Line bundles with partially vanishing cohomology

Burt Totaro

Displaying similar documents to “Line bundles with partially vanishing cohomology”

A classification theorem on Fano bundles

Roberto Muñoz, Luis E. Solá Conde, Gianluca Occhetta (2014)

Annales de l’institut Fourier

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In this paper we classify rank two Fano bundles $ℰ$ on Fano manifolds satisfying $H^{2} (X, ℤ) ≅ H^{4} (X, ℤ) ≅ ℤ$ . The classification is obtained via the computation of the nef and pseudoeffective cones of the projectivization $ℙ (ℰ)$ , that allows us to obtain the cohomological invariants of $X$ and $ℰ$ . As a by-product we discuss Fano bundles associated to congruences of lines, showing that their varieties of minimal rational tangents may have several linear components.

Lifting to the r-frame bundle by means of connections

J. Kurek, W. M. Mikulski (2010)

Annales Polonici Mathematici

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Let m and r be natural numbers and let $P^{r} : ℳ f_{m} \to ℱ ℳ$ be the rth order frame bundle functor. Let $F : ℳ f_{m} \to ℱ ℳ$ and $G : ℳ f_{k} \to ℱ ℳ$ be natural bundles, where $k = d i m (P^{r} ℝ^{m})$ . We describe all $ℳ f_{m}$ -natural operators A transforming sections σ of $F M \to M$ and classical linear connections ∇ on M into sections A(σ,∇) of $G (P^{r} M) \to P^{r} M$ . We apply this general classification result to many important natural bundles F and G and obtain many particular classifications.

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...

Geometric stability of the cotangent bundle and the universal cover of a projective manifold

Frédéric Campana, Thomas Peternell (2011)

Bulletin de la Société Mathématique de France

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We first prove a strengthening of Miyaoka’s generic semi-positivity theorem: the quotients of the tensor powers of the cotangent bundle of a non-uniruled complex projective manifold $X$ have a pseudo-effective (instead of generically nef) determinant. A first consequence is that $X$ is of general type if its cotangent bundle contains a subsheaf with ‘big’ determinant. Among other applications, we deduce that if the universal cover of $X$ is not covered by compact positive-dimensional analytic...

Semistability of Frobenius direct images over curves

Vikram B. Mehta, Christian Pauly (2007)

Bulletin de la Société Mathématique de France

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Let $X$ be a smooth projective curve of genus $g \geq 2$ defined over an algebraically closed field $k$ of characteristic $p > 0$ . Given a semistable vector bundle $E$ over $X$ , we show that its direct image $F_{*} E$ under the Frobenius map $F$ of $X$ is again semistable. We deduce a numerical characterization of the stable rank- $p$ vector bundles $F_{*} L$ , where $L$ is a line bundle over $X$ .

The natural transformations between r-tangent and r-cotangent bundles over Riemannian manifolds

Jan Kurek, Włodzimierz Mikulski (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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If $(M, g)$ is a Riemannian manifold, we have the well-known base preserving vector bundle isomorphism $T M \tilde{=} T^{*} M$ given by $v \to g (v, -)$ between the tangent $T M$ and the cotangent $T^{*} M$ bundles of $M$ . In the present note, we generalize this isomorphism to the one $T^{(r)} M \tilde{=} T^{r *} M$ between the $r$ -th order vector tangent $T^{(r)} M = {(J^{r} {(M, R)}_{0})}^{*}$ and the $r$ -th order cotangent $T^{r *} M = J^{r} {(M, R)}_{0}$ bundles of $M$ . Next, we describe all base preserving vector bundle maps $C_{M} (g) : T^{(r)} M \to T^{r *} M$ depending on a Riemannian metric $g$ in terms of natural (in $g$ ) tensor fields on $M$ .

Complex structures on product of circle bundles over complex manifolds

Parameswaran Sankaran, Ajay Singh Thakur (2013)

Annales de l’institut Fourier

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Let ${\bar{L}}_{i} \to X_{i}$ be a holomorphic line bundle over a compact complex manifold for $i = 1, 2$ . Let $S_{i}$ denote the associated principal circle-bundle with respect to some hermitian inner product on ${\bar{L}}_{i}$ . We construct complex structures on $S = S_{1} \times S_{2}$ which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that ${\bar{L}}_{i}$ are equivariant ${(ℂ^{*})}^{n_{i}}$ -bundles satisfying some additional conditions....

