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Moduli space

In mathematics, particularly within algebraic geometry and related fields, a moduli space is a geometric space—often structured as a scheme, algebraic variety, or stack—whose points parametrize isomorphism classes of mathematical objects of a fixed type, such as smooth projective curves, vector bundles, or Riemann surfaces, thereby providing a geometric framework for classifying these objects up to equivalence.^[1]^[2] These spaces arise from the need to study families of objects that vary continuously or algebraically, capturing not only discrete classifications but also the topology and geometry inherent in their deformations.^[2] Moduli spaces can be fine or coarse, depending on whether they admit a universal family that rigidly represents all families of the objects over test schemes; a fine moduli space is representable by a scheme with a universal family establishing a bijection between families and morphisms to the space, while a coarse one provides only a bijection with isomorphism classes over algebraically closed fields but may lack such a universal structure.^[1] Prominent examples include the moduli space of elliptic curves M_{1,1}, which is the quotient of the upper half-plane by the modular group \mathrm{PSL}(2,\mathbb{Z}) and classifies elliptic curves up to isomorphism, and the moduli space of genus-g curves \overline{M}_g, a compactification introduced by Deligne and Mumford that is a smooth, proper Deligne-Mumford stack of dimension $3g-3 for g \geq 2, enabling the study of stable curves with nodal singularities.^[1]^[2] Other foundational instances are Grassmannians \mathrm{Gr}(k,n), which parametrize k-dimensional subspaces of an n-dimensional vector space and serve as projective schemes over \mathrm{Spec}(\mathbb{Z}), and Teichmüller spaces, which classify marked Riemann surfaces of genus g.^[1]^[2] The construction of moduli spaces often involves tools like Hilbert schemes, which parametrize subschemes of a fixed length or degree on a variety and are projective over the base scheme, ensuring compactness and facilitating enumerative geometry applications such as counting curves on surfaces.^[1] These spaces exhibit rich geometric properties, including smoothness under conditions like the vanishing of obstruction sheaves, and play crucial roles across disciplines: in algebraic geometry for deformation theory, in number theory via connections to the Langlands program, in topology through Donaldson invariants, and even in physics for string theory compactifications.^[1]^[2] Challenges in their study include achieving representability, handling automorphisms that lead to stacky structures, and developing compactifications to include limits of degenerating families, as exemplified by the Deligne-Mumford compactification \overline{M}_g.^[1]

Motivation and Fundamentals

Defining Moduli Problems

A moduli problem in algebraic geometry is formalized as a contravariant functor from the category of schemes to the category of sets, which associates to each scheme S (serving as a base) the set of isomorphism classes of families of geometric objects of a fixed type over S.^[1]^[3] This setup parameterizes the geometric objects up to isomorphism, capturing how they vary continuously over the base scheme, with morphisms in the category inducing pullbacks of families to preserve the functorial structure.^[4] The naive approach to a moduli space views it as a parameter space that classifies pairs consisting of a geometric object X and some additional structure, up to isomorphism, often constructed as a scheme whose points correspond to such classes without fully accounting for automorphisms.^[3] For instance, this might involve a scheme P equipped with a family F \to P, where the fibers over points of P represent the objects, but equivalences between fibers are handled only coarsely.^[1] When the geometric objects are rigid—meaning they possess no nontrivial automorphisms—the moduli space coincides simply with the parameter space, as there are no equivalences to quotient by beyond the trivial ones.^[3] In such cases, the forgetful functor from the category of objects over S to the base schemes is representable without complications.^[1] However, non-rigid objects with nontrivial automorphisms lead to representability issues for the moduli functor, as the action of automorphism groups prevents a scheme from faithfully representing the set of isomorphism classes via its points.^[4] Forgetful functors, which map to coarser moduli problems by discarding structure, highlight these obstructions, often requiring group actions on parameter spaces to identify isomorphic objects through orbits.^[3] Projective spaces exemplify simple cases where the functor is representable due to minimal automorphisms.^[1]

Historical Development

The concept of moduli spaces originated in the mid-19th century with Bernhard Riemann's foundational work on Riemann surfaces. In his 1857 paper "Theorie der Abel’schen Functionen," Riemann determined that the equivalence classes of Riemann surfaces of genus p \geq 2 under biholomorphic maps are parametrized by $3p - 3 independent complex parameters, which he termed "moduli" to capture the essential degrees of freedom in their conformal structure. This count arose from analyzing the periods of Abelian integrals and the branching of algebraic functions, laying the groundwork for classifying surfaces up to isomorphism.^[5] Building on Riemann's ideas, Alfred Clebsch advanced the parametrization of algebraic curves in 1872 through his study of binary algebraic forms. In "Theorie der binären algebraischen Formen," Clebsch provided explicit invariants and parametrizations for plane quartic curves, which are genus 3 Riemann surfaces, effectively describing a 6-dimensional moduli space via absolute invariants under projective transformations. His approach bridged complex analysis and classical invariant theory, offering concrete tools for enumerating isomorphism classes of curves.^[5] The 20th century saw a shift toward algebraic constructions of moduli spaces. In the 1950s, Wei-Liang Chow developed constructions for parametrizing algebraic cycles, introducing the Chow variety as a projective scheme that parametrizes effective algebraic cycles of fixed dimension and degree, providing an early algebro-geometric framework for moduli problems. David Mumford's Geometric Invariant Theory (GIT) in 1965 formalized the construction of moduli spaces as quotients of projective varieties by reductive group actions, using stability conditions to ensure good geometric properties. Key milestones in the late 20th century included Pierre Deligne and David Mumford's 1969 compactification of the moduli space of genus g curves, \overline{\mathcal{M}}_g, as a Deligne-Mumford stack, incorporating stable curves to achieve properness and irreducibility for g \geq 2. Michael Artin's 1971 introduction of algebraic spaces provided a category intermediate between schemes and stacks, essential for representing moduli functors that are not representable by schemes. The 1990s brought the Keel-Mori theorem, which guarantees the existence of coarse moduli spaces for algebraic stacks with finite stabilizers, as quotients by groupoids. The transition to stacky perspectives began in the 1990s with contributions from Kai Behrend, who developed tools for algebraic stacks in moduli theory, including trace formulas and cohomology computations for stacky quotients. More recently, in the 2010s, Jacob Lurie's work on derived algebraic geometry extended moduli spaces to derived stacks, accommodating infinitesimal thickenings and homotopy-theoretic structures for problems like derived deformations. The influence of physics emerged prominently from the 1980s onward, with Edward Witten applying moduli spaces of Calabi-Yau manifolds in string theory to parametrize vacua and mirror symmetry, linking geometric invariants to physical phenomena like supersymmetry breaking.

