Injectivity Theorem Research Articles

Vanishing theorems for Mather–Jacobian multiplier ideals on a Gorenstein projective variety

Abstract In this paper, we establish several results related to vanishing theorems for Mather–Jacobian multiplier ideals on a Gorenstein projective variety, including an injectivity theorem, a Nadel-type vanishing theorem, a Griffiths-type vanishing theorem for vector bundles, and a vanishing theorem for the asymptotic Mather–Jacobian multiplier ideals.

Quasianalyticity, uncertainty, and integral transforms on higher grassmannians

We investigate the support of a distribution f on the real grassmannian Grk(Rn) whose spectrum, namely its nontrivial O(n)-components, is restricted to a subset Λ of all O(n)-types. We prove that unless Λ is co-sparse, f cannot be supported at a point. We utilize this uncertainty principle to prove that if 2≤k≤n−2, then the cosine transform of a distribution on the grassmannian cannot be supported inside any single open Schubert cell Σk. The same holds for certain more general α-cosine transforms and for the Radon transform between grassmannians, and more generally for various GLn(R)-modules. These results are then applied to convex geometry and geometric tomography, where sharper versions of the Aleksandrov projection theorem, Funk section theorem, and Klain's and Schneider's injectivity theorems for convex valuations are obtained.

Some Kollár–Enoki Type Injectivity Theorems on Compact Kähler Manifolds

Some Kollár–Enoki Type Injectivity Theorems on Compact Kähler Manifolds

On an injectivity theorem for log-canonical pairs with analytic adjoint ideal sheaves

As an application of the residue functions corresponding to the lc-measures developed by the authors, the proof of the injectivity theorem on compact Kähler manifolds for plt pairs by Matsumura is improved in this article to allow multiplier ideal sheaves of plurisubharmonic functions with neat analytic singularities in the coefficients of the relevant cohomology groups. With the use of a refined version of analytic adjoint ideal sheaves, a plan towards a solution to the generalised version of Fujino’s conjecture (i.e. an injectivity theorem on compact Kähler manifolds for lc pairs with multiplier ideal sheaves) is laid down and, in addition to the result for plt pairs, a proof for lc pairs in dimension 2 2 , which is also an improvement of Matsumura’s result, is given.

Basic kirwan injectivity and its applications

Abstract Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem and use it to study Hamiltonian torus actions on transversely Kähler foliations. Among other things, we prove a foliated analogue of the Carrell–Liberman theorem. As an application, this confirms a conjecture raised by Battaglia–Zaffran on the basic Hodge numbers of symplectic toric quasifolds. Our methods also allow us to present a symplectic approach to the calculation of the basic Betti numbers of symplectic toric quasifolds.

Injectivity theorems with multiplier ideal sheaves for higher direct images under Kähler morphisms

The purpose of this paper is to establish injectivity theorems for higher direct image sheaves of canