Line Bundle Research Articles

On the moment measure conjecture

AbstractThe moment measure conjecture of Białynicki-Birula and Sommese gives a combinatorial characterization of all open substacks of a global quotient stack for a torus action on a normal projective variety that admit a proper good moduli space, in other words it characterizes the invariant open subvarieties that admit a proper quotient. In this article we prove the conjecture for actions on smooth varieties. This gives an instance of stability conditions defined by cohomological invariants that are not given by Chern classes of line bundles.

Equivariant orientation of vector bundles over disconnected base spaces

AbstractIn this paper, we view the equivariant orientation theory of equivariant vector bundles from the lenses of equivariant Picard spectra. This viewpoint allows us to identify, for a finite group , a precise condition under which an ‐orientation of a ‐equivariant vector bundle is encoded by a Thom class. Consequently, we are able to construct a generalization of the first Stiefel–Whitney class of a “homogeneous” ‐equivariant bundle with respect to an ‐ring spectrum . As an application, we show that the 2‐fold direct sum of any homogeneous bundle is ‐orientable, where is the Burnside Mackey functor. We notice that ‐orientability is equivalent to ‐orientability when the order of is odd. When the order of is even, we show that a ‐equivariant analog of the tautological line bundle over is ‐orientable but not ‐orientable.

Convex Fujita numbers are not determined by the fundamental group

Abstract We study effective global generation of adjoint line bundles on smooth projective varieties. To measure the effectivity we introduce the concept of the convex Fujita number of a smooth projective variety and compute its value for a class of varieties with prescribed dimension d ≥ 2 and an arbitrary projective group as fundamental group.

On a B-field transform of generalized complex structures over complex tori

Let (Xn,Xˇn) be a mirror pair of an n-dimensional complex torus Xn and its mirror partner Xˇn. Then, by SYZ transform, we can construct a holomorphic line bundle with an integrable connection from each pair of a Lagrangian section of Xˇn→Rn/Zn and a unitary local system along it, and those holomorphic line bundles with integrable connections form a dg-category DGXn. In this paper, we focus on a certain B-field transform of the generalized complex structure induced from the complex structure on Xn, and interpret it as the deformation XGn of Xn by a flat gerbe G. Moreover, we construct the deformation of DGXn associated to the deformation from Xn to XGn, and also discuss the homological mirror symmetry between XGn and its mirror partner on the object level.

Approximation of semistable bundles on smooth algebraic varieties

We prove some strong results on approximation of strongly semistable bundles with vanishing numerical Chern classes by filtrations, whose quotients are line bundles of similar slope. This generalizes some earlier results of Parameswaran–Subramanian in the curve case and Koley–Parameswaran in the surface case and it confirms the conjecture posed by Koley and Parameswaran.

Non-Density of the Exceptional Components of the Noether–Lefschetz Locus

Abstract We study when the Picard group of smooth surfaces of degree $d\geq 5$ in $\mathbb{P}^{3}$ acquires extra classes. In particular we show that the so-called exceptional components of the Noether–Lefschetz locus are not Zariski dense. This answers a 1991 question of C. Voisin. We also obtain similar results for the Noether–Lefschetz locus for suitable $(Y,L)$, where $Y$ is a smooth projective three-fold and $L$ a very ample line bundle. Both results are applications of the Zilber–Pink viewpoint recently developed by the authors for arbitrary (polarized, integral) variations of Hodge structures.

Limit of geometric quantizations on Kähler manifolds with T‐symmetry

AbstractA compact Kähler manifold with ‐symmetry admits a natural mixed polarization whose real directions come from the ‐action. In Leung and Wang [Adv. Math. 450 (2024), 109756], we constructed a one‐parameter family of Kähler structures ’s with the same underlying Kähler form and , such that (i) there is a ‐equivariant biholomorphism between and and (ii) Kähler polarizations ’s corresponding to ’s converge to as goes to infinity. In this paper, we study the quantum analog of above results. Assume that is a prequantum line bundle on . Let and be quantum spaces defined using polarizations and , respectively. In particular, . They are both representations of . We show that (i) there is a ‐equivariant isomorphism between and and (ii) for regular ‐weight , corresponding ‐weight spaces ’s converge to as goes to infinity.

