Improvement of Grauert-Riemenschneider's theorem for a normal surface

Jean Giraud

Displaying similar documents to “Improvement of Grauert-Riemenschneider's theorem for a normal surface”

On Verlinde sheaves and strange duality over elliptic Noether-Lefschetz divisors

Alina Marian, Dragos Oprea (2014)

Annales de l’institut Fourier

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We extend results on generic strange duality for $K 3$ surfaces by showing that the proposed isomorphism holds over an entire Noether-Lefschetz divisor in the moduli space of quasipolarized $K 3$ s. We interpret the statement globally as an isomorphism of sheaves over this divisor, and also describe the global construction over the space of polarized $K 3 s$ .

Degree of the fibres of an elliptic fibration

Alexandru Buium (1983)

Annales de l'institut Fourier

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Let $X \to B$ an elliptic fibration with general fibre $F$ . Let $n_{e}, n_{s}, n_{a}, n_{v}$ be the minima of the non-zero intersection numbers $(ℒ, F)$ where $ℒ$ runs successively through the following sets: effective divisors on $X$ , invertible sheaves spanned by global sections, ample divisors and very ample divisors. Let $m$ be the maximum of the multiplicities of the fibres of $X \to B$ . We prove that $n_{e} = n_{s}$ if and only if $n_{e} \geq 2 m$ and that $n_{a} = n_{v}$ if and only if $n_{a} \geq 3 m$ .

Sheaves associated to holomorphic first integrals

Alexis García Zamora (2000)

Annales de l'institut Fourier

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Let $ℱ : L \to T S$ be a foliation on a complex, smooth and irreducible projective surface $S$ , assume $ℱ$ admits a holomorphic first integral $f : S \to ℙ^{1}$ . If $h^{0} (S, 𝒪_{S} (- n 𝒦_{S})) > 0$ for some $n \geq 1$ we prove the inequality: $(2 n - 1) (g - 1) \leq h^{1} (S, ℒ^{' - 1} (- (n - 1) K_{S})) + h^{0} (S, ℒ^{'}) + 1$ . If $S$ is rational we prove that the direct image sheaves of the co-normal sheaf of $ℱ$ under $f$ are locally free; and give some information on the nature of their decomposition as direct sum of invertible sheaves.

Stein open subsets with analytic complements in compact complex spaces

Jing Zhang (2015)

Annales Polonici Mathematici

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Let Y be an open subset of a reduced compact complex space X such that X - Y is the support of an effective divisor D. If X is a surface and D is an effective Weil divisor, we give sufficient conditions so that Y is Stein. If X is of pure dimension d ≥ 1 and X - Y is the support of an effective Cartier divisor D, we show that Y is Stein if Y contains no compact curves, $H^{i} (Y,_{Y}) = 0$ for all i > 0, and for every point x₀ ∈ X-Y there is an n ∈ ℕ such that $Φ_{| n D |}^{- 1} (Φ_{| n D |} (x ₀)) \cap Y$ is empty or has dimension 0, where $Φ_{| n D |}$ ...

A group law on smooth real quartics having at least $3$ real branches

Johan Huisman (2002)

Journal de théorie des nombres de Bordeaux

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Let $C$ be a smooth real quartic curve in $ℙ^{2}$ . Suppose that $C$ has at least $3$ real branches $B_{1}, B_{2}, B_{3}$ . Let $B = B_{1} \times B_{2} \times B_{3}$ and let $O \in B$ . Let $τ_{O}$ be the map from $B$ into the neutral component Jac $(C) {(ℝ)}^{0}$ of the set of real points of the jacobian of $C$ , defined by letting $τ_{O} (P)$ be the divisor class of the divisor $\sum P_{i} - O_{i}$ . Then, $τ_{O}$ is a bijection. We show that this allows an explicit geometric description of the group law on Jac $(C) {(ℝ)}^{0}$ . It generalizes the classical geometric description of the group law on the neutral component of the set of real...

