Answering a Question Raised by a Famous Mathematician in Algebraic Geometry
- IBS researchers prove that the study of complex abstract spaces can be simplified -
We are familiar with the 3D world we live in, but does the entire universe look like this? How many other possible shapes would exist? Seeking to expand our understanding of complicated spaces and possibly simplify them, researchers at the Center for Geometry and Physics, within the Institute for Basic Science (IBS), have studied an invariant, a property that remains unchanged when transformations of a certain type are applied to objects, called 'derived category of coherent sheaves'. It plays an important role in a lot of fairly new branches of mathematics and physics fields, like: algebraic geometry, symplectic geometry, representation theory, string theory, and mirror symmetry. The first part of this ongoing research was published in Advances in Mathematics and provides the answer to an attempt to simplify one algebraic geometry problem.
In our 3D world, each point can be parametrized with three real numbers: latitude, longitude, and height. Spaces of more than three dimensions and abstract spaces cannot be easily visualized, neither imagined, nevertheless they can be studied via mathematics.
Mathematicians use the term "manifold" to describe spaces that can be bent, and locally look like our familiar spaces (Euclidean spaces). This can be understood via an intuitive example. While you walk or drive on the road, the world seems to be a flat surface. However, if you ask the same question to astronauts in space, miles above the Earth, they would reply that the world is more similar to a sphere, without any doubt. Therefore the same space can be seen in different ways depending on the point of view. This is a simple example of 2D manifold that can be described by real numbers, but spaces can be also linked to complex numbers, like a+bi, where i is the imaginary unit and i2=−1. In this study, IBS mathematicians have dealt with spaces that have special properties and structures, known as complex projective manifolds.
One dimensional complex projective manifolds are roughly classified into three classes, one of them is the so-called Fano manifolds. Fano manifolds enjoy several distinguished properties. For example, in complex higher dimensions, any two points can be connected by spheres.
Mathematicians have developed several mathematical tools, called topology, metric, sheaves, cohomology, and derived categories to understand spaces. Among these tools, derived categories draw lots of attention these days. Invented by the great mathematician Alexander Grothendieck, and his student Jean-Louis Verdier, this invariant was further studied to understand complex manifolds by Russian mathematicians, such as Alexei Igorewitsch Bondal, Dmitri Orlov, and Mikhail Kapranov. And now they draw the attention of mathematicians worldwide as they are believed to be one of the main ingredients of new mathematics.
One pending doubt about Fano manifolds was raised by Bondal: is it enough to study Fano manifolds to understand derived categories of the other classes of complex manifolds? (Or in more technical words, for every complex projective manifold Y, is there a Fano manifold X such that the derived category of Y is contained in the derived category of X?) IBS mathematicians proved that the answer is "yes", for a large class of complex manifolds.
The corresponding author of the study, LEE Kyoung-Seog, explains it with a simple analogy. Imagine that all complex manifolds are represented by the set of all living creatures. Now divide them into three sections: bacteria, animals and plants. The question would be, could you study the creatures of one subset to know information about the other creatures? For example, is it enough to study bacteria to know about plants and animals? IBS researchers have demonstrated that it is true for large classes of complex manifolds. For example, it would be enough to study the large set of bacteria in the sea to be knowledgeable about marine flora and fauna.
"By studying only the Fano manifolds, instead of all complex manifolds, researchers might simplify some algebraic geometry problems," says Lee. "Recently we proved that information of much larger complex manifolds can be embedded into information of the realm of Fano if we consider orbifold instead of manifolds". They also studied how information interacts between Fano varieties and their embedded (in derived sense) varieties.
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