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Infinite Analysis Lab (Odaka)
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The lab studies Kähler-Einstein manifolds and moduli spaces, with K-stability as its central theme.
AboutAbout the lab
Connecting the geometry of canonical metrics to algebraic geometry
The Odaka Lab conducts research related to moduli spaces, in the fields of algebraic geometry and complex geometry.
It connects the geometry of Kähler-Einstein manifolds, which include many classical and important algebraic varieties, to modern algebraic geometry such as birational geometry.
The lab has studied K-stability, a concept equivalent to the existence of canonical metrics on algebraic varieties, from an algebro-geometric perspective.
The lab asks students who are considering Prof. Odaka as a graduate school advisor to contact him before applying.
Key terms
- Algebraic variety
- A geometric object represented as the set of solutions of polynomial equations, and the main object of study in algebraic geometry
- Moduli space
- A space obtained by collecting all geometric objects of a given kind and arranging each of them as a point
- K-stability
- A condition, expressed in the language of algebraic geometry, for whether a canonical metric (such as a Kähler-Einstein metric) exists
NewsLab news
View More ↗ResearchResearch themes
The lab connects the geometry of Kähler-Einstein manifolds to birational geometry, moduli spaces and their compactifications, singularity theory, Arakelov geometry, and more. In its early work, using birational geometry based on minimal model theory and the theory of algebraic stacks, it laid the groundwork for treating K-stability algebro-geometrically.
*Illustrative imageCriteria for K-stability and the K-moduli conjecture
The lab showed that the Donaldson-Futaki invariant can be described in terms of intersection numbers, building the foundation of the theory of criteria for K-stability. Based on this, it introduced the "K-moduli conjecture."
*Illustrative imageK-moduli of Fano varieties
For Fano varieties with positive Ricci curvature, the lab introduced the δ-invariant (stability threshold), which completely determines K-stability.
*Illustrative imageModuli and metrics of Calabi-Yau varieties
The lab studies the relationship between the moduli and the metrics of (broad-sense) Calabi-Yau varieties. It extended the framework to the compactification of the entire moduli space.
*Illustrative imageArakelov-geometric formulation of K-stability theory
In the arithmetic direction, the lab developed an Arakelov-geometric version of the theory of K-stability. It has introduced the K-modular height (arithmetic Mabuchi functional).
MembersFaculty
- ProfessorYuji Odaka
VoicesStudent voices
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Quartic Fermat curve” Jacob Rus / CC BY-SA 4.0 (cropped)
- Criteria for K-stability and the K-moduli conjecture: “Gaussian 2d surface” Kopak999 / CC BY-SA 4.0 (cropped)
- K-moduli of Fano varieties: “Clebsch” Rectas / CC0 (cropped)
- Moduli and metrics of Calabi-Yau varieties: “Dimensions enroulées (Espaces de CY)” Historicair, Rogilbert / Public domain (cropped)
- Arakelov-geometric formulation of K-stability theory: “Riemann zeta-function x greater 1” The original uploader was Thin at Polish Wikipedia. / Public domain (cropped)
- Related lab (Infinite Analysis Lab (Namikawa)): “Consani-Scholten” Ursulageorges / CC BY 4.0 (cropped)
- Related lab (Tamagawa Lab): “Klein-map” PSL27 / CC BY-SA 4.0 (cropped)
- Related lab (Infinite Analysis Lab (Mochizuki)): “Kyoto University RIMS” KATSURA Roen / CC BY 3.0 (cropped)
- Related lab (Infinite Analysis Research Division (Hoshi) Lab): “Kleinian group limit set on sphere” Adam majewski / CC BY-SA 4.0 (cropped)
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