Elementary Examples

Projective Spaces

Projective space \mathbb{P}^n over a field k serves as the simplest example of a moduli space, parameterizing the 1-dimensional subspaces of the vector space k^{n+1}. Each point in \mathbb{P}^n corresponds to a line through the origin in k^{n+1}, with two vectors representing the same point if one is a scalar multiple of the other. This structure makes \mathbb{P}^n the moduli space for such lines, where the geometry arises naturally from quotienting the nonzero vectors by the multiplicative group k^\times.^[6]^[7] The dimension of \mathbb{P}^n is n, reflecting the n degrees of freedom after accounting for scaling in the (n+1)-dimensional ambient space. Points are described using homogeneous coordinates [x_0 : \dots : x_n], where (x_0, \dots, x_n) \in k^{n+1} \setminus \{0\} and scaling does not change the equivalence class. This coordinate system facilitates algebraic descriptions, such as equations defining subvarieties within \mathbb{P}^n.^[6]^[8] Equivalently, \mathbb{P}^n parameterizes the effective Cartier divisors of degree 1 on itself, which are precisely the hyperplanes. Each such divisor corresponds to a linear equation a_0 x_0 + \dots + a_n x_n = 0, up to scalar multiple, yielding a point in the dual projective space isomorphic to \mathbb{P}^n. These divisors generate the Picard group of \mathbb{P}^n, consisting of \mathbb{Z} generated by the class of a hyperplane.^[8]^[9] In terms of very ample line bundles, \mathbb{P}^n can be realized as \mathbb{P}(V) for a vector space V of dimension n+1, where points parameterize the 1-dimensional subspaces of V. The tautological line bundle \mathcal{O}_{\mathbb{P}(V)}(-1) has global sections isomorphic to V^*, and its dual \mathcal{O}(1) is very ample, embedding \mathbb{P}(V) via the complete linear system | \mathcal{O}(1) |. This construction highlights how \mathbb{P}^n arises as the parameter space for lines in the space of sections.^[6]^[10] The automorphism group of \mathbb{P}^n is the projective general linear group \mathrm{PGL}(n+1, k), which acts transitively on the points, reflecting the homogeneity of the space. This group consists of invertible linear transformations of k^{n+1} modulo scalars, inducing projective transformations that preserve the moduli structure.^[10]^[11]

Grassmannians

The Grassmannian \mathrm{Gr}(k, n) is a fundamental example of a moduli space that parameterizes the set of all k-dimensional linear subspaces of an n-dimensional vector space V \cong \mathbb{K}^n, where \mathbb{K} is a field such as \mathbb{C} or \mathbb{R}.^[12] This space arises naturally as the solution to the moduli problem of classifying such subspaces up to the action of the general linear group, providing a geometric framework for linear algebra objects in higher dimensions. As a smooth projective variety, \mathrm{Gr}(k, n) has dimension k(n-k), which reflects the degrees of freedom in choosing a k-plane in n-space after accounting for the \mathrm{GL}(k)-automorphisms stabilizing it.^[13] Points in \mathrm{Gr}(k, n) can be represented concretely as the row spaces of k \times n matrices of full rank k, where two such matrices define the same point if one is obtained from the other by left multiplication by an invertible k \times k matrix, i.e., under the left action of \mathrm{GL}(k).^[14] This quotient construction highlights the moduli interpretation, as it identifies isomorphic configurations. The projective space \mathbb{P}^{n-1} emerges as the special case \mathrm{Gr}(1, n).^[13] A canonical embedding of \mathrm{Gr}(k, n) into projective space is provided by the Plücker embedding, which maps a k-dimensional subspace U \subset V to the line in \mathbb{P}(\wedge^k V) spanned by the wedge product of a basis of U, yielding an embedding into \mathbb{P}^{\binom{n}{k} - 1}.^[15] The image satisfies the Plücker relations, a system of quadratic equations derived from the antisymmetry and multilinearity of the wedge product; for instance, for the Grassmannian \mathrm{Gr}(2,4), the relation is p_{12} p_{34} - p_{13} p_{24} + p_{14} p_{23} = 0, generating the ideal of the embedded Grassmannian.^[15] Associated to \mathrm{Gr}(k, n) are the tautological subbundle \mathcal{S} and the universal quotient bundle \mathcal{Q}. The subbundle \mathcal{S} is a rank-k vector bundle whose fiber over a point [U] \in \mathrm{Gr}(k, n) is the subspace U itself, while \mathcal{Q} is the rank-(n-k) quotient bundle fitting into the exact sequence $0 \to \mathcal{S} \to \mathrm{Gr}(k, n) \times V \to \mathcal{Q} \to 0, capturing the universal property of the Grassmannian in bundle theory.^[16]

Formal Frameworks

Fine Moduli Spaces

In algebraic geometry, a fine moduli space provides a scheme-theoretic solution to a moduli problem by representing the associated functor. Consider a moduli functor \mathcal{M} from the opposite category of schemes over a base S to sets, where \mathcal{M}(T) denotes the set of isomorphism classes of families of objects (such as varieties or sheaves) over T, up to isomorphism over T. A scheme M over S is a fine moduli space for \mathcal{M} if there exists a natural isomorphism \mathcal{M} \cong \Hom_S(-, M), meaning that for every T over S, the isomorphism classes of families over T are in bijection with morphisms T \to M. This representability ensures that M rigidly parameterizes the objects, capturing their structure without ambiguity.^[17] Associated to this representing scheme is a universal family \mathcal{U} \to M, which is the family over M corresponding to the identity morphism \id_M \in \Hom_S(M, M). The universal property guarantees that for any family \mathcal{F} \to T over another scheme T, there exists a unique morphism f: T \to M such that \mathcal{F} \cong f^*\mathcal{U} as families over T. The fiber of \mathcal{U} over a point m \in M recovers the object classified by m, making the fine moduli space a geometric parameter space that encodes both discrete isomorphism classes and continuous deformations of families. This structure distinguishes fine moduli spaces as ideal solutions when they exist.^[17] The existence of a fine moduli space is obstructed primarily by non-trivial automorphisms of the objects in the moduli problem. If the objects possess non-constant automorphism groups, the moduli functor typically fails to be representable, as distinct families related by automorphisms may induce the same morphism to a potential moduli scheme, violating the bijection. For example, non-trivial automorphisms prevent a fine moduli space for unordered collections of points on a curve or for elliptic curves without additional structure. In contrast, fine moduli spaces exist when automorphisms are trivialized or rigidified, as occurs for principally polarized abelian varieties, where the polarization ensures the automorphism group is finite and the functor becomes representable.^[18]^[19] A concrete example is the Jacobian variety J(C) of a smooth projective curve C over a field k, which serves as a fine moduli space for the functor parameterizing degree-zero line bundles on C. The Jacobian J(C) is an abelian variety representable over k, with points corresponding to isomorphism classes of such line bundles, and it admits a universal Poincar'e bundle as the universal family whose restriction to C \times \{ \mathcal{L} \} yields \mathcal{L}. This representability holds because line bundles of fixed degree have rigid automorphism groups, allowing the Picard scheme to fully represent the functor.^[20]

