Anwendungen spezieller reeller Geometrie in der Kähler-Geometrie
Zusammenfassung der Projektergebnisse
The first goal of the project was to gain insight into which projective special real (PSR) manifolds, respectively their corresponding real hyperbolic cubic polynomials, can be realised as level sets in Kähler cones of compact Kähler three-folds. These polynomials are the volume on the real (1, 1)-cohomology. The second goal was to understand what kind of information we can extract from the geometry of the PSR and generalised PSR (GPSR) manifolds that can be constructed in this fashion about the underlying Kähler manifolds. Our work programme was set up with realistic goals in mind, for example by restricting the considered class of Kähler manifolds to the well-studied class of smooth projective toric varieties, and on the level of polynomials to cubics and quartics. In each point of the work programme, we have obtained new results. On the level of real hyperbolic polynomials, our first main result is a complete classification of all quartic real hyperbolic polynomials in two variables, or, equivalently, all maximal quartic GPSR curves. This extends our previous work to cover the case of geodesically incomplete cases, and in all cases classifies the asymptotic geometries. Furthermore, we have been able to use our results to show that homogeneous GPSR manifold have irregular boundary behaviour in the sense that the real projective algebraic variety corresponding to their respective defining polynomial is singular, or that the centro-affine metric degenerates along said variety. Our second main result covers the classification of homogeneous (G)PSR curves and surfaces, the latter in collaboration with Andrew Swann. While we initially planned to restrict our research to cubics and quartics, it turned out that with the right approach, this was not necessary when restricting the dimension instead. The reason for us to study the homogeneous cases were surprising results regarding the real hyperbolic polynomials that were constructed in examples as volume polynomials turned out to define homogeneous (G)PSR manifolds quite regularly. Before our findings, only a classification of homogeneous PSR manifolds was known. Lastly, in another collaboration with Andrew Swann we have studied smooth projective toric three-folds. The paper is still in preparation. The main results are as follows. First, we found a new classification method for homogeneous PSR manifolds, using former results to prove that certain types of blowups preserve the property that if the volume polynomial of some smooth projective toric three-fold defines a homogeneous PSR manifold, the same holds true for the volume polynomial of the blowup. The second main result is a relation between strictly Lorentzian polynomials and real hyperbolic polynomials, namely that every real hyperbolic polynomial is linearly equivalent to a strictly Lorentzian polynomial. Until now, only the opposite direction was known. The third main result relates the global geometry of the PSR manifolds of volume polynomials to properties of the defining toric. We found that the PSR manifold in the Kähler cone is geodesically complete if and only if the toric three-fold is a product of CP1, CP2 , and CP3 . This is an unexpectedly strong restriction.
Projektbezogene Publikationen (Auswahl)
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Special geometry of quartic curves
D. Lindemann
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Special homogeneous surfaces. Mathematical Proceedings of the Cambridge Philosophical Society, 177(2), 333-362.
LINDEMANN, DAVID & SWANN, ANDREW
