Klaus Künnemann
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Klaus Künnemann has a total of 14 co-authors. Here are the top co-authors by number of shared publications:
- Walter Gubler
Fakultät für Mathematik Universität Regensburg 93040 Regensburg Germany
7 shared publications - José Ignacio Burgos Gil
Instituto de Ciencias Matemáticas (ICMAT‐CSIC‐UAM‐UCM‐UC3), Consejo Superior de Investigaciones Científicas (CSIC), Spain
3 shared publications Jean-Benoît Bost
2 shared publications- Philipp Jell
Univ Regensburg, Math, D-93040 Regensburg, Germany
2 shared publications - Yanbo Fang
Fakultät für Mathematik Universität Regensburg 93040 Regensburg Germany
2 shared publications Burgos Gil
1 shared publication
Latest Publications
- Open Access
- 4 March 2025
- Published by Cambridge University Press (CUP) in Nagoya Mathematical Journal
- p. 1-14
ON THE NON-ARCHIMEDEAN MONGE–AMPÈRE EQUATION IN MIXED CHARACTERISTIC
Let X be a smooth projective variety over a complete discretely valued field of mixed characteristic. We solve non-Archimedean Monge–Ampère equations on X assuming resolution and embedded resolution of singularities. We follow the variational approach of Boucksom, Favre, and Jonsson proving the continuity of the plurisubharmonic envelope of a continuous metric on an ample line bundle on X. We replace the use of multiplier ideals in equicharacteristic zero by the use of perturbation friendly test ideals introduced by Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron, and Witaszek building upon previous constructions by Hacon, Lamarche, and Schwede.
- 4 March 2025
- Published by Cambridge University Press (CUP) in Nagoya Mathematical Journal
- p. 1-14
Open AccessON THE NON-ARCHIMEDEAN MONGE–AMPÈRE EQUATION IN MIXED CHARACTERISTIC
Let X be a smooth projective variety over a complete discretely valued field of mixed characteristic. We solve non-Archimedean Monge–Ampère equations on X assuming resolution and embedded resolution of singularities. We follow the variational approach of Boucksom, Favre, and Jonsson proving the continuity of the plurisubharmonic envelope of a continuous metric on an ample line bundle on X. We replace the use of multiplier ideals in equicharacteristic zero by the use of perturbation friendly test ideals introduced by Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron, and Witaszek building upon previous constructions by Hacon, Lamarche, and Schwede.
- Open Access
On the non-archimedean Monge-Ampère equation in mixed characteristic
Let X be a smooth projective variety over a complete discretely valued field of mixed characteristic. We solve non-archimedean Monge-Amp\`ere equations on X assuming resolution and embedded resolution of singularities. We follow the variational approach of Boucksom, Favre, and Jonsson proving the continuity of the plurisubharmonic envelope of a continuous metric on an ample line bundle on X. We replace the use of multiplier ideals in equicharacteristic zero by the use of perturbation friendly test ideals introduced by Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron, and Witaszek building upon previous constructions by Hacon, Lamarche, and Schwede.
Open AccessOn the non-archimedean Monge-Ampère equation in mixed characteristic
Let X be a smooth projective variety over a complete discretely valued field of mixed characteristic. We solve non-archimedean Monge-Amp\`ere equations on X assuming resolution and embedded resolution of singularities. We follow the variational approach of Boucksom, Favre, and Jonsson proving the continuity of the plurisubharmonic envelope of a continuous metric on an ample line bundle on X. We replace the use of multiplier ideals in equicharacteristic zero by the use of perturbation friendly test ideals introduced by Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron, and Witaszek building upon previous constructions by Hacon, Lamarche, and Schwede.
- Open Access
- 8 March 2021
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 299 (3-4) , 1199-1255
A comparison of positivity in complex and tropical toric geometry
Given a smooth complex toric variety we will compare real Lagerberg forms and currents on its tropicalization with invariant complex forms and currents on the toric variety. Our main result is a correspondence theorem which identifies the cone of invariant closed positive currents on the complex toric variety with closed positive currents on the tropicalization. In a subsequent paper, this correspondence will be used to develop a Bedford–Taylor theory of plurisubharmonic functions on the tropicalization.
- 8 March 2021
- Published by Springer Nature in Mathematische Zeitschrift
- Vol. 299 (3-4) , 1199-1255
Open AccessA comparison of positivity in complex and tropical toric geometry
Given a smooth complex toric variety we will compare real Lagerberg forms and currents on its tropicalization with invariant complex forms and currents on the toric variety. Our main result is a correspondence theorem which identifies the cone of invariant closed positive currents on the complex toric variety with closed positive currents on the tropicalization. In a subsequent paper, this correspondence will be used to develop a Bedford–Taylor theory of plurisubharmonic functions on the tropicalization.
- Open Access
A comparison of positivity in complex and tropical toric geometry
Given a smooth complex toric variety we will compare real Lagerberg forms and currents on its tropicalization with invariant complex forms and currents on the toric variety. Our main result is a correspondence theorem which identifies the cone of invariant closed positive currents on the complex toric variety with closed positive currents on the tropicalization. In a subsequent paper, this correspondence will be used to develop a Bedford-Taylor theory of plurisubharmonic functions on the tropicalization.
Open AccessA comparison of positivity in complex and tropical toric geometry
Given a smooth complex toric variety we will compare real Lagerberg forms and currents on its tropicalization with invariant complex forms and currents on the toric variety. Our main result is a correspondence theorem which identifies the cone of invariant closed positive currents on the complex toric variety with closed positive currents on the tropicalization. In a subsequent paper, this correspondence will be used to develop a Bedford-Taylor theory of plurisubharmonic functions on the tropicalization.
- Open Access
- 16 September 2019
- Published by MathDoc/Centre Mersenne in Annales de l'institut Fourier
- Vol. 69 (5) , 2331-2376
Continuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals
Let be an ample line bundle on a smooth projective variety over a non-archimedean field . For a continuous metric on , we show in the following two cases that the semipositive envelope is a continuous semipositive metric on and that the non-archimedean Monge–Ampère equation has a solution. First, we prove it for curves using results of Thuillier. Second, we show it under the assumption that is a surface defined geometrically over the function field of a curve over a perfect field of positive characteristic. The second case holds in higher dimensions if we assume resolution of singularities over . The proof follows a strategy from Boucksom, Favre and Jonsson, replacing multiplier ideals by test ideals. Finally, the appendix by Burgos and Sombra provides an example of a semipositive metric whose retraction is not semipositive. The example is based on the construction of a toric variety which has two SNC-models which induce the same skeleton but different retraction maps.- 16 September 2019
- Published by MathDoc/Centre Mersenne in Annales de l'institut Fourier
- Vol. 69 (5) , 2331-2376
Open AccessContinuity of Plurisubharmonic Envelopes in Non-Archimedean Geometry and Test Ideals
Let be an ample line bundle on a smooth projective variety over a non-archimedean field . For a continuous metric on , we show in the following two cases that the semipositive envelope is a continuous semipositive metric on and that the non-archimedean Monge–Ampère equation has a solution. First, we prove it for curves using results of Thuillier. Second, we show it under the assumption that is a surface defined geometrically over the function field of a curve over a perfect field of positive characteristic. The second case holds in higher dimensions if we assume resolution of singularities over . The proof follows a strategy from Boucksom, Favre and Jonsson, replacing multiplier ideals by test ideals. Finally, the appendix by Burgos and Sombra provides an example of a semipositive metric whose retraction is not semipositive. The example is based on the construction of a toric variety which has two SNC-models which induce the same skeleton but different retraction maps. - Open Access
Differentiability of non-archimedean volumes and non-archimedean Monge-Ampère equations (with an appendix by Robert Lazarsfeld)
Let $X$ be a normal projective variety over a complete discretely valued field and $L$ a line bundle on $X$. We denote by $X^\textrm{an}$ the analytification of $X$ in the sense of Berkovich and equip the analytification $L^\textrm{an}$ of $L$ with a continuous metric $\| \ \|$. We study non-archimedean volumes, a tool which allows us to control the asymptotic growth of small sections of big powers of $L$. We prove that the non-archimedean volume is differentiable at a continuous semipositive metric and that the derivative is given by integration with respect to a Monge-Amp\`ere measure. Such a differentiability formula had been proposed by M. Kontsevich and Y. Tschinkel. In residue characteristic zero, it implies an orthogonality property for non-archimedean plurisubharmonic functions which allows us to drop an algebraicity assumption in a theorem of S. Boucksom, C. Favre and M. Jonsson about the solution to the non-archimedean Monge-Amp\`{e}re equation. The appendix by R. Lazarsfeld establishes the holomorphic Morse inequalities in arbitrary characteristic.Open AccessDifferentiability of non-archimedean volumes and non-archimedean Monge-Ampère equations (with an appendix by Robert Lazarsfeld)
