Algebra, fundamentally, is the study of polynomial equations. If a certain geometric object, a variety or manifold, is defined as the set of roots of a polynomial equation, then it is called an algebraic variety.
To study and classify the characteristics of geometric objects using algebraic methods, birational morphism is used. Kähler complex manifolds are classified into three types: general type, Calabi-Yau, and Fano. This research mainly focuses on Fano varieties, which are close to projective spaces and can be birationally transformed into projective spaces. Additionally, research is underway to derive algebraic propositions using geometric properties.
One of the most important recent problems in complex geometry or algebraic geometry is whether a certain Fano variety has a Kähler-Einstein metric. This problem originated from the existence problem of the solution of the Monge-Ampère equation in partial differential equations in either complex or differential geometry. However, it was recently revealed that this existence problem is equivalent to the complete algebraic stability problem called K-stability. Therefore, research is being conducted by measuring algebraic invariants called alpha or delta invariants, which are tools for revealing K-stability. The ultimate goal is to classify the existence of Kähler-Einstein metrics for all Fano varieties.
One of the famous conjectures is that rational points on a Fano variety are potentially dense. "Potentially dense" means that the rational points of an algebraic variety defined over a given field K have dense rational points when extended to a finite extension of K, essentially a problem in number theory. This problem is studied based on geometric characteristics.