Coarse Moduli Spaces

A coarse moduli space provides an approximation to a moduli problem by classifying isomorphism classes of objects without necessarily representing families over the space itself. Given a moduli functor \mathcal{M} associating to each scheme S the set of isomorphism classes of families of objects over S, a coarse moduli space is a scheme M equipped with a natural transformation \pi: \mathcal{M} \to h_M (where h_M is the functor represented by M) such that \pi is universal among maps to functors represented by schemes: for any scheme N and natural transformation \phi: \mathcal{M} \to h_N, there exists a unique morphism f: M \to N making the diagram commute.^[21] The map \pi is typically proper and identifies points corresponding to objects that are isomorphic or lie in the same closure in the moduli stack, but it may contract families with nontrivial automorphisms, losing information about stabilizers.^[21] This makes coarse moduli spaces particularly useful in birational geometry, where they serve as geometric models for studying invariants like canonical divisors or ample cones, despite not parametrizing deformations precisely. For instance, the Deligne-Mumford compactification \overline{\mathcal{M}}_g is a fine moduli stack for stable curves of genus g, with a universal family over the stack. Its coarse moduli space \bar{M}_g, a scheme, classifies isomorphism classes of stable curves, capturing nodal degenerations while forgetting the stack structure that tracks automorphisms.^[22] The existence of coarse moduli spaces is guaranteed under suitable stability conditions by the Keel-Mori theorem, which states that for an Artin stack \mathcal{X} locally of finite presentation over a base scheme with finite inertia (i.e., finite stabilizers), there exists a proper, separated morphism \phi: \mathcal{X} \to Y to an algebraic space Y that is a coarse moduli space, universal for maps to algebraic spaces.^[23] Such quotients often arise from geometric invariant theory under stability conditions that bound automorphisms and ensure properness.^[24] In the context of stacks, the coarse moduli space of a Deligne-Mumford stack \mathcal{X} is denoted |\mathcal{X}| and obtained by quotienting by the étale equivalence relation generated by the inertia, yielding a scheme or algebraic space that coarse-classifies objects while the stack retains full automorphism data. Fine moduli spaces, when they exist, are special cases of coarse ones where the map is representable and families descend universally.^[21]

Stacky Perspectives

Moduli Stacks

A moduli stack addresses the limitations of scheme-theoretic moduli spaces by incorporating automorphisms through the language of category theory. Specifically, it is a stack in groupoids fibered over the category of schemes equipped with the fppf topology, where for any scheme S, the fiber category over S has objects consisting of families of the geometric objects parametrized by S (such as curves or vector bundles), and morphisms given by isomorphisms of these families. This fibered category satisfies the descent condition for effective epimorphisms, ensuring that families over covers glue appropriately up to isomorphism. The structure captures the full groupoid of objects and their symmetries, providing a more refined parametrization than schemes, which would collapse isomorphic families to points.^[25] Unlike classical representable functors to sets, moduli stacks represent functors from the opposite category of schemes to the 2-category of groupoids, allowing for non-trivial automorphism groups. For instance, the moduli stack \overline{\mathcal{M}}_{g,n} of stable n-pointed curves of genus g assigns to each scheme S the groupoid whose objects are stable families of n-pointed genus g curves over S (proper flat morphisms f: \mathcal{C} \to S with nodal connected fibers of arithmetic genus g and n distinct marked sections satisfying the stability condition), and whose morphisms are isomorphisms of such families over S. This stack is not representable by a scheme due to non-trivial automorphisms (e.g., hyperelliptic involutions for even g), but it faithfully encodes the moduli problem. The example \overline{\mathcal{M}}_{g,n} exists as a Deligne-Mumford stack for $2g + n \geq 3, highlighting how stacks resolve representability issues in classical algebraic geometry.^[26]^[27] Deligne-Mumford stacks form a distinguished class of moduli stacks suitable for problems with finite automorphisms, defined as algebraic stacks that are étale (admitting an étale surjective morphism from a scheme) and separated (with proper diagonal). An algebraic stack has a diagonal morphism representable by algebraic spaces and is locally of finite presentation, while the étale and separated conditions ensure it behaves like a scheme orbifold, with an atlas given by a scheme U via an étale representable morphism U \to \mathcal{X}. These properties guarantee that the stack is "tame," with finite stabilizers, facilitating geometric constructions like quotients. For \overline{\mathcal{M}}_{g,n}, the Deligne-Mumford conditions hold, as proven by showing the diagonal is unramified and the stack has a smooth scheme cover.^[28] The inertia stack of a moduli stack \mathcal{X}, denoted I\mathcal{X}, encodes the automorphisms of its objects and is defined as the fiber product \mathcal{X} \times_{\Delta, \mathcal{X} \times \mathcal{X}} \mathcal{X}, where \Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X} is the diagonal. Over a geometric point, the automorphism group of an object corresponds to the fiber of I\mathcal{X} at that point, making the inertia stack a key tool for analyzing symmetries in moduli problems. In Deligne-Mumford stacks, the inertia stack is finite over \mathcal{X}, reflecting the finite nature of automorphisms. This construction is central to understanding stacky phenomena, such as orbifold structures in the moduli of curves.^[29]^[1]