Let $X$ be a normal projective variety over a complete discretely valued field and $L$ a line bundle on $X$. We denote by $X^\textrm{an}$ the analytification of $X$ in the sense of Berkovich and equip the analytification $L^\textrm{an}$ of $L$ with a continuous metric $\| \ \|$. We study non-archimedean volumes, a tool which allows us to control the asymptotic growth of small sections of big powers of $L$. We prove that the non-archimedean volume is differentiable at a continuous semipositive metric and that the derivative is given by integration with respect to a Monge-Amp\`ere measure. Such a differentiability formula had been proposed by M. Kontsevich and Y. Tschinkel. In residue characteristic zero, it implies an orthogonality property for non-archimedean plurisubharmonic functions which allows us to drop an algebraicity assumption in a theorem of S. Boucksom, C. Favre and M. Jonsson about the solution to the non-archimedean Monge-Amp\`{e}re equation. The appendix by R. Lazarsfeld establishes the holomorphic Morse inequalities in arbitrary characteristic. - 23 January 2017
- Published by Mathematical Sciences Publishers in Algebra & Number Theory
- Vol. 11 (1) , 77-180
A tropical approach to nonarchimedean Arakelov geometry
Chambert-Loir and Ducros have recently introduced a theory of real valued differential forms and currents on Berkovich spaces. In analogy to the theory of forms with logarithmic singularities, we enlarge the space of differential forms by so called delta-forms on the non-archimedean analytification of an algebraic variety. This extension is based on an intersection theory for tropical cycles with smooth weights. We prove a generalization of the Poincar\'e-Lelong formula which allows us to represent the first Chern current of a formally metrized line bundle by a delta-form. We introduce the associated Monge-Amp\`ere measure $\mu$ as a wedge-power of this first Chern delta-form and we show that $\mu$ is equal to the corresponding Chambert-Loir measure. The star-product of Green currents is a crucial ingredient in the construction of the arithmetic intersection product. Using the formalism of delta-forms, we obtain a non-archimedean analogue at least in the case of divisors. We use it to compute non-archimedean local heights of proper varieties.Comment: 91 pages, revised and updated version, new example 4.2- 23 January 2017
- Published by Mathematical Sciences Publishers in Algebra & Number Theory
- Vol. 11 (1) , 77-180
A tropical approach to nonarchimedean Arakelov geometry
Chambert-Loir and Ducros have recently introduced a theory of real valued differential forms and currents on Berkovich spaces. In analogy to the theory of forms with logarithmic singularities, we enlarge the space of differential forms by so called delta-forms on the non-archimedean analytification of an algebraic variety. This extension is based on an intersection theory for tropical cycles with smooth weights. We prove a generalization of the Poincar\'e-Lelong formula which allows us to represent the first Chern current of a formally metrized line bundle by a delta-form. We introduce the associated Monge-Amp\`ere measure $\mu$ as a wedge-power of this first Chern delta-form and we show that $\mu$ is equal to the corresponding Chambert-Loir measure. The star-product of Green currents is a crucial ingredient in the construction of the arithmetic intersection product. Using the formalism of delta-forms, we obtain a non-archimedean analogue at least in the case of divisors. We use it to compute non-archimedean local heights of proper varieties.Comment: 91 pages, revised and updated version, new example 4.2- 8 November 2016
- Published by Walter de Gruyter GmbH in Journal für die reine und angewandte Mathematik (Crelles Journal)
- Vol. 2019 (752) , 141-177
Positivity properties of metrics and delta-forms
In previous work, we have introduced δ-forms on the Berkovich analytification of an algebraic variety in order to study smooth or formal metrics via their associated Chern δ-forms. In this paper, we investigate positivity properties of δ-forms and δ-currents. This leads to various plurisubharmonicity notions for continuous metrics on line bundles. In the case of a formal metric, we show that many of these positivity notions are equivalent to Zhang’s semipositivity. For piecewise smooth metrics, we prove that plurisubharmonicity can be tested on tropical charts in terms of convex geometry. We apply this to smooth metrics, to canonical metrics on abelian varieties and to toric metrics on toric varieties.
