8.3 Geometric invariant theory (GIT)

Geometric invariant theory (GIT) is a powerful tool for constructing quotients of algebraic varieties by group actions. It introduces stability conditions to determine which points contribute to the quotient, allowing for the creation of well-behaved spaces with desirable geometric properties.

GIT plays a crucial role in constructing moduli spaces, which parameterize algebraic or geometric objects. By applying GIT to parameter spaces and group actions, mathematicians can create compact moduli spaces that include both original and degenerate objects, providing insights into their structure and properties.

Geometric Invariant Theory

Fundamental Concepts and Goals

• Geometric invariant theory (GIT) provides a framework for constructing quotients of algebraic varieties by group actions, particularly when the quotient space is not well-behaved or the group action is not free
• The main goal of GIT is to construct a "good" quotient space, called the GIT quotient, which retains desirable geometric properties and is more tractable than the original quotient
• GIT introduces the notion of stability conditions, which determine the locus of points in the variety where the group action is well-behaved and the quotient can be constructed
• The GIT quotient is obtained by removing the unstable points and then taking the ordinary quotient of the remaining semistable points by the group action
• GIT quotients have applications in various areas of algebraic geometry, such as the construction of moduli spaces (moduli space of curves, vector bundles) and the study of invariant theory

Stability Conditions and Hilbert-Mumford Criterion

• Stability conditions play a crucial role in GIT as they determine which points in the variety contribute to the GIT quotient and how the quotient space is constructed
• A point $x$ in $X$ is called stable if its $G$-orbit is closed in the semistable locus and its stabilizer is finite. Stable points have a well-defined image in the GIT quotient
• A point $x$ in $X$ is called semistable if there exists a $G$-invariant section of some power of $L$ that does not vanish at $x$. The GIT quotient is constructed using semistable points
• A point $x$ in $X$ is called unstable if it is not semistable. Unstable points are excluded from the GIT quotient construction
• The Hilbert-Mumford criterion provides a numerical characterization of stability in terms of one-parameter subgroups of $G$, simplifying the verification of stability conditions

Quotients of Algebraic Varieties

Applying GIT to Construct Quotients

• To apply GIT, one needs to specify an algebraic variety $X$, a reductive algebraic group $G$ acting on $X$, and a linearization of the action, which is a lifting of the action to an ample line bundle $L$ on $X$
• The choice of the linearization determines the stability conditions and, consequently, the GIT quotient. Different linearizations may lead to different GIT quotients
• The stability of a point $x$ in $X$ is determined by the behavior of the $G$-orbit of $x$ under the linearized action on the space of global sections of powers of $L$
• Points in $X$ are classified as stable, semistable, or unstable based on the growth of the dimensions of the space of $G$-invariant sections of powers of $L$
• The GIT quotient is constructed by taking the projective spectrum of the graded algebra of $G$-invariant sections of powers of $L$, after removing the unstable points

Variation of GIT Quotients

• Variation of GIT quotients studies how the GIT quotient changes as the linearization varies, leading to different stability conditions and birational transformations between quotients
• By varying the linearization, one can obtain a family of GIT quotients related by birational transformations, such as blow-ups and blow-downs
• The variation of GIT quotients can be used to study the relationship between different compactifications of moduli spaces and to understand the birational geometry of these spaces
• Wall-crossing phenomena occur when the stability condition changes abruptly as the linearization crosses certain walls in the space of linearizations, leading to different GIT quotients on either side of the wall

Moduli Spaces and Compactifications

Constructing Moduli Spaces using GIT

• GIT is a powerful tool for constructing moduli spaces, which are spaces that parameterize algebraic or geometric objects of a certain type, such as curves, surfaces, or vector bundles
• Moduli spaces often arise as quotients of parameter spaces by group actions, where the group encodes the symmetries or isomorphisms of the objects being parameterized
• GIT provides a way to construct compact moduli spaces by enlarging the space of objects to include certain degenerate or singular objects, corresponding to semistable points in the parameter space
• The GIT quotient of the parameter space, with a suitable choice of linearization, yields a compactification of the moduli space, which includes both the original moduli space and its boundary

Examples and Applications

• Examples of moduli spaces studied using GIT include the moduli space of curves (Deligne-Mumford compactification), the moduli space of vector bundles on a curve (Gieseker-Maruyama compactification), and the moduli space of stable maps (Kontsevich compactification)
• The geometry and topology of GIT quotients used in the construction of moduli spaces provide insights into the structure and properties of the moduli spaces themselves
• Moduli spaces constructed using GIT have applications in various areas of algebraic geometry, such as the study of the cohomology of moduli spaces, the computation of intersection numbers, and the formulation of enumerative problems
• GIT compactifications of moduli spaces also play a role in the study of degenerations and limit behavior of algebraic objects, as well as in the construction of invariants and characteristic classes associated with these objects