Algebraic Stacks in Moduli Theory

Algebraic stacks provide a foundational framework in moduli theory by generalizing schemes and algebraic spaces to account for automorphisms and stacky phenomena, allowing for the precise formulation of moduli problems that lack fine moduli spaces. An algebraic stack over a base scheme S is a stack in groupoids over the big fppf site of S-schemes, equipped with a representable diagonal morphism that is representable by algebraic spaces, and admitting a smooth and surjective representable morphism from a scheme.^[25] This structure, introduced by Artin, ensures that algebraic stacks behave well under base change and descent, making them suitable for geometric constructions in moduli theory.^[30] Quotient stacks exemplify algebraic stacks in moduli problems involving group actions, where the stack [X/G] classifies principal G-torsors equipped with an X-structure over schemes. Here, X is an algebraic space acted upon by a group scheme G, and the stack [X/G] is the stackification of the presheaf associating to each scheme the category of G-torsors over it with compatible X-structures.^[31] A key example arises in the moduli of principal bundles: for a reductive group G, the quotient stack [*/G] (or more generally, stacks of G-bundles over a fixed base) parametrizes isomorphism classes of principal G-bundles, capturing the stacky nature due to nontrivial automorphisms while providing a geometric object for further study.^[31] In deformation theory, algebraic stacks handle rigidity and obstructions more robustly than schemes, with infinitesimal deformations controlled by cohomology groups and versal families providing local models. For an object in an algebraic stack \mathcal{X}, obstructions to lifting deformations to higher order lie in H^2 of the cotangent complex or associated sheaves, generalizing the scheme case where such obstructions appear in Ext groups.^[32] Artin's framework establishes the existence of versal deformations for algebraic stacks satisfying Schlessinger's criteria, ensuring that formal versal deformations algebraize to algebraic families, thus enabling the construction of smooth presentations and the proof of algebraicity via deformation properties.^[30] Recent advancements extend algebraic stacks to derived settings, incorporating homotopical data essential for modern moduli problems like those in mirror symmetry. Derived algebraic stacks, which resolve singularities via simplicial or dg enhancements, admit shifted symplectic structures—generalizations of classical symplectic forms shifted by an integer degree.^[33] Seminal work shows that classifying stacks of reductive groups and the derived moduli stack of perfect complexes carry canonical 2-shifted symplectic structures, facilitating quantization and Lagrangian correspondences in mirror symmetry contexts post-2010.^[33] These structures equip derived moduli spaces, such as those of local systems on Calabi-Yau varieties, with tools for studying virtual invariants and homological mirror symmetry.^[33]

Key Examples in Algebraic Geometry

Moduli of Curves

The moduli space M_g parametrizes isomorphism classes of smooth projective curves of genus g \geq 2 over \mathbb{C}. It has complex dimension $3g-3, reflecting the $3g-3 independent moduli needed to specify such a curve up to isomorphism.^[27] However, M_g fails to be a fine moduli space because curves with non-trivial automorphism groups, such as hyperelliptic curves, prevent the existence of a universal family over it; instead, M_g is realized as a smooth Deligne–Mumford stack of finite type.^[27] To obtain a compactification, Deligne and Mumford constructed \overline{M}_g, which includes stable nodal curves—connected, projective curves with at worst nodal singularities where every rational component has at least three special points (marked points or nodes) and every elliptic component has at least one. This compactification is a smooth, proper Deligne-Mumford stack of dimension $3g-3, with M_g as a dense open subset.^[27] The complement \overline{M}_g \setminus M_g, known as the boundary or degenerate locus, is a normal crossings divisor consisting of irreducible components \Delta_i for i = 0, \dots, \lfloor g/2 \rfloor; here, \Delta_i parametrizes stable curves with a separating node joining irreducible components of arithmetic genera i and g-i. Additionally, there is the irreducible nodal divisor \Delta_{\rm irr} for curves with a single node but irreducible normalization of genus g. These boundary divisors encode the ways in which smooth curves degenerate in families.^[27] A key line bundle on \overline{M}_g is the lambda bundle \lambda, defined as the determinant of the Hodge bundle \mathbb{E}, a rank-g vector bundle whose fiber over a point [C] \in \overline{M}_g is H^0(C, \omega_C), the space of holomorphic differentials on C.^[34] For g \geq 3, the Picard group of the open moduli space M_g is \mathbb{Z} and generated by \lambda, which pulls back from \overline{M}_g and plays a central role in the intersection theory and birational geometry of these spaces.^[35] The intersection theory of \overline{M}_g features prominently in enumerative geometry, exemplified by Witten's 1990 conjecture relating intersection numbers of psi classes (first Chern classes of the cotangent bundles at marked points) on \overline{M}_{g,n} to correlators in two-dimensional quantum gravity, equivalently predicting closed formulas for these numbers via the KdV hierarchy. Kontsevich proved this conjecture in 1992 using matrix integrals and graph combinatorics, establishing explicit recursive relations for the integrals \int_{\overline{M}_{g,n}} \prod_{i=1}^n \psi_i^{k_i}. These numbers provide deep insights into the tautological ring of \overline{M}_g and its compactifications.

Moduli of Abelian Varieties

The moduli space \mathcal{A}_g parametrizes isomorphism classes of principally polarized abelian varieties of dimension g. Over the complex numbers, \mathcal{A}_g is realized as the quotient \mathbb{H}_g / \mathrm{Sp}(2g, \mathbb{Z}), where \mathbb{H}_g denotes the Siegel upper half-space consisting of g \times g complex symmetric matrices with positive definite imaginary part. This construction provides a coarse moduli space, as every point corresponds to a unique principally polarized abelian variety up to isomorphism. The dimension of \mathcal{A}_g is \frac{g(g+1)}{2}.^[36] To achieve a fine moduli structure with level information, one considers level-n covers such as \mathcal{A}_g(n), which parametrize principally polarized abelian varieties equipped with a level-n structure, resolving the obstruction from the action of the symplectic group. These covers are finite étale over \mathcal{A}_g and facilitate the study of torsion points on the abelian varieties. For an arithmetic perspective, \mathcal{A}_g admits toroidal compactifications defined over \mathrm{Spec} \mathbb{Z}, which extend the moduli problem to include semi-abelian degenerations with toric parts; prominent examples include the perfect cone and second Voronoi compactifications. These constructions, pioneered in the analytic setting and later algebraicized, ensure the compactifications carry universal families.^[36] Siegel modular forms are scalar-valued automorphic forms on \mathrm{Sp}(2g, \mathbb{Z}) \backslash \mathbb{H}_g, defined as holomorphic functions f: \mathbb{H}_g \to \mathbb{C} satisfying the transformation law f\left( \begin{pmatrix} A & B \\ C & D \end{pmatrix} \tau \right) = \det(C\tau + D)^k f(\tau) for \begin{pmatrix} A & B \\ C & D \end{pmatrix} \in \mathrm{Sp}(2g, \mathbb{Z}) and weight k, with suitable behavior at the cusps. They arise naturally as sections of powers of the determinant line bundle on \mathcal{A}_g, providing invariants that distinguish points in the moduli space. For g=1, the ring of Siegel modular forms coincides with the ring of elliptic modular forms on the modular group \mathrm{SL}(2, \mathbb{Z}). Seminal results on their structure and dimension formulas were established for small genera.^[37] The Torelli theorem asserts that the map sending a smooth projective curve of genus g to its Jacobian abelian variety with the induced principal polarization embeds the moduli space M_g of curves into \mathcal{A}_g. This injectivity highlights the role of abelian varieties in reconstructing curve data from period matrices. The result, originally proved analytically, extends to the algebraic category and underscores the interplay between curve and abelian moduli.