- 8 November 2016
- Published by Walter de Gruyter GmbH in Journal für die reine und angewandte Mathematik (Crelles Journal)
- Vol. 2019 (752) , 141-177
Positivity properties of metrics and delta-forms
In previous work, we have introduced δ-forms on the Berkovich analytification of an algebraic variety in order to study smooth or formal metrics via their associated Chern δ-forms. In this paper, we investigate positivity properties of δ-forms and δ-currents. This leads to various plurisubharmonicity notions for continuous metrics on line bundles. In the case of a formal metric, we show that many of these positivity notions are equivalent to Zhang’s semipositivity. For piecewise smooth metrics, we prove that plurisubharmonicity can be tested on tropical charts in terms of convex geometry. We apply this to smooth metrics, to canonical metrics on abelian varieties and to toric metrics on toric varieties.
- 1 February 2010
- Published by Elsevier in Advances in Mathematics
- Vol. 223 (3) , 987-1106
Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers
- 1 February 2010
- Published by Elsevier in Advances in Mathematics
- Vol. 223 (3) , 987-1106
Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers
- 30 September 2006
- Published by European Mathematical Society - EMS - Publishing House GmbH in Oberwolfach Reports
- Vol. 2 (4) , 2447-2492
Arakelov Geometry
Arakelov geometry studies the geometry and arithmetic of schemes of finite type over Spec {\bf Z} , i.e. systems of polynomial equations with integer coefficients. It combines methods from algebraic geometry, number theory, and hermitian differential geometry. The workshop was organized by Jean-Benoît Bost (Orsay), Klaus Künnemann (Regensburg) and Damian Roessler (Paris). It brought together internationally leading experts in the area as well as a considerable number of young researchers. The talks covered various aspects of Arakelov geometry from analytic torsion over adelic and non-archimedean analytic spaces to modular forms and diophantine geometry. A non-mathematical complement was a piano recital by Harry Tamvakis on Thursday night featuring Bach, Beethoven and Chopin.
- 30 September 2006
- Published by European Mathematical Society - EMS - Publishing House GmbH in Oberwolfach Reports
- Vol. 2 (4) , 2447-2492
Arakelov Geometry
Arakelov geometry studies the geometry and arithmetic of schemes of finite type over Spec {\bf Z} , i.e. systems of polynomial equations with integer coefficients. It combines methods from algebraic geometry, number theory, and hermitian differential geometry. The workshop was organized by Jean-Benoît Bost (Orsay), Klaus Künnemann (Regensburg) and Damian Roessler (Paris). It brought together internationally leading experts in the area as well as a considerable number of young researchers. The talks covered various aspects of Arakelov geometry from analytic torsion over adelic and non-archimedean analytic spaces to modular forms and diophantine geometry. A non-mathematical complement was a piano recital by Harry Tamvakis on Thursday night featuring Bach, Beethoven and Chopin.