The Motivic Anabelian Geometry of Local Heights on Abelian Varieties

We study the problem of describing local components of Néron–Tate height functions on abelian varieties over characteristic 0 0 local fields as functions on spaces of torsors under various realisations of the 2 2 -step unipotent motivic fundamental group of the G m \mathbb G_m -torsor corresponding to the defining line bundle. To this end, we present three main theorems giving such a description in terms of the Q ℓ \mathbb Q_\ell - and Q p \mathbb Q_p -pro-unipotent étale realisations when the base field is p p -adic, and in terms of the R \mathbb R -pro-unipotent Betti–de Rham realization when the base field is archimedean. In the course of proving the p p -adic instance of these theorems, we develop a new technique for studying local nonabelian Bloch–Kato Selmer sets, working with certain explicit cosimplicial group models for these sets and using methods from homotopical algebra. Among other uses, these models enable us to construct a pro-unipotent generalization of the Bloch–Kato exponential sequence under minimal assumptions.

GCD inequalities inspired by Vojta’s conjecture

AbstractWe prove several GCD inequalities inspired by Vojta’s deep conjecture in Diophantine geometry. These inequalities work with all algebraic numbers, so they can be thought of as attempts to extend GCD inequalities for S-units obtained by Corvaja and Zannier. Some of the inequalities we derive are weakened versions of inequalities obtained by Silverman under the assumption of Vojta’s conjecture. The upper bounds for the GCD’s in these inequalities all involve some contributions of heights and some contributions coming from the local heights outside S. The main ingredient of the proofs is the recent Diophantine approximation result of Ru and Vojta which is based on Schmidt subspace theorem and which involves a birational invariant coming from the asymptotic growth of the number of global sections of certain line bundles.

Singularities of secant varieties from a Hodge theoretic perspective

AbstractWe study the singularities of secant varieties of smooth projective varieties using methods from birational geometry when the embedding line bundle is sufficiently positive. More precisely, we study the Du Bois complex of secant varieties and its relationship with the sheaves of differential forms. Through this analysis, we give a necessary and sufficient condition for these varieties to have ‐Du Bois singularities (in a sense that was proposed in Shen, Venkatesh, and Vo [On k‐Du Bois and k‐rational singularities, arXiv e‐prints (June 2023), arXiv:2306.03977]). In addition, we show that the singularities of these varieties are never higher rational, by giving a classification of the cases when they are pre‐1‐rational. From these results, we deduce several consequences, including a Kodaira–Akizuki–Nakano type vanishing result for the reflexive differential forms of the secant varieties.

Umbilical Line Securement Bundle to Reduce Line Loss in the Neonate.

Umbilical line migration not only increases the risks of complications but also results in malposition and, ultimately, loss of the umbilical line. To evaluate the use of an umbilical line securement bundle to reduce unintended line discontinuation after line adjustment in the neonate at a single 40-bed Level IV neonatal intensive care unit. A pre-post design of 75 neonates, preimplementation (n=50) and postimplementation (n=25), was analyzed using data collection from the electronic health record. There was a 37.5% absolute reduction in removal of the umbilical line due to malposition after line adjustment utilizing the umbilical line bundle, standardizing the adjustment order, nursing process, and follow-up x-ray evaluation. This absolute reduction has clinical significance although not statistically significant. Provider compliance rates with line adjustment order bundle were 75%, decreasing with additional adjustments (50%). Nursing staff reported comfort with umbilical line management, ranging from 63% to 87% on different tasks. The use of umbilical line bundles reduces rates of line discontinuation due to malposition. The adoption of umbilical line bundles in neonatal intensive care unit practice may help to prevent unintended line discontinuation. There is a need for continued research regarding the use of secondary securement devices for decreased rate of malposition and the timing and methods for surveillance of umbilical line position.

A Study to Assess the Effectiveness of Informative Booklet Regarding Knowledge, Attitude and Practice Among Intensive Care Unit (ICU) Staff Nurses Regarding Prevention of Central Line Associated Blood Stream Infection (CLABSI)