Constructions and Techniques

Geometric Invariant Theory

Geometric Invariant Theory (GIT), developed by David Mumford, offers a foundational approach to constructing moduli spaces in algebraic geometry by forming geometric quotients of projective varieties under actions of reductive algebraic groups. For a projective variety X over an algebraically closed field equipped with an action by a reductive group G, one selects an ample line bundle L on X together with a G-linearization, which endows the powers of L with compatible G-actions. The GIT quotient X // G is constructed as the Proj of the ring of invariants \bigoplus_{n \geq 0} H^0(X, L^{\otimes n})^G, which yields a projective variety parameterizing closed G-orbits in the semistable locus of X. This quotient captures invariants of the group action and provides a coarse moduli space for isomorphism classes of objects parameterized by X. Central to GIT are the notions of stability, which determine the points included in the quotient. A point x \in X is semistable if for every one-parameter subgroup \lambda: \mathbb{G}_m \to G, the limit \lim_{t \to 0} \lambda(t) \cdot x exists in X. More restrictively, x is properly stable if it is semistable, its stabilizer in G is finite, and its orbit is closed in the semistable locus. The semistable locus X^{ss}(L) consists of all semistable points with respect to the linearization of L, and the quotient X^{ss}(L) // G is a geometric quotient on the open subset of properly stable points, where orbits correspond bijectively to points in the quotient. This is equivalent to the Hilbert-Mumford numerical criterion: for a point x \in X and a 1-PS \lambda, define the numerical function \mu(x, \lambda) as the minimum weight of the action of \lambda on the fiber L_x, normalized appropriately (specifically, \mu(x, \lambda) = -\min \{ r_i \mid x lies in the span of basis elements with weights r_i under \lambda)). Then x is semistable if and only if \mu(x, \lambda) \geq 0 for all 1-PS \lambda. This criterion reduces the geometric condition to a combinatorial computation of weights, facilitating the identification of stable loci in explicit examples. A prominent application of GIT arises in the construction of the moduli space of stable curves. The Deligne-Mumford compactification \overline{\mathcal{M}}_g of the moduli space of genus-g curves (g \geq 2) is obtained as a GIT quotient of the Hilbert scheme of tri-canonically embedded stable curves in \mathbb{P}^{5g-6}, where the projective linear group \mathrm{PGL}(5g-5) acts via the linear system |3K| (the complete linear series of the canonical bundle to the third power).^[27] Stable curves, defined as those with finite automorphism groups and only nodal singularities, embed as GIT-stable points under this linearization, yielding \overline{\mathcal{M}}_g as the projective quotient that parameterizes isomorphism classes of such curves. This construction proves the irreducibility of \overline{\mathcal{M}}_g and extends the moduli problem to a compact space.^[27]

Hilbert and Chow Schemes

The Hilbert scheme \mathrm{Hilb}^d_{\mathbb{P}^n} provides a moduli space for zero-dimensional subschemes of length d in projective n-space \mathbb{P}^n. Introduced by Grothendieck, it represents the functor that assigns to any scheme S the set of flat families of such subschemes over S, and is an irreducible projective scheme of dimension d(n+1).^[38] For linear subspaces, the Hilbert scheme recovers the Grassmannian when considering appropriate dimensions. A key feature is the universal family \mathcal{Z} \subset \mathrm{Hilb}^d_{\mathbb{P}^n} \times \mathbb{P}^n, which is flat over the Hilbert scheme and parametrizes all such subschemes universally, enabling the study of deformations within flat families. This flatness ensures that the Hilbert scheme captures infinitesimal deformations, making it a fundamental tool in deformation theory for resolving singularities in subschemes, such as smoothing multiple points into reduced configurations.^[38] The Chow variety, constructed by Chow and van der Waerden, parameterizes effective zero-cycles of degree d on \mathbb{P}^n, serving as a coarser moduli space focused on cycle classes rather than scheme structures. It arises as the quotient of the Hilbert scheme via the Hilbert-Chow morphism, a birational resolution that contracts strata corresponding to non-reduced subschemes to their underlying cycles, thus providing a normalization of the symmetric product of \mathbb{P}^n.^[39] Applications of these schemes extend to enumerative invariants; for instance, Göttsche's formula computes the refined Euler characteristic of the Hilbert scheme of points on a smooth projective surface, expressing it in terms of the eta function and surface invariants.

Properties and Invariants

Dimensions and Volumes

The dimension of the moduli space \mathcal{M}_g of smooth genus-g curves over \mathbb{C} is $3g-3 for g \geq 2. This follows from classical deformation theory, where the Zariski tangent space to \mathcal{M}_g at the point corresponding to a smooth curve X is isomorphic to the first cohomology group H^1(X, T_X), with T_X denoting the tangent sheaf of X. For a smooth projective curve of genus g \geq 2, the Riemann-Roch theorem yields \dim H^1(X, T_X) = 3g-3, since \dim H^0(X, T_X) = 0 (as the automorphism group is finite) and the Euler characteristic \chi(T_X) = -(3g-3).^[22]^[40] In general, for a moduli space parametrizing families of geometric objects, the local dimension at a point [X] is given by \dim H^1(X, T_X) minus the dimension of the automorphism group, reflecting the obstructions and infinitesimal deformations. This framework extends beyond curves; for example, the moduli space of abelian varieties of dimension d has dimension d(d+1)/2, derived similarly from the cohomology of the tangent sheaf.^[40] The Weil-Petersson (WP) metric on \mathcal{M}_g, induced from the hyperbolic metric on Teichmüller space via the mapping class group action, defines a natural Riemannian structure whose associated volume form yields finite orbifold volumes for \mathcal{M}_g. These WP volumes encode deep geometric information, including counts of simple closed geodesics on hyperbolic surfaces. In a seminal 2007 paper, Mirzakhani established a recursive formula for the WP volumes V_{g,n}(b_1, \dots, b_n) of the moduli space \mathcal{M}_{g,n}(b) of genus-g hyperbolic surfaces with n geodesic boundary components of fixed lengths b_i, expressing them as polynomials in the b_i whose coefficients are weighted intersection numbers on \mathcal{M}_{g,n}. This recursion relates directly to hyperbolic geometry through Wolpert's magic formula for the WP metric and symplectic structure on Teichmüller space.^[41] For large genus g, the WP volume V_g of \mathcal{M}_g exhibits asymptotic growth V_g \sim C \cdot \kappa^g / g^{1/2} for some constants C > 0 and \kappa > 0, reflecting the exponential proliferation of hyperbolic structures tempered by polynomial factors from the metric's curvature properties. This asymptotic refines earlier estimates and confirms conjectures on the leading behavior, with applications to random hyperbolic surfaces.^[42] In the stacky perspective, the moduli stack [\mathcal{M}_g / \mathrm{Aut}] carries an orbifold structure where volumes are computed by adjusting for stabilizers: the orbifold WP volume integrates the volume form over the coarse moduli space \mathcal{M}_g, dividing locally by the order of the automorphism group \mathrm{Aut}(X) at each point [X] with non-trivial stabilizers (e.g., hyperelliptic curves with |\mathrm{Aut}(X)| = 2). This adjustment ensures the stack volume matches the orbifold volume of \mathcal{M}_g, preserving invariance under stacky isomorphisms and facilitating computations via intersection theory on the stack.^[41]

