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Complex Geometry Lab
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The lab explores the geometry of spaces without Kähler structures and of homogeneous spaces.
AboutAbout the lab
Bringing asymmetric geometry to symmetric spaces.
The Kasuya Lab studies differential geometry and complex geometry.
It works on the geometry of spaces that do not have a Kähler structure, noncommutative Hodge theory, and the geometry of Lie groups and homogeneous spaces.
Beyond linear algebra and set theory, no prior knowledge is required when joining the lab, and students who have never studied geometry are accepted.
The lab encourages students to consult with it before submitting the lab preference form, and online meetings and visits to its seminar are available.
Key terms
- Kähler structure
- A very well-behaved structure that combines Riemannian geometry, complex geometry, and symplectic geometry
- Lie group
- A group made up of smooth, continuous transformations, such as rotations
- Homogeneous space
- A space in which every point is equal and indistinguishable from the others, like a plane or a sphere
- Hodge theory
- A theory that captures how the geometric structure of a shape is reflected in information about its holes (cohomology)
ResearchResearch themes
The lab studies continuous spaces (mainly manifolds) and the geometric structures that govern them as a whole, using groups, Lie algebras, complexes, and similar tools. The central themes are non-Kähler geometry, noncommutative Hodge theory, and the non-invariant geometry of homogeneous spaces.
*Illustrative imageGeometry of Solvmanifolds
Homogeneous spaces of solvable Lie groups are called solvmanifolds, and they are an essential generalization of nilmanifolds. Nilmanifolds have been studied extensively, but extending to solvmanifolds makes things more complicated, so the lab investigates their properties.
*Illustrative imageNoncommutative Hodge Theory
Ordinary Hodge theory looks at how geometric structures are reflected in cohomology. The lab studies Hodge theory related to noncommutative fundamental groups, from the viewpoint of rational homotopy theory and from the viewpoint of the correspondence between flat bundles and Higgs bundles.
*Illustrative imageComplex Homogeneous Spaces and Their Extensions
The lab aims to extend homogeneous spaces from the viewpoint of complex geometry. Using deformation theory, foliations, and similar tools, it considers how to construct spaces that are not bound to the conventional quotient spaces.
*Illustrative imageNon-Kähler Geometry
The lab studies the geometry of complex manifolds and symplectic manifolds without Kähler structures, and of odd-dimensional manifolds such as CR manifolds and Sasakian manifolds. It does so through invariants such as Bott-Chern cohomology, special metrics other than Kähler metrics, and deformation theory.
MembersFaculty
The Kasuya Lab consists of faculty members, master's students, and undergraduate students.As of 2026
- ProfessorHisashi Kasuya
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: “Triangle general surface” Gregors / CC BY-SA 3.0 (cropped)
- Geometry of Solvmanifolds: “E10 e8coxeterplane height120” Teake Nutma / CC BY 3.0 (cropped)
- Noncommutative Hodge Theory: “Sculpture town park” Danrok / CC BY-SA 3.0 (cropped)
- Complex Homogeneous Spaces and Their Extensions: “Hopf Fibration” Niles Johnson / CC BY 4.0 (cropped)
- Non-Kähler Geometry: “CalabiYau5” Andrew J. Hanson / CC BY-SA 3.0 (cropped)
- Related lab (Topology, Differential Geometry, and Global Analysis Lab (Hitoshi Moriyoshi)): “Surface of revolution illustration” Oleg Alexandrov / Public domain (cropped)
- Related lab (Nakamura Lab): “Willmore Surface sculpture Durham” Chemical Engineer / CC BY-SA 4.0 (cropped)
- Related lab (Combinatorics, Theoretical Computer Science, Algorithm Theory, and Optimization Theory Lab (Hiroshi Hirai)): “Imperialist-competitive-algorithm-flowchart-bg” Original author: Icasite Author of the derivative work: Spiritia / CC BY-SA 3.0 (cropped)
- Related lab (Naito Lab): “Carbon lattice diamond” GYassineMrabetTalk✉ This structure was created with PyMOL. / Public domain (cropped)
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