Surfaces on fourfolds

Introduction

Enumerative geometry is the field of algebraic geometry dealing with counting geometric objects satisfying constraints. For instance, in Ancient Greece, Apollonius asked how many circles are tangent to three given circles in the plane. It is a very active area due to unexpected connections with other fields of mathematics and physics.

So far, modern enumerative geometry is largely about counting curves. Recently, I worked on foundations for a theory for counting surfaces in 4-dimensional spaces. This is the starting point of this proposal, which is about discovering new properties of 4-dimensional spaces using surface counting.

Project A: Surface Counting in 4D

Project A explores surface counting in Calabi-Yau, hyper-Kähler, and Abelian fourfolds in a series of concrete settings.

Impact of Project A

The impact is this: when the count is non-zero for some (2,2) class γ on X, then it implies the variational Hodge conjecture for (X,γ). The Hodge conjecture is one of the millennium prize problems, and the first open case is for (2,2) classes on 4-dimensional spaces.

Project B: Investigating 4D Singularities

Project B investigates 4-dimensional singularities. It is about discovering a connection between the geometry and algebra hidden in the singularity called "crepant resolution conjecture."

Impact of Project B

The impact is this: for 3-dimensional singularities, the crepant resolution conjecture does not work when surfaces get contracted. By embedding 3-dimensional singularities in 4 dimensions, I expect to solve this open case.

Project C: Counting Representations of Non-Commutative Rings

Project C shifts from counting surfaces in 4-dimensional space to counting representations of 4-dimensional non-commutative rings.

Significance of Project C

The same move for 3-dimensional rings opened up an entire field, and this project will do the same for 4-dimensional rings. Interesting examples include:

• Sklyanin algebras
• Non-commutative resolutions of 4D Gorenstein singularities
• Quantum Fermat sextic fourfolds

Conclusion

The common denominator of these projects is that they involve 4D phenomena that could previously not be explored and are made accessible by this proposal.