Stability Conditions

Stability conditions provide criteria to select a well-behaved subset of objects in a moduli problem, ensuring the resulting moduli space is compact and projective. These conditions filter out unstable objects, such as those with excessive automorphisms or unbounded invariants, allowing the construction of geometric quotients via techniques like geometric invariant theory (GIT). In algebraic geometry, they are crucial for parametrizing families of curves, sheaves, and other geometric structures while maintaining desirable properties like Hausdorff separation and properness.^[43] In GIT, stability is defined for points in a projective variety under a reductive group action linearized by an ample line bundle. A point x is \mu-semistable if for every one-parameter subgroup \lambda of the group, the Hilbert-Mumford weight \mu(\lambda, x) \leq 0, ensuring the point lies in the affine cone over the projective space. Properly stable points further require that \mu(\lambda, x) = 0 only for the trivial subgroup and that the orbit closure does not contain fixed points, which guarantees finite stabilizers and closed orbits in the quotient, facilitating the formation of a good moduli space. For vector bundles on curves, slope stability uses the slope \mu(E) = \deg(E)/\rk(E), where \deg(E) is the degree and \rk(E) the rank. A bundle E is stable if for every proper subbundle F \subset E, \mu(F) < \mu(E), and semistable if \mu(F) \leq \mu(E); this condition bounds the possible extensions and ensures the moduli space of stable bundles of fixed rank and degree is a projective variety.^[44] Gieseker stability refines slope stability for coherent sheaves on higher-dimensional varieties by incorporating the Hilbert polynomial P_E(m), which encodes the dimensions of cohomology groups via the Riemann-Roch theorem. A sheaf E is Gieseker-semistable if for every proper subsheaf F \subset E, the reduced Hilbert polynomial p_E(t) = P_E(t)/\rk(E) satisfies p_F(t) \leq p_E(t) in the sense of leading coefficients and degrees; this weighting by polynomial terms addresses limitations of pure slope stability in higher dimensions, yielding bounded families and projective moduli spaces for semistable sheaves.^[45] Bridgeland stability generalizes these notions to the derived category of coherent sheaves, introducing stability conditions as pairs (Z, \mathcal{P}) on a triangulated category, where Z is a central charge function assigning complex phases to objects, and \mathcal{P} is a slicing by phase. An object is stable if its phase exceeds that of any quotient, with semistability allowing direct sum decompositions into stables of equal phase; introduced in 2007, this framework incorporates tilting to relate classical and derived stabilities and underpins Donaldson-Thomas invariants in enumerative geometry.^[46] These stability conditions are applied, for instance, in the moduli of stable vector bundles on curves, where slope stability yields compactifications parametrizing S-equivalence classes.^[44]

Applications Beyond Geometry

In Differential Geometry

In differential geometry, moduli spaces parametrize geometric structures on manifolds up to diffeomorphisms or other equivalence relations, often arising from solutions to partial differential equations, in contrast to the algebraic geometry setting where they classify objects up to birational transformations or isomorphisms in a projective category. These smooth moduli spaces frequently exhibit non-compactness and may be infinite-dimensional before quotienting by symmetry groups, reflecting the analytic nature of the underlying problems without the stabilizing compactifications typical in algebraic contexts. A key example is the Teichmüller space, which serves as a foundational model for such constructions. The Teichmüller space \mathcal{T}_g for a closed orientable surface of genus g \geq 2 parametrizes marked Riemann surfaces, that is, equivalence classes of pairs (X, f) where X is a Riemann surface of genus g and f: S \to X is a diffeomorphism from a fixed reference surface S, up to homotopy. This space is contractible, as established by Teichmüller's theorem, providing a universal cover for the moduli space of Riemann surfaces. The mapping class group \Gamma_g, consisting of isotopy classes of diffeomorphisms of S, acts properly discontinuously on \mathcal{T}_g, yielding the moduli space \mathcal{M}_g = \mathcal{T}_g / \Gamma_g as an orbifold. The real dimension of \mathcal{T}_g is $6g - 6, derived from the local coordinates given by Fenchel-Nielsen or Beltrami differentials, which count the degrees of freedom in deforming the complex structure. Another prominent example is the moduli space of flat connections on a principal G-bundle over a compact Riemann surface, where G is a compact Lie group. This space arises as the quotient of the infinite-dimensional space of all connections by the gauge group action, with critical points of the Yang-Mills functional corresponding to flat connections. Atiyah and Bott analyzed this using Morse theory on the space of connections, showing that the moduli space carries a natural symplectic structure and that its cohomology can be computed via equivariant techniques. The prequotient space of connections is infinite-dimensional, and the resulting moduli space is finite-dimensional, often stratified by topological invariants like the Chern class. In gauge theory, moduli spaces of instantons on compact 4-manifolds provide invariants for smooth topology. These are solutions to the anti-self-dual Yang-Mills equations on principal SU(2)-bundles, forming the moduli space \mathcal{M}_k(X) of dimension $8k - 3(b_2^+(X) + 1) for second Chern number k, after quotienting by the gauge group.^[47] Donaldson introduced these in the 1980s to construct polynomial invariants distinguishing exotic smooth structures on 4-manifolds, such as showing that the E_8 plumbing is not diffeomorphic to \mathbb{CP}^2 \# 9\overline{\mathbb{CP}^2}. The Uhlenbeck compactification addresses bubbling phenomena at infinity, ensuring a stratified compactification unlike the Deligne-Mumford compactification in the algebraic case. Unlike algebraic moduli spaces, which are often compact via stable reduction and benefit from stability conditions to ensure properness, smooth moduli spaces like those above are typically non-compact due to the absence of analogous bounding criteria, leading to asymptotic behaviors such as necks degenerating in Teichmüller space or bubbles in instanton moduli. Over the complex numbers, the moduli space of smooth Riemann surfaces coincides analytically with that of algebraic curves, but the smooth perspective emphasizes infinite-dimensional function spaces and PDE solutions.

