KontsevichSoibelman

Kontsevich-Soibelman is a mathematical theory that provides a framework for understanding the structure of moduli spaces of stable maps in algebraic geometry. Developed by Maxim Kontsevich and Alexander Soibelman, the theory is a significant advancement in the study of Gromov-Witten invariants and quantum cohomology.

The Kontsevich-Soibelman theory introduces the concept of a "virtual fundamental class" for moduli spaces of stable maps. This class is a virtual cycle that captures the expected dimension of the moduli space, even when the space is not smooth. The virtual fundamental class is constructed using a combination of techniques from algebraic geometry, symplectic geometry, and homological algebra.

One of the key features of the Kontsevich-Soibelman theory is its ability to handle moduli spaces of stable maps with arbitrary genus and multiple marked points. This allows for a more comprehensive study of the geometry of these spaces and their contributions to Gromov-Witten invariants.

The theory has had a profound impact on the field of algebraic geometry, leading to new insights into the structure of moduli spaces and the behavior of Gromov-Witten invariants. It has also inspired further research in related areas, such as the study of quantum cohomology and the enumeration of rational curves on algebraic varieties.

In summary, the Kontsevich-Soibelman theory is a powerful tool for studying the geometry of moduli spaces of stable maps and their contributions to Gromov-Witten invariants. Its development has opened up new avenues for research in algebraic geometry and has had a significant impact on the field.