$𝒟$ -bundles and integrable hierarchies

David Ben-Zvi, Thomas Nevins (2011)

Journal of the European Mathematical Society

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We study the geometry of $𝒟$ -bundles—locally projective $𝒟$ -modules—on algebraic curves, and apply them to the study of integrable hierarchies, specifically the multicomponent Kadomtsev–Petviashvili (KP) and spin Calogero–Moser (CM) hierarchies. We show that KP hierarchies have a geometric description as flows on moduli spaces of $𝒟$ -bundles; in particular, we prove that the local structure of $𝒟$ -bundles is captured by the full Sato Grassmannian. The rational, trigonometric, and elliptic solutions...

On coverings of simple abelian varieties

Olivier Debarre (2006)

Bulletin de la Société Mathématique de France

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To any finite covering $f : Y \to X$ of degree $d$ between smooth complex projective manifolds, one associates a vector bundle $E_{f}$ of rank $d - 1$ on $X$ whose total space contains $Y$ . It is known that $E_{f}$ is ample when $X$ is a projective space ([Lazarsfeld 1980]), a Grassmannian ([Manivel 1997]), or a Lagrangian Grassmannian ([Kim Maniel 1999]). We show an analogous result when $X$ is a simple abelian variety and $f$ does not factor through any nontrivial isogeny $X^{'} \to X$ . This result is obtained by showing that $E_{f}$ is $M$ -regular...

Holomorphic line bundles and divisors on a domain of a Stein manifold

Makoto Abe (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let $D$ be an open set of a Stein manifold $X$ of dimension $n$ such that $H^{k} (D, 𝒪) = 0$ for $2 \leq k \leq n - 1$ . We prove that $D$ is Stein if and only if every topologically trivial holomorphic line bundle $L$ on $D$ is associated to some Cartier divisor $𝔡$ on $D$ .

Pseudo-real principal Higgs bundles on compact Kähler manifolds

Indranil Biswas, Oscar García-Prada, Jacques Hurtubise (2014)

Annales de l’institut Fourier

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Let $X$ be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let $G$ be a connected complex reductive affine algebraic group equipped with a real form $σ_{G}$ . We define pseudo-real principal $G$ -bundles on $X$ . These are generalizations of real algebraic principal $G$ -bundles over a real algebraic variety. Next we define stable, semistable and polystable pseudo-real principal $G$ -bundles. Their relationships with the usual...

Equivalence bundles over a finite group and strong Morita equivalence for unital inclusions of unital $C^{*}$ -algebras

Kazunori Kodaka (2022)

Mathematica Bohemica

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Let $𝒜 = {A_{t}}_{t \in G}$ and $ℬ = {B_{t}}_{t \in G}$ be $C^{*}$ -algebraic bundles over a finite group $G$ . Let $C = ⨁_{t \in G} A_{t}$ and $D = ⨁_{t \in G} B_{t}$ . Also, let $A = A_{e}$ and $B = B_{e}$ , where $e$ is the unit element in $G$ . We suppose that $C$ and $D$ are unital and $A$ and $B$ have the unit elements in $C$ and $D$ , respectively. In this paper, we show that if there is an equivalence $𝒜 - ℬ$ -bundle over $G$ with some properties, then the unital inclusions of unital $C^{*}$ -algebras $A \subset C$ and $B \subset D$ induced by $𝒜$ and $ℬ$ are strongly Morita equivalent. Also, we suppose that $𝒜$ and $ℬ$ are saturated and that $A^{'} \cap C = 𝐂 1$ . We show that...

Steinness of bundles with fiber a Reinhardt bounded domain

Karl Oeljeklaus, Dan Zaffran (2006)

Bulletin de la Société Mathématique de France

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Let $E$ denote a holomorphic bundle with fiber $D$ and with basis $B$ . Both $D$ and $B$ are assumed to be Stein. For $D$ a Reinhardt bounded domain of dimension $d = 2$ or $3$ , we give a necessary and sufficient condition on $D$ for the existence of a non-Stein such $E$ (Theorem $1$ ); for $d = 2$ , we give necessary and sufficient criteria for $E$ to be Stein (Theorem $2$ ). For $D$ a Reinhardt bounded domain of any dimension not intersecting any coordinate hyperplane, we give a sufficient criterion for $E$ to be Stein (Theorem...