In Physics

In theoretical physics, moduli spaces play a central role in string theory, where the moduli space of Calabi-Yau threefolds parameterizes the different possible vacua of type II string compactifications, determining the low-energy effective theory including the gauge groups and matter content.^[48] The complex structure moduli control the deformations of the threefold's holomorphic structure, while the Kähler moduli govern the sizes of its cycles, both contributing to the overall vacuum landscape.^[49] Mirror symmetry, discovered in the 1990s, establishes a profound duality between pairs of topologically distinct Calabi-Yau threefolds, under which the complex structure moduli space of one manifold is exchanged with the Kähler moduli space of its mirror, leading to isomorphic superconformal field theories despite different geometries.^[50] This symmetry not only equates the number of vacua but also provides computational tools for enumerating Hodge numbers and understanding non-perturbative effects in string theory.^[48] In the context of superconformal field theories (SCFTs), the moduli space encompasses exactly solvable models such as Gepner models, which construct the internal sector of string compactifications using tensor products of N=2 minimal models with total central charge c=9 to match the requirements for Calabi-Yau threefolds. The dimension of this moduli space can be counted from the conformal weights and fusion rules of the minimal models, ensuring consistency with the anomaly cancellation and supersymmetry preservation in the full string theory. Donaldson-Thomas invariants, generalized to quiver representations, provide a mathematical framework for counting BPS states in three-dimensional N=4 gauge theories arising from M-theory compactifications on Calabi-Yau threefolds, as developed by Joyce and Song in 2009.^[51] These invariants capture the protected spin characters of stable quiver representations, offering invariants of the BPS spectrum that are independent of the choice of stability condition within the chambers of the moduli space.^[51] In the 2010s, F-theory compactifications on elliptically fibered K3 surfaces have utilized moduli spaces to describe configurations of 7-branes, where the complex structure of the elliptic fibration encodes the positions and types of intersecting 7-branes, leading to realistic grand unified theories with controlled Yukawa couplings. The moduli space integrates the geometric deformations of the base and the non-perturbative effects from 7-brane monodromies, facilitating model-building for particle physics phenomenology.

