Moduli Spaces, Virtual Invariants and Shifted Symplectic Structures

Enumerative geometry is a core area of algebraic geometry that dates back to Apollonius in the second century BCE. It asks for the number of geometric figures with desired properties and has many applications from classical geometry to modern physics. Typically, an enumerative geometry problem is solved by first constructing the space of all geometric figures of fixed type, called the moduli space, and then finding the subspace of objects satisfying the desired properties. Unfortunately, many moduli spaces from nature are highly singular, and an intersection theory is difficult to make sense of. However, they come with deeper structures, such as perfect obstruction theories, which enable us to define nice subsets, called virtual fundamental classes. Now, enumerative numbers, called virtual invariants, are defined as integrals against the virtual fundamental classes.

Derived algebraic geometry is a relatively new area of algebraic geometry that is a natural generalization of Serre's intersection theory in the 1950s and Grothendieck's scheme theory in the 1960s. Many moduli spaces in enumerative geometry admit natural derived structures as well as shifted symplectic structures.

The book covers foundations on derived algebraic and symplectic geometry. Then, it covers foundations on virtual fundamental classes and moduli spaces from a classical algebraic geometry point of view. Finally, it fuses derived algebraic geometry with enumerative geometry and covers the cutting-edge research topics about DonaldsonThomas invariants in dimensions three and four.

Shows a comprehensive review of derived algebraic and symplectic geometry Presents a comprehensive review of virtual fundamental class from classical and derived geometry viewpoint Provides a survey of recent advances in DonaldsonThomas theory for dimension three and four

Auteur

Young-Hoon Kiem completed B.S., Mathematics, Seoul National University, in February 1993; Ph.D., M.S., M.Phil., Mathematics, Yale University, in May 2000. Young-Hoon Kiem was Szego Assistant Professor, Department of Mathematics, Stanford University, September 2000-August 2002; Professor, Department of Mathematics, Seoul National University, September 2002-December 2022; Professor, School of Mathematics, Korea Institute for Advanced Study, December 2022-present; and Director, June E Huh Center for Mathematical Challenges, January 2023-February 2024.

Contenu

An Introduction to Derived Algebraic Geometry.- An Introduction to Shifted Symplectic Structures.- An Introduction to Virtual Cycles via Classical Algebraic Geometry.- An Introduction to Virtual Cycles via Derived Algebraic Geometry.- An Introduction to Cohomological Donaldson Thomas Theory.- Moduli Spaces of Sheaves: An Overview, Curves and Surfaces.- Sheaf Counting Theory in Dimension Three and Four.