Relative subanalytic sheaves

Teresa Monteiro Fernandes, Luca Prelli (2014)

Fundamenta Mathematicae

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Given a real analytic manifold Y, denote by $Y_{s a}$ the associated subanalytic site. Now consider a product Y = X × S. We construct the endofunctor $ℱ \mapsto ℱ^{S}$ on the category of sheaves on $Y_{s a}$ and study its properties. Roughly speaking, $ℱ^{S}$ is a sheaf on $X_{s a} \times S$ . As an application, one can now define sheaves of functions on Y which are tempered or Whitney in the relative sense, that is, only with respect to X.

The end curve theorem for normal complex surface singularities

Walter D. Neumann, Jonathan Wahl (2010)

Journal of the European Mathematical Society

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We prove the “End Curve Theorem,” which states that a normal surface singularity $(X, o)$ with rational homology sphere link $Σ$ is a splice quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree. An “end curve function” is an analytic function $(X, o) \to (ℂ, 0)$ whose zero set intersects $Σ$ in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf. A “splice quotient singularity” $(X, o)$ is described by giving an explicit set of...

Stratified Whitney jets and tempered ultradistributions on the subanalytic site

N. Honda, G. Morando (2011)

Bulletin de la Société Mathématique de France

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In this paper we introduce the sheaf of stratified Whitney jets of Gevrey order on the subanalytic site relative to a real analytic manifold $X$ . Then, we define stratified ultradistributions of Beurling and Roumieu type on $X$ . In the end, by means of stratified ultradistributions, we define tempered-stratified ultradistributions and we prove two results. First, if $X$ is a real surface, the tempered-stratified ultradistributions define a sheaf on the subanalytic site relative to $X$ . Second,...

Tilting Bundles on Rational Surfaces and Quasi-Hereditary Algebras

Lutz Hille, Markus Perling (2014)

Annales de l’institut Fourier

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Let $X$ be any rational surface. We construct a tilting bundle $T$ on $X$ . Moreover, we can choose $T$ in such way that its endomorphism algebra is quasi-hereditary. In particular, the bounded derived category of coherent sheaves on $X$ is equivalent to the bounded derived category of finitely generated modules over a finite dimensional quasi-hereditary algebra $A$ . The construction starts with a full exceptional sequence of line bundles on $X$ and uses universal extensions. If $X$ is any smooth projective...

Algebraic tori as Nisnevich sheaves with transfers

Bruno Kahn (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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We relate $R$ -equivalence on tori with Voevodsky’s theory of homotopy invariant Nisnevich sheaves with transfers and effective motivic complexes.

Zariski surfaces

Piotr Blass

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CONTENTSAcknowledgements...................................................................................................5Introduction..............................................................................................................6Notations..................................................................................................................8Chapter I. Zariski surfaces: definition and general properties................................10Chapter II. The theory...

Algebraic bounds on analytic multiplier ideals

Brian Lehmann (2014)

Annales de l’institut Fourier

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Given a pseudo-effective divisor $L$ we construct the diminished ideal $𝒥_{σ} (L)$ , a “continuous” extension of the asymptotic multiplier ideal for big divisors to the pseudo-effective boundary. Our main theorem shows that for most pseudo-effective divisors $L$ the multiplier ideal $𝒥 (h_{\min})$ of the metric of minimal singularities on $𝒪_{X} (L)$ is contained in $𝒥_{σ} (L)$ . We also characterize abundant divisors using the diminished ideal, indicating that the geometric and analytic information should coincide.

Syzygies and logarithmic vector fields along plane curves

Alexandru Dimca, Edoardo Sernesi (2014)

Journal de l’École polytechnique — Mathématiques

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We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a plane curve $C$ and the stability of the sheaf of logarithmic vector fields along $C$ , the freeness of the divisor $C$ and the Torelli properties of $C$ (in the sense of Dolgachev-Kapranov). We show in particular that curves with a small number of nodes and cusps are Torelli in this sense.