References

[1]
[PDF] Math 259x: Moduli Spaces in Algebraic Geometry
Aug 26, 2020 · One of the characterizing features of algebraic geometry is that the set of all geometric objects of a fixed type (e.g. smooth projective ...
[2]
[PDF] Moduli Spaces - UT Math
Moduli spaces can be thought of as geometric solu- tions to geometric classification problems. In this arti- cle we shall illustrate some of the key features ...
[3]
[PDF] Moduli Problems and Geometric Invariant Theory.
Abstract. In this course, we study moduli problems in algebraic geometry and the construction of moduli spaces using geometric invariant theory.
[4]
[PDF] My Introduction to Schemes and Functors
Feb 24, 2009 · In modern terms, this means calculating the dimension of Γ(X, L), the global sec- tions of a line bundle on a variety X. The Italians had worked ...
[5]
None
Below is a merged summary of the key historical milestones in the development of moduli spaces, combining all the information from the provided segments into a single, comprehensive response. To retain as much detail as possible, I will use a table in CSV format for clarity and density, followed by a narrative summary that integrates additional context and references not easily captured in the table. The table will focus on key contributors, their contributions, years, and references, while the narrative will address broader themes like physics influence and additional details.
[6]
[PDF] math 631 notes algebraic geometry - karen smith
Definition 5.1. The projective space of V , denoted P(V ), is the set of 1-dimensional subspaces of. V . We denote Pn.
[7]
[PDF] Algebraic Geometry - UCR Math Department
For any vector space V over k, the projective space of V , denoted. P(V ), is the set of all 1-dimensional subspaces of V . Alternatively one can write P n k =.
[8]
[PDF] Algebraic Geometry (Math 6130)
projective space. This allows us to define algebraic sets and varieties in ... So P(V ) may be thought of as the set of hyperplanes in V . (b) Clearly ...
[9]
[PDF] FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 28 AND 29
Effective Cartier divisors sometimes restrict too: if you have effective ... hyperplanes on Pn. Show that Pn − Y is affine, and. Pn − Y − Hi is a ...
[10]
[PDF] Projective Varieties and their Sheaves of Regular Functions
(d) Find an isomorphism of groups: Aut(CPn) ∼= PGL(n + 1, C) between the group Aut(CPn) of regular automorphisms of CPn and the group PGL(n + 1,C) of ...
[11]
[2405.13916] Projective Space in Synthetic Algebraic Geometry - arXiv
May 22, 2024 · We show that the Picard group of projective n-space is the integers, the automorphism group of projective n-space is PGL(n+1) and morphisms between projective ...
[12]
[PDF] grassmannians: the first example of a moduli space
Let G(k, n) denote the classical Grassmannian that parameterizes k-dimensional linear subspaces of a fixed n-dimensional vector space V . G(k, n) naturally ...
[13]
[PDF] Lecture 4 Grassmannians, Finite and Affine Morphisms
The Grassmannian Gr(k, n) is the set of linear subspaces of dimension k in the n-dimensional vector space Kn := V . For example, Gr(1,n) = Pn−1. Here, we ...
[14]
[PDF] Projective Spaces, Grassmannians and the Plücker Embedding
Here, the geometric objects are subspaces of any dimension, and the space of k-dimensional subspaces in n-space will be called the Grassmannian Gr(k, n). 5.1 ...
[15]
[PDF] from Algebraic Geometry - A First Course - Anand Deopurkar
The Ea are thus quadratic polynomials whose common zero locus is the Grass- mannian G(k, V). They are called the Plücker relations, and they do in fact generate.
[16]
[PDF] Degeneracy loci in Grassmannians - UC Irvine
We call the quotient bundle Q = (Gr(k, n) × Cn)/S the universal quotient bundle or the tautological quotient bundle on Gr(k, n). When k = 1, the universal ...
[17]
[PDF] 2.3. Fine moduli spaces. The ideal situation is when there is a ...
Fine moduli spaces. The ideal situation is when there is a scheme that represents our given moduli functor. Definition 2.15. Let M : Sch → Set be a moduli ...
[18]
[PDF] Notes on Moduli theory, Stacks and 2-Yoneda's Lemma - arXiv
Feb 14, 2022 · fine moduli space corresponds to the representability of this moduli functor. More details are to be discussed below. Definition 1.0.1. Let ...
[19]
[PDF] Abelian Varieties and Moduli - Purdue Math
Apr 19, 2012 · 4.5 Moduli space of principally polarized abelian varieties . . . . . . ... The property of being a fine moduli space characterize the space up to.
[20]
[PDF] Jacobian Varieties
This article contains a detailed treatment of Jacobian varieties. Sections. 2, 5, and 6 prove the basic properties of Jacobian varieties starting from the.
[21]
Subsection 112.5.2 (04UX): Coarse moduli spaces
A general coarse moduli space for an Artin stack with finite inertia will only commute with flat base change.
[22]
[PDF] The moduli space of curves
Sep 30, 2020 · Functors in the image are called representable and have a fine moduli space. However, there are bigger classes of moduli functors, having ...
[23]
Quotients by Groupoids - jstor
This paper discusses quotients by groupoids, showing that a flat groupoid with a finite stabilizer can form a geometric and categorical quotient.Missing: original | Show results with:original
[24]
Section 106.13 (0DUK): The Keel-Mori theorem—The Stacks project
106.13 The Keel-Mori theorem. In this section we start discussing the theorem of Keel and Mori in the setting of algebraic stacks.
[25]
[PDF] The irreducibility of the space of curves of given genus - Numdam
THE IRREDUCIBILITY OF THE SPACE OF CURVES OF GIVEN GENUS. 101 separated, and locally of finite presentation. An algebraic stack is noetherian, if it is quasi ...
[26]
Section 94.12 (026N): Algebraic stacks—The Stacks project
Here is the definition of an algebraic stack. ... To distinguish the two cases one sees the terms “Deligne-Mumford stack” and “Artin stack” used in the literature ...
[27]
Versal deformations and algebraic stacks | Inventiones mathematicae
Versal deformations and algebraic stacks. Published: September 1974. Volume 27, pages 165–189, (1974); Cite this article. Download PDF ... Artin, M.: Algebraic ...
[28]
Lemma 94.16.1 (04T4)—The Stacks project
### Summary of Quotient Stacks [X/G] and Their Role
[29]
Deformation theory and algebraic stacks—The Stacks project
Artin: Versal deformations and algebraic stacks [ArtinVersal]. This momentous ... Then if we assume \mathcal{X} admits a deformation and obstruction theory ...
[30]
[1111.3209] Shifted Symplectic Structures - arXiv
Nov 14, 2011 · We prove that classifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-shifted symplectic structures.Missing: mirror post- 2010
[31]
[PDF] Towards an Enumerative Geometry of the Moduli Space of Curves
The goal of this paper is to formulate and to begin an exploration of the enumerative geometry of the set of all curves of arbitrary genus g.
[32]
On the Kodaira dimension of the moduli space of curves
Harris, J., Mumford, D. On the Kodaira dimension of the moduli space of curves. Invent Math 67, 23–86 (1982).
[33]
[PDF] arXiv:0711.0094v4 [math.AG] 2 Sep 2010
Sep 2, 2010 · We denote by Ag the moduli space of principally po- larized abelian varieties (or ppavs for short) of dimension g, up to isomorphisms preserving ...
[34]
The Geometry of Siegel Modular Varieties - Project Euclid
Siegel modular varieties are moduli spaces for abelian varieties with polarization and level structure, and are locally symmetric varieties.
[35]
[PDF] Connectedness of the Hilbert scheme - Numdam
In this paper we study the Hilbert scheme of subschemes of projective space, as defined by Grothendieck [FGA, p. 221-01 ft] (2). Let S be a noetherian prescheme ...Missing: original | Show results with:original
[36]
[1009.5898] The Hilbert-Chow morphism and the incidence divisor
Sep 29, 2010 · There is a natural definition of the corresponding line bundle on a product of Hilbert schemes, and we show this bundle descends to the Chow ...
[37]
[PDF] Exercises on the moduli spaces and the Torelli map
Definition 2.1.2. The moduli stack of curves of genus g Mg is defined as the algebraic stack whose S points are the groupoid of smooth proper f : C → S of ...
[38]
weil-petersson volumes and intersection theory on the moduli space ...
Mar 8, 2006 · ON THE MODULI SPACE OF CURVES. MARYAM MIRZAKHANI. 1. Introduction. In this paper, we establish a relationship between the Weil-Petersson volume.
[39]
None
### Summary of Asymptotic Formula for Weil-Petersson Volume
[40]
[PDF] MODULI SPACES AND GEOMETRIC INVARIANT THEORY
Mumford's geometric invariant theory (GIT) [74] for reductive groups provides a method for constructing quotients of reductive group actions in algebraic ...
[41]
Stable and Unitary Vector Bundles on a Compact Riemann Surface
This paper, 'Stable and Unitary Vector Bundles on a Compact Riemann Surface', is by M.S. Narasimhan and C.S. Seshadri, published in Annals of Mathematics.
[42]
On the Moduli of Vector Bundles on an Algebraic Surface - jstor
The set of such sheaves is bounded. We have established Theorem 0.7 i). 2. We wish to analyze the definition of semi-stability in a special case.
[43]
[PDF] Stability conditions on triangulated categories - Annals of Mathematics
This paper introduces the notion of a stability condition on a triangu- lated category. The motivation comes from the study of Dirichlet branes in.
[44]
[hep-th/9309097] Calabi-Yau Moduli Space, Mirror Manifolds and ...
Oct 5, 1993 · We analyze the moduli spaces of Calabi-Yau threefolds and their associated conformally invariant nonlinear sigma-models and show that they are described by an ...Missing: 1990s | Show results with:1990s
[45]
[PDF] Moduli Spaces of Calabi-Yau Compactifications (PDF)
Calabi-Yau compactifications have played an important role in studying su- persymmetric vacua of string theory. More recently they have also featured.
[46]
Duality in Calabi-Yau moduli space - ScienceDirect.com
We describe a duality in the moduli space of string vacua which pairs topologically distinct Calabi-Yau manifolds and shows that the yield isomorphic conformal ...
[47]
[0810.5645] A theory of generalized Donaldson-Thomas invariants
Oct 31, 2008 · This book defines and studies a generalization of Donaldson-Thomas invariants. Our new invariants \bar{DT}^a(t) are rational numbers, defined for all Chern ...Missing: BPS states