Background: Nurses in Intensive Care Units (ICU) have a vital role in reducing Central line associated blood stream infections (CLABSI) if they are well educated to use strategies to decrease Central line associated blood stream infections (CLABSI) rate which are based on the center for disease control and prevention guidelines (CDC) to improve patient’s outcomes. Health-care-associated infections are a huge concern in the intensive care units (ICUs) across the globe. Nearly 90% of catheter-related bloodstream infections are caused by central venous catheters (CVCs). Central venous catheters (CVC) are commonly inserted in critically ill patients and play an important role for hemodynamic monitoring and delivery of crucial treatments. Central line-associated bloodstream infection (CLABSI) is a laboratory confirmed bloodstream infection occurring in a patient with a CVC in place for >48 hours that is not related to an infection at any other site. Preventing nosocomial infections pose a great challenge in ICU, because of the multi-drug resistant organisms treated with broad-spectrum antibiotics. Clinical practice guidelines from various organizations are available for prevention of vascular catheter related infections. The Institute for Healthcare Improvement has introduced a “central line bundle” that includes the five best evidence-based practices to prevent CLABSI. This bundle includes hand hygiene, maximal barrier precautions during catheter insertion, skin antisepsis with chlorhexidine, optimal selection of the catheter site, with avoidance of the femoral vein and a daily assessment of line necessity, including the removal of an unnecessary line. Of these components, “hand hygiene” (HH) is the most convenient and cost effective to follow. “Hand Hygiene” HH prevents the transmission of pathogens and is a universal strategy to prevent all nosocomial infections. The World Health Organization has proposed a multimodal strategy; which includes five elements called the “Five Moments for Hand Hygiene”, as an evidence-based approach to prevent hospital-acquired infections. The five moments include, before touching the patient, before performing any a clean/aseptic procedure, after exposure/risk of body fluid, after touching the patient and after touching the patient surroundings. Therefore, the researcher was curious to know the effectiveness of informative booklet regarding knowledge, attitude and practice of intensive care unit staff nurses regarding prevention of central line associated blood stream infection (CLABSI).

The Yang–Mills–Higgs functional on complex line bundles: Asymptotics for critical points

Abstract We consider a gauge-invariant Ginzburg–Landau functional (also known as Abelian Yang–Mills–Higgs model), on Hermitian line bundles over closed Riemannian manifolds of dimension n ≥ 3 {n\geq 3} . Assuming a logarithmic energy bound in the coupling parameter, we study the asymptotic behaviour of critical points in the London limit. After a convenient choice of the gauge, we show compactness of finite-energy critical points in Sobolev norms. Moreover, thanks to a suitable monotonicity formula, we prove that the energy densities of critical points, rescaled by the logarithm of the coupling parameter, converge to the weight measure of a stationary, rectifiable varifold of codimension 2.

Assessment of the Level of Awareness and Degree of Implementation of Central Line Bundles for Prevention of Central Line-associated Blood Stream Infection: A Questionnaire-based Observational Study.

The objective of this study was to assess the extent of knowledge and application of central line bundles in the intensive care unit (ICU) of a tertiary care hospital for the purpose of avoiding central line-associated bloodstream infections (CLABSI). This assessment was conducted through the use of a questionnaire. A cross-sectional study was conducted in the ICU, involving doctors and nurses. The study was observational in nature. The study employed a methodical validated questionnaire to evaluate the level of knowledge, attitude, and practice of central line bundles for the prevention of central line-associated bloodstream infections (CLABSI). The questionnaire was designed using preexisting awareness surveillance systems, infection control measures, and patient care practices that were specifically relevant to CLABSIs in the ICU. The data were analyzed utilizing SPSS. The research involved a total of 93 healthcare professionals, consisting of 67 physicians and 26 nurses. The mean knowledge score among participants was 82%, with higher scores reported in individuals who had training in central line bundles. Healthcare professionals exhibited robust compliance with hand cleanliness, antiseptic skin preparation prior to insertion, aseptic draping of the patient, utilization of utmost sterile barriers, verification of central venous catheter (CVC) tip placement using chest X-ray or fluoroscopy, and preservation of a sterile environment. The study emphasized the significance of training in enhancing understanding and adherence to central line bundling protocols in ICUs. Participants exhibited a high level of knowledge and commitment to recommended practices, indicating that this training can have a favorable effect on CLABSI rates. Singh S, Sharma A, Dhawan M, Sharma SP. Assessment of the Level of Awareness and Degree of Implementation of Central Line bundles for Prevention of Central Line-associated Blood Stream Infection: A Questionnaire-based Observational Study. Indian J Crit Care Med 2024;28(9):847-853.

On equidistribution theorem for multi-sequences of holomorphic line bundles

On equidistribution theorem for multi-sequences of holomorphic line bundles

Generating Functions for Line Bundle Cohomology Dimensions on Complex Projective Varieties

This paper explores the possibility of constructing multivariate generating functions for all cohomology dimensions of all holomorphic line bundles on certain complex projective varieties of Fano, Calabi-Yau and general type in various dimensions and Picard numbers. Most of the results are conjectural and rely on explicit cohomology computations. We first propose a generating function for the Euler characteristic of all holomorphic line bundles on complete intersections in products of projective spaces and toric varieties. This generating function is constructed by expanding the Hilbert-Poincaré series associated with the coordinate ring of the variety around all possible combinations of zero and infinity and then summing up the resulting contributions with alternating signs. Similar generating functions are proposed for the individual cohomology dimensions of all holomorphic line bundles on certain complete intersections, including examples of Mori and non-Mori dream spaces. Surprisingly, the examples studied indicate that a single generating function encodes both the zeroth and all higher cohomologies upon expansion around different combinations of zero and infinity, raising the question whether such generating functions determine the variety uniquely.

