mo>)</mml:mo> </mml:math> , is algebraic. On the other hand we prove that in level 1 or 2, the Hodge locus is analytically dense in $S^{\mbox{an}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>S</mml:mi> <mml:mtext>an</mml:mtext> </mml:msup> </mml:math> as soon as it contains one typical special subvariety. These results follow from a complete elucidation of the distribution in $S$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>S</mml:mi> </mml:math> of the special subvarieties in terms of typical/atypical intersections, with the exception of the atypical special subvarieties of zero period dimension.
Commentarii Mathematici Helvetici · 16 Zitationen · DOI
Let X be an irreducible Hermitian symmetric space of non-compact type of dimension greater than 1 and G be the group of biholomorphisms of X ; let {\rm M} = \Gamma \backslash X be a quotient of X by a torsion-free discrete subgroup \Gamma of G such that M is of finite volume in the canonical metric. Then, due to the G-equivariant Borel embedding of X into its compact dual Xc, the locally symmetric structure of M can be considered as a special kind of a (G_{\Bbb C} , X_c) -structure on M, a maximal atlas of Xc-valued charts with locally constant transition maps in the complexified group {\rm G}_{\Bbb C} . By Mostow's rigidity theorem the locally symmetric structure of M is unique. We prove that the ({\rm G}_{\Bbb C} , X_c) -structure of M is the unique one compatible with its complex structure. In the rank one case this result is due to Mok and Yeung.
Journal of the Institute of Mathematics of Jussieu · 14 Zitationen · DOI
Abstract Let $\smash{\sGa\stackrel{i}{\hookrightarrow}L}$ be a lattice in the real simple Lie group L . If L is of rank at least 2 (respectively locally isomorphic to Sp( n , 1)) any unbounded morphism ρ : Γ → G into a simple real Lie group G essentially extends to a Lie morphism ρ L : L → G (Margulis's superrigidity theorem, respectively Corlette's theorem). In particular any such morphism is infinitesimally, thus locally, rigid. On the other hand, for L = SU( n , 1) even morphisms of the form $\smash{\rho:\sGa\stackrel{i}{\hookrightarrow}L \rightarrow G}$ are not infinitesimally rigid in general. Almost nothing is known about their local rigidity. In this paper we prove that any cocompact lattice Γ in SU( n , 1) is essentially locally rigid (while in general not infinitesimally rigid) in the quaternionic groups Sp( n , 1), SU(2 n , 2) or SO(4 n , 4) (for the natural sequence of embeddings SU( n , 1) ⊂ Sp( n , 1) ⊂ SU(2 n , 2) ⊂ SO(4 n , 4)).
Inventiones mathematicae · 13 Zitationen · DOI
Abstract Given $${{\mathbb {V}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> a polarizable variation of $${{\mathbb {Z}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Z</mml:mi> </mml:math> -Hodge structures on a smooth connected complex quasi-projective variety S , the Hodge locus for $${{\mathbb {V}}}^\otimes $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mo>⊗</mml:mo> </mml:msup> </mml:math> is the set of closed points s of S where the fiber $${{\mathbb {V}}}_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> has more Hodge tensors than the very general one. A classical result of Cattani, Deligne and Kaplan states that the Hodge locus for $${{\mathbb {V}}}^\otimes $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mo>⊗</mml:mo> </mml:msup> </mml:math> is a countable union of closed irreducible algebraic subvarieties of S , called the special subvarieties of S for $${{\mathbb {V}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> . Under the assumption that the adjoint group of the generic Mumford–Tate group of $${{\mathbb {V}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> is simple we prove that the union of the special subvarieties for $${{\mathbb {V}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> whose image under the period map is not a point is either a closed algebraic subvariety of S or is Zariski-dense in S . This implies for instance the following typical intersection statement: given a Hodge-generic closed irreducible algebraic subvariety S of the moduli space $${{\mathcal {A}}}_g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math> of principally polarized Abelian varieties of dimension g , the union of the positive dimensional irreducible components of the intersection of S with the strict special subvarieties of $${{\mathcal {A}}}_g$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math> is either a closed algebraic subvariety of S or is Zariski-dense in S .
arXiv (Cornell University) · 13 Zitationen · DOI
We present a conjecture on the geometry of the Hodge locus of a (graded polarizable, admissible) variation of mixed Hodge structure over a complex smooth quasi-projective base, generalizing to this context the Zilber-Pink Conjecture for mixed Shimura varieties (in particular the André-Oort conjecture).
arXiv (Cornell University) · 12 Zitationen · DOI
The hyperbolic Ax-Lindemann-Weierstrass conjecture is a functional algebraic independence statement for the uniformizing map of an arithmetic variety. In this paper we provide a proof of this conjecture, generalizing previous work of Pila-Tsimerman and Peterzil-Starchenko.
Inventiones mathematicae · 12 Zitationen · DOI
Annals of Mathematics · 11 Zitationen · DOI
Journal of the European Mathematical Society · 7 Zitationen · DOI
We equip integral graded-polarized mixed period spaces with a natural \mathbb{R}_{\mathrm{alg}} -definable analytic structure, and prove that any period map associated to an admissible variation of integral graded-polarized mixed Hodge structures is definable in \mathbb{R}_{\mathrm{an,exp}} with respect to this structure. As a consequence we re-prove that the zero loci of admissible normal functions are algebraic.
Mathematische Annalen · 7 Zitationen · DOI
Abstract We prove in this paper, the Ax–Schanuel conjecture for all admissible variations of mixed Hodge structures.
Geometric and Functional Analysis · 7 Zitationen · DOI
Annales Scientifiques de l École Normale Supérieure · 6 Zitationen · DOI
A polarizable variation of Hodge structure over a smooth complex quasi projective variety $S$ is said to be defined over a number field $L$ if $S$ and the algebraic connection associated to the variation are both defined over $L$. Conjecturally any special subvariety (also called an irreducible component of the Hodge locus) for such variations is defined over $\overline{\mathbb{Q}}$, and its Galois conjugates are also special subvarieties. We prove this conjecture for special subvarieties satisfying a simple monodromy condition. As a corollary we reduce the conjecture that special subvarieties for variation of Hodge structures defined over a number field are defined over $\overline{\mathbb{Q}}$ to the case of special points.
arXiv (Cornell University) · 5 Zitationen · DOI
arXiv (Cornell University) · 5 Zitationen · DOI
Given a variation of Hodge structures on a quasi-projective base $S$, whose generic Mumford-Tate group is non-product, we prove that the (countable) union of positive components of the Hodge locus is either an algebraic subvariety of $S$, or is Zariski-dense in $S$.
arXiv (Cornell University) · 5 Zitationen · DOI
We prove that the uniformizing map of any arithmetic quotient, as well as the period map associated to any pure polarized $\mathbb{Z}$-variation of Hodge structure $\mathbb{V}$ on a smooth complex quasi-projective variety $S$, are topologically tame. As an easy corollary of these results and of Peterzil-Starchenko's o-minimal GAGA theorem we obtain that the Hodge locus of $(S, \mathbb{V})$ is a countable union of algebraic subvarieties of $S$ (a result originally due to Cattani-Deligne-Kaplan).
Expositiones Mathematicae · 3 Zitationen · DOI
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- Prof. Dr. Bruno Klingler
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- Institut für Mathematik
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