Philipp Jell
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Philipp Jell has a total of 6 co-authors. Here are the top co-authors by number of shared publications:
- Florent Martin
Univ Regensburg, Fak Math, D-93040 Regensburg, Germany
1 shared publication - Johannes Rau
Universidad de los Andes, Bogota, Colombia
1 shared publication - José Ignacio Burgos Gil
Instituto de Ciencias Matemáticas (ICMAT‐CSIC‐UAM‐UCM‐UC3), Consejo Superior de Investigaciones Científicas (CSIC), Spain
1 shared publication - Klaus Künnemann
Fakultät für Mathematik Universität Regensburg 93040 Regensburg Germany
1 shared publication Kristin Shaw
1 shared publication- Walter Gubler
Fakultät für Mathematik Universität Regensburg 93040 Regensburg Germany
1 shared publication
Latest Publications
- 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 October 2018
- Published by Springer Nature in Israel Journal of Mathematics
- Vol. 229 (1) , 287-305
Tropical Hodge numbers of non-archimedean curves
- 23 October 2018
- Published by Springer Nature in Israel Journal of Mathematics
- Vol. 229 (1) , 287-305
Tropical Hodge numbers of non-archimedean curves
- Open Access
Lefschetz (1,1)-theorem in tropical geometry
For a tropical manifold of dimension n we show that the tropical homology classes of degree (n-1, n-1) which arise as fundamental classes of tropical cycles are precisely those in the kernel of the eigenwave map. To prove this we establish a tropical version of the Lefschetz (1, 1)-theorem for rational polyhedral spaces that relates tropical line bundles to the kernel of the wave homomorphism on cohomology. Our result for tropical manifolds then follows by combining this with Poincar\'e duality for integral tropical homology.
Open AccessLefschetz (1,1)-theorem in tropical geometry
For a tropical manifold of dimension n we show that the tropical homology classes of degree (n-1, n-1) which arise as fundamental classes of tropical cycles are precisely those in the kernel of the eigenwave map. To prove this we establish a tropical version of the Lefschetz (1, 1)-theorem for rational polyhedral spaces that relates tropical line bundles to the kernel of the wave homomorphism on cohomology. Our result for tropical manifolds then follows by combining this with Poincar\'e duality for integral tropical homology.
- Open Access
Tropical Hodge numbers of non-archimedean curves
We study the tropical Dolbeault cohomology of non-archimedean curves as defined by Chambert-Loir and Ducros. We give a precise condition for when this cohomology satisfies Poincar\'e duality. The condition is always satisfied when the residue field of the non-archimedean base field is the algebraic closure of a finite field. We also show that for curves over fields where the residue field is the field of complex numbers, the tropical (1,1)-Dolbeault cohomology can be infinite dimensional. Our main new ingredient is an exponential type sequence that relates tropical Dolbeault cohomology to the cohomology of the sheaf of harmonic functions. As an application of our Poincar\'e duality result, we calculate the dimensions of the tropical Dolbeault cohomology, called tropical Hodge numbers, for (open subsets of) curves.
Open AccessTropical Hodge numbers of non-archimedean curves
We study the tropical Dolbeault cohomology of non-archimedean curves as defined by Chambert-Loir and Ducros. We give a precise condition for when this cohomology satisfies Poincar\'e duality. The condition is always satisfied when the residue field of the non-archimedean base field is the algebraic closure of a finite field. We also show that for curves over fields where the residue field is the field of complex numbers, the tropical (1,1)-Dolbeault cohomology can be infinite dimensional. Our main new ingredient is an exponential type sequence that relates tropical Dolbeault cohomology to the cohomology of the sheaf of harmonic functions. As an application of our Poincar\'e duality result, we calculate the dimensions of the tropical Dolbeault cohomology, called tropical Hodge numbers, for (open subsets of) curves.