Crepant Resolutions of Stratified Varieties via Gluing

Abstract Let $X$ be a variety with a stratification ${\mathcal{S}}$ into smooth locally closed subvarieties such that $X$ is locally a product along each stratum (e.g., a symplectic singularity). We prove that assigning to each open subset $U \subset X$ the set of isomorphism classes of locally projective crepant resolutions of $U$ defines an ${\mathcal{S}}$-constructible sheaf of sets. Thus, for each stratum $S$ and basepoint $s \in S$, the fundamental group acts on the set of germs of projective crepant resolutions at $s$, leaving invariant the germs extending to the entire stratum. Global locally projective crepant resolutions correspond to compatible such choices for all strata. For example, if the local projective crepant resolutions are unique, they automatically glue uniquely. We give criteria for a locally projective crepant resolution $\rho : \tilde X \to X$ to be globally projective. We show that the sheafification of the presheaf $U \mapsto \operatorname{Pic}(\rho ^{-1}(U)/U)$ of relative Picard classes is also constructible. The resolution is globally projective only if there exist local relatively ample bundles whose classes glue to a global section of this sheaf. The obstruction to lifting this section to a global ample line bundle is encoded by a gerbe on the singularity $X$. We show the gerbes are automatically trivial if $X$ is a symplectic quotient singularity. Our main results hold in the more general setting of partial crepant resolutions, that need not have smooth source. We apply the theory to symmetric powers and Hilbert schemes of surfaces with du Val singularities, finite quotients of tori, multiplicative, and Nakajima quiver varieties, as well as to canonical three-fold singularities.

Ng\^o support theorem and polarizability of quasi-projective commutative group schemes

We prove that any commutative group scheme over an arbitrary base scheme of finite type over a field with connected fibers and admitting a relatively ample line bundle is polarizable in the sense of Ng\^o. This extends the applicability of Ng\^o's support theorem to new cases, for example to Lagrangian fibrations with integral fibers and has consequences to the construction of algebraic classes.

Mutations of numerically exceptional collections on surfaces

A conjecture of Bondal–Polishchuk states that, in particular for the bounded derived category of coherent sheaves on a smooth projective variety, the action of the braid group on full exceptional collections is transitive up to shifts. We show that the braid group acts transitively on the set of maximal numerically exceptional collections on rational surfaces up to isometries of the Picard lattice and twists with line bundles. Considering the blow-up of the projective plane in up to 9 points in very general position, these results lift to the derived category. More precisely, we prove that, under these assumptions, a maximal numerically exceptional collection consisting of line bundles is a full exceptional collection and any two of them are related by a sequence of mutations and shifts. The former extends a result of Elagin–Lunts and the latter a result of Kuleshov–Orlov, both concerning del Pezzo surfaces. In contrast, we show in concomitant work (Krah in Invent Math 235(3):1009–1018, 2024) that the blow-up of the projective plane in 10 points in general position admits a non-full exceptional collection of maximal length consisting of line bundles.

On the Geometry of Quantum Spheres and Hyperboloids

We study two classes of quantum spheres and hyperboloids, one class consisting of homogeneous spaces, which are ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}-quantum spaces for the quantum orthogonal group O(SOq(3))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(SO_q(3))$$\\end{document}. We construct line bundles over the quantum homogeneous space associated with the quantum subgroup SO(2) of SOq(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$SO_q(3)$$\\end{document}. The line bundles are associated to the quantum principal bundle via representations of SO(2) and are described dually by finitely-generated projective modules En\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {E}_n$$\\end{document} of rank 1 and of degree computed to be an even integer -2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-2n$$\\end{document}. The corresponding idempotents, that represent classes in the K-theory of the base space, are explicitly worked out and are paired with two suitable Fredhom modules that compute the rank and the degree of the bundles. For q real, we show how to diagonalise the action (on the base space algebra) of the Casimir operator of the Hopf algebra Uq1/2(sl2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {U}_{q^{1/2}}(sl_2)}$$\\end{document} which is dual to O(SOq(3))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(SO_q(3))$$\\end{document}.