Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties

In this paper, we define the virtual moduli cycle of moduli spaces with perfect tangent-obstruction theory. The two interesting moduli spaces of this type are moduli spaces of vector bundles over surfaces and moduli spaces of stable morphisms from curves to projective varieties. As an application, we define the Gromov-Witten invariants of smooth projective varieties and prove all its basic properties.

     

Injective resolutions of BG and derived moduli spaces of local systems

It was suggested on several occasions by Deligne, Drinfeld and Kontsevich that all the moduli spaces arising in the classical problems of deformation theory should be extended to natural “derived” moduli spaces which are always smooth in an appropriate sense and whose tangent spaces involve the entire cohomology of the sheaf of infinitesimal automorphisms, not just H¹. In this note we give an algebraic construction of such an extension for the simplest class of moduli spaces, namely for moduli of local systems (representations of the fundamental group).     

Pullback Moduli Spaces

Geometric invariant theory can be used to construct moduli spaces associated to representations of finite dimensional algebras. One difficulty which occurs in various natural cases is that nonisomorphic modules are sent to the same point in the moduli spaces which arise. In this article, we study how this collapsing phenomenon can sometimes be reduced by considering pullbacks of modules for an auxiliary algebra. One application is a geometric proof that the twisting action of an algebra automorphism induces an algebraic isomorphism between moduli spaces.     

Morse theory, Higgs fields, and Yang–Mills–Higgs functionals

In this mostly expository paper we describe applications of Morse theory to moduli spaces of Higgs bundles. The moduli spaces are finite-dimensional analytic varieties but they arise as quotients of infinite-dimensional spaces. There are natural functions for Morse theory on both the infinite-dimensional spaces and the finite-dimensional quotients. The first comes from the Yang?CMills?CHiggs energy, while the second is provided by the Hitchin function. After describing what Higgs bundles are, we explore these functions and how they may be used to extract topological information about the moduli spaces.     

On the Euler characteristic of Kronecker moduli spaces

Combining the MPS degeneration formula for the Poincaré polynomial of moduli spaces of stable quiver representations and localization theory, it turns that the determination of the Euler characteristic of these moduli spaces reduces to a combinatorial problem of counting certain trees. We use this fact in order to obtain an upper bound for the Euler characteristic in the case of the Kronecker quiver. We also derive a formula for the Euler characteristic of some of the moduli spaces appearing in the MPS degeneration formula.     

Tilting Theoretical Approach to Moduli Spaces Over Preprojective Algebras

We apply tilting theory over preprojective algebras Λ to the study of moduli spaces of Λ-modules. We define the categories of semistable modules and give equivalences, so-called reflection functors, between them by using tilting modules over Λ. Moreover we prove that the equivalence induces an isomorphism of K-schemes between moduli spaces. In particular, we study the case when the moduli spaces are related to Kleinian singularities, and generalize some results of Crawley-Boevey (Am J Math 122:1027–1037, 2000).     

Harmonic maps between complex projective spaces

Using the DoCarmo-Wallach theory, we classify (homogeneous) polynomial harmonic maps of complex projective spaces into spheres and complex projective spaces in terms of finite dimensional moduli spaces. We make use of representation theory of the (special) unitary group to give, for a spherical range, the exact dimension and, for complex projective spaces, a lower bound of the dimension of the moduli spaces.Supported in part by NSF Grant DMS 8603172.Research done in part while the third named author was on leave at University of California, Berkeley.     

Cobordism for Hamiltonian loop group actions and flat connections on the punctured two-sphere

We develop a theory of symplectic cobordism and a Duistermaat-Heckman principle for Hamiltonian loop group actions. As an application, we construct a symplectic cobordism between moduli spaces of flat connections on the three holed sphere and disjoint unions of toric varieties. This cobordism yields formulas for the mixed Pontrjagin numbers of the moduli spaces, equivalent to Witten's formulas in the case of symplectic volumes. Received June 15, 1998     

Some results on the moduli spaces of quiver bundles

This article is concerned with the study of gauge theory, stability and moduli for twisted quiver bundles in algebraic geometry. We review natural vortex equations for twisted quiver bundles and their link with a stability condition. Then we provide a brief overview of their relevance to other geometric problems and explain how quiver bundles can be viewed as sheaves of modules over a sheaf of associative algebras and why this view point is useful, e.g., in their deformation theory. Next we explain the main steps of an algebro-geometric construction of their moduli spaces. Finally, we focus on the special case of holomorphic chains over Riemann surfaces, providing some basic links with quiver representation theory. Combined with the analysis of the homological algebra of quiver sheaves and modules, these links provide a criterion for smoothness of the moduli spaces and tools to study their variation with respect to stability.      

Stability and equivalence of admissible pairs of arbitrary dimension for a compactification of the moduli space of stable vector bundles

Theoretical and Mathematical Physics - Moduli spaces of stable vector bundles and compactifications of these moduli spaces are closely related to Yang–Mills gauge field theory. This paper,...     

Representation theory and ADHM-construction on quaternion symmetric spaces

We determine all irreducible homogeneous bundles with anti-self-dual canonical connections on compact quaternion symmetric spaces. To deform the canonical connections, we give a relation between the representation theory and the theory of monads on the twistor space. The moduli spaces are described via the Bott-Borel-Weil Thereom. The Horrocks bundle is also generalized to higher-dimensional projective spaces.

     

Moduli Stacks of Permutation Classes of Pointed Stable Curves

The notion of m/Γ-pointed stable curves is introduced. It should be viewed as a generalization of the notion of m-pointed stable curves of a given genus, where the labels of the marked points are only determined up to the action of a group of permutations Γ. The classical moduli spaces and moduli stacks are generalized to this wider setting. Finally, an explicit construction of the new moduli stack of m/Γ-pointed stable curves as a quotient stack is given. Received: February 2008     

Eigenvalues and Entropy Moduli of Operators in Interpolation Spaces

The paper approaches in an abstract way the spectral theory of operators in abstract interpolation spaces. We introduce entropy numbers and spectral moduli of operators, and prove a relationship between them and eigenvalues of operators. We also investigate interpolation variants of the moduli, and offer a contribution to the theory of eigenvalues of operators. Specifically, we prove an interpolation version of the celebrated Carl–Triebel eigenvalue inequality. Based on these results we are able to prove interpolation estimates for single eigenvalues as well as for geometric means of absolute values of the first n eigenvalues of operators. In particular, some of these estimates may be regarded as generalizations of the classical spectral radius formula. We give applications of our results to the study of interpolation estimates of entropy numbers as well as of the essential spectral radius of operators in interpolation spaces.     

Yang-Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics

Atiyah and Bott used equivariant Morse theory applied to theYang–Mills functional to calculate the Betti numbers ofmoduli spaces of vector bundles over a Riemann surface, rederivinginductive formulae obtained from an arithmetic approach whichinvolved the Tamagawa number of SL_n. This article attempts tosurvey and extend our understanding of this link between Yang–Millstheory and Tamagawa numbers, and to explain how methods usedover the last three decades to study the singular cohomologyof moduli spaces of bundles on a smooth projective curve over can be adapted to the setting of ¹-homotopy theory to studythe motivic cohomology of these moduli spaces over an algebraicallyclosed field.     

Trace Functionals on Noncommutative Deformations of Moduli Spaces of Flat Connections

We describe an efficient construction of a canonical noncommutative deformation of the algebraic functions on the moduli spaces of flat connections on a Riemann surface. The resulting algebra is a variant of the quantum moduli algebra introduced by Alekseev, Grosse, and Schomerus and Buffenoir and Roche. We construct a natural trace functional on this algebra and show that it is related to the canonical trace in the formal index theory of Fedosov and Nest and Tsygan via Verlinde's formula.     

Moduli Spaces of Maslov Complex Germs

Maslov complex germs (complex vector bundles, satisfying certain additional conditions, over isotropic submanifolds of the phase space) are one of the central objects in the theory of semiclassical quantization. To these bundles one assigns spectral series (quasimodes) of partial differential operators. We describe the moduli spaces of Maslov complex germs over a point and a closed trajectory and find the moduli of complex germs generated by a given symplectic connection over an invariant torus.