Let X be a smooth projective variety over C. Let G be the group O(r,C), or Sp(r,C). We find the natural notion of semistable principal G-bundle and construct the moduli space, which we compactify by considering also principal G-sheaves, i.e., pairs (E,φ), where E is a torsion free sheaf on X and φ is a symmetric (if G is orthogonal) or antisymmetric (if G is symplectic) bilinear form on E. If G...

Each infinitesimally faithful representation of a reductive complex connected algebraic group G induces a dominant morphism Φ from the group to its Lie algebra g by orthogonal projection in the endomorphism ring of the representation space. The map Φ identifies the field Q(G) of rational functions on G with an algebraic extension of the field Q(g) of rational functions on g. For the spin repres...

Let S be a Noetherian scheme, φ : X → Y a surjective S-morphism of S-schemes, with X of finite type over S. We discuss what makes Y of finite type. First, we prove that if S is excellent, Y is reduced, and φ is universally open, then Y is of finite type. We apply this to understand Fogarty’s theorem in “Geometric quotients are algebraic schemes, Adv. Math. 48 (1983), 166–171” for the special ca...

For any C-algebra A let A denote the set of all positive elements in A. A state on a unital C-algebra A is a linear functional ω : A → C such that ω(a) ≥ 0 for every a ∈ A and ω(I) = 1 where I is the unit of A. By S(A) we will denote the set of all states on A. For any Hilbert space H we denote by B(H) the set of all bounded linear operators on H . A linear map φ : A → B between C-algebras is c...

Let S be a Noetherian scheme, φ : X → Y a surjective S-morphism of S-schemes, with X of finite type over S. We discuss what makes Y of finite type. First, we prove that if S is excellent, Y is reduced, and φ is universally open, then Y is of finite type. We apply this to understand Fogarty’s theorem in “Geometric quotients are algebraic schemes, Adv. Math. 48 (1983), 166–171” for the special ca...

Assuming that φ : Sym(E) → F is a morphism of vector bundles on a variety X, we compute the class of the locus in X where Ker(φ) contains a quadric of prescribed rank. Our formulas have many applications to moduli theory: (i) we find a simple proof of Borcherds’ result that the Hodge class on the moduli space of polarized K3 surfaces of fixed genus is of Noether-Lefschetz type, (ii) we construc...

Let G = SpecA be a linearly reductive group and let wG ∈ A ∗ be the invariant integral on G. We establish the algebraic harmonic analysis on G and we compute wG when G = Sln, Gln, On, Sp2n by geometric arguments and by means of the Fourier transform. Introduction An affine k-group G = SpecA is linearly semisimple (that is, linearly reductive) if and only if A splits into the form A = k × B as k...

We define and study a lift of the Boardman-Vogt tensor product of operads to bimodules over operads. Introduction Let Op denote the category of symmetric operads in the monoidal category S of simplicial sets. The Boardman-Vogt tensor product [3] −⊗− : Op× Op→ Op, which endows the category Op with a symmetric monoidal structure, codifies interchanging algebraic structures. For all P,Q ∈ Op, a (P...

Themain purpose of this note is to present a characterization of rational singularities in characteristic 0. The essence of the characterization is that it is enough to require less than the usual definition. Theorem 1. Let φ : Y → X be a morphism of varieties over C, and let ρ : X → Rφ∗ Y be the associated natural morphism. Assume that Y has rational singularities and there exists a morphism (...

We introduce the complete lifts of maps between (real and complex) Euclidean spaces and study their properties concerning holomorphicity, harmonicity and horizontal weakly conformality. As applications, we are able to use this concept to characterize holomorphic maps φ : C ⊃ U −→ C (Proposition 2.3) and to construct many new examples of harmonic morphisms (Theorem 3.3). Finally we show that the...