A note on the cohomology ring of the oriented Grassmann manifolds ${\tilde{G}}_{n, 4}$

Tomáš Rusin (2019)

Archivum Mathematicum

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We use known results on the characteristic rank of the canonical $4$ –plane bundle over the oriented Grassmann manifold ${\tilde{G}}_{n, 4}$ to compute the generators of the $ℤ_{2}$ –cohomology groups $H^{j} ({\tilde{G}}_{n, 4})$ for $n = 8, 9, 10, 11$ . Drawing from the similarities of these examples with the general description of the cohomology rings of ${\tilde{G}}_{n, 3}$ we conjecture some predictions.

On lifting of connections to Weil bundles

Jan Kurek, Włodzimierz M. Mikulski (2012)

Annales Polonici Mathematici

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We prove that the problem of finding all $ℳ f_{m}$ -natural operators $B : Q ⟿ Q T^{A}$ lifting classical linear connections ∇ on m-manifolds M to classical linear connections $B_{M} (\nabla)$ on the Weil bundle $T^{A} M$ corresponding to a p-dimensional (over ℝ) Weil algebra A is equivalent to the one of finding all $ℳ f_{m}$ -natural operators $C : Q ⟿ (T ¹_{p - 1}, T * \otimes T * \otimes T)$ transforming classical linear connections ∇ on m-manifolds M into base-preserving fibred maps $C_{M} (\nabla) : T ¹_{p - 1} M = ⨁_{M}^{p - 1} T M \to T * M \otimes T * M \otimes T M$ .

Monodromy of a family of hypersurfaces

Vincenzo Di Gennaro, Davide Franco (2009)

Annales scientifiques de l'École Normale Supérieure

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Let $Y$ be an $(m + 1)$ -dimensional irreducible smooth complex projective variety embedded in a projective space. Let $Z$ be a closed subscheme of $Y$ , and $δ$ be a positive integer such that $ℐ_{Z, Y} (δ)$ is generated by global sections. Fix an integer $d \geq δ + 1$ , and assume the general divisor $X \in | H^{0} (Y, ℐ_{Z, Y} (d)) |$ is smooth. Denote by $H^{m} {(X; ℚ)}_{⊥ Z}^{van}$ the quotient of $H^{m} (X; ℚ)$ by the cohomology of $Y$ and also by the cycle classes of the irreducible components of dimension $m$ of $Z$ . In the present paper we prove that the monodromy representation on $H^{m} {(X; ℚ)}_{⊥ Z}^{van}$ for the family...

A finiteness theorem for holomorphic Banach bundles

Jürgen Leiterer (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let $E$ be a holomorphic Banach bundle over a compact complex manifold, which can be defined by a cocycle of holomorphic transition functions with values of the form $id + K$ where $K$ is compact. Assume that the characteristic fiber of $E$ has the compact approximation property. Let $n$ be the complex dimension of $X$ and $0 \leq q \leq n$ . Then: If $V \to X$ is a holomorphic vector bundle (of finite rank) with $H^{q} (X, V) = 0$ , then $dim H^{q} (X, V \otimes E) < \infty$ . In particular, if $dim H^{q} (X, 𝒪) = 0$ , then $dim H^{q} (X, E) < \infty$ .

On lifts of projectable-projectable classical linear connections to the cotangent bundle

Anna Bednarska (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We describe all $ℱ^{2} ℳ_{m_{1}, m_{2}, n_{1}, n_{2}}$ -natural operators $D : Q_{p r o j - p r o j}^{τ} ⇝ Q T^{*}$ transforming projectable-projectable classical torsion-free linear connections $\nabla$ on fibred-fibred manifolds $Y$ into classical linear connections $D (\nabla)$ on cotangent bundles $T^{*} Y$ of $Y$ . We show that this problem can be reduced to finding $ℱ^{2} ℳ_{m_{1}, m_{2}, n_{1}, n_{2}}$ -natural operators $D : Q_{p r o j - p r o j}^{τ} ⇝ (T^{*}, \otimes^{p} T^{*} \otimes \otimes^{q} T)$ for $p = 2$ , $q = 1$ and $p = 3$ , $q = 0$ .

Grauert's line bundle convexity, reduction and Riemann domains

Viorel Vâjâitu (2016)

Czechoslovak Mathematical Journal

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We consider a convexity notion for complex spaces $X$ with respect to a holomorphic line bundle $L$ over $X$ . This definition has been introduced by Grauert and, when $L$ is analytically trivial, we recover the standard holomorphic convexity. In this circle of ideas, we prove the counterpart of the classical Remmert’s reduction result for holomorphically convex spaces. In the same vein, we show that if $H^{0} (X, L)$ separates each point of $X$ , then $X$ can be realized as a Riemann domain over the complex projective...