Roch Theorem Research Articles

Adjoints and pluricanonical adjoints to an algebraic hypersurface

. We develop the theory of canonical and pluricanonical adjoints, of global canonical and pluricanonical adjoints, and of adjoints and global adjoints to an irreducible, algebraic hypersurface V⊂ℙ n , under certain hypotheses on the singularities of V. We subsequently apply the results of the theory to construct a non-singular threefold of general type X, desingularization of a hypersurface V of degree six in ℙ4, having the birational invariants q 1=q 2=p g =0, P 2=P 3=5. We demonstrate that the bicanonical map ϕ |2KX| is birational and finally, as a consequence of the Riemann–Roch theorem and vanishing theorems, we prove that any non-singular model Y, birationally equivalent to X, has the canonical divisors K Y that do not (simultaneously) satisfy the two properties: (K Y 3)>0 and K Y numerically effective.

Galois Module Structure and the γ-Filtration

We describe a general method for calculating equivariant Euler characteristics. The method exploits the fact that the γ-filtration on the Grothendieck group of vector bundles on a Noetherian quasi-projective scheme has finite length; it allows us to capture torsion information which is usually ignored by equivariant Riemann–Roch theorems. As applications, we study the G-module structure of the coherent cohomology of schemes with a free action by a finite group G and, under certain assumptions, we give an explicit formula for the equivariant Euler characteristic \(\chi (\mathcal{O}_X ) = {\text{H}}^{\text{0}} (X,\mathcal{O}_X ) - {\text{H}}^{\text{1}} (X,\mathcal{O}_X )\) in the Grothendieck group of finitely generated Z[G]-modules, when X is a curve over Z and G has prime order.

Calcul d'un Invariant de Star-Produit Fermé sur une Variété Symplectique

Let M be a symplectic manifold over $ℝ. In [CFS] the authors construct an invariant ϕ in the cyclic cohomology of M for any closed star-product. They compute this invariant in the de Rham complex of M when M=T*V. We generalize this result by computing the image of ϕ in the de Rham complex for any symplectic manifold and any star-product and we show how this invariant is related to the general classification of Kontsevich. The proof uses the Riemann–Roch theorem for periodic cyclic chains of Nest–Tsygan.

Chern classes and Hirzebruch–Riemann–Roch theorem for coherent sheaves on complex-projective orbifolds with isolated singularities

Chern classes and Hirzebruch–Riemann–Roch theorem for coherent sheaves on complex-projective orbifolds with isolated singularities

A Generalized Grothendieck–RIemann–Roch Theorem for Hirzebruch's $χ_y$-Characteristic and $T_y$-Characteristic

The Grothendieck-Riemann-Roch (abbr. GRR) is a relative version of the Hirzebruch-Riemann-Roch (abbr. HRR), %(X, E)=T(X, E). Hirzebruch's %ycharacteristic Xy(X, E) and ^-characteristic Ty(X, E) are a generalization of the Euler-Poincare characterisitc %(X, E) and the Todd characteristic T(X, £*) such that when>? = 0 Xo(X, E) = %(X, £) and T0(X, £) =T(X, £), and they are equal; Xy(X, E} = Ty(X, E), which is called the generalized Hirzebruch-Riemann-Roch (abbr. g-HRR). In this short note we show that we can get a generalized GRR version (abbr. g-GRR) such that when y = 0 our g-GRR specializes to the original GRR and such that the Hirzebruch's g-HRR is induced from our g-GRR by mapping X to a point, just like HRR is induced from GRR by mapping X to a point. For the statements and the proofs of our main theorems see § 2.

Homolohy Group and Generalized Riemann -Roch Theorem

In this paper we show in particular th a t if X is a connected com plex analytic m anifold w ith fundam ental group F =|= 1, th en the com m utator subgroup [F, F ] determ ines com plete’y the vector space A(X) of holom orphic functions on X. As a consequence, sim ilarly and generally every norm al subgroup D of F such th a t F /D is com m utative determ ines com pletely an ideal of A(X). In this paper we recollect and expand on th e results obtained in [1, 2 ] and deduce thereof th e fundam ental Theorem . The paper is nevertheless self-contained.

The Riemann–Roch theorem for analytic embeddings

The Riemann–Roch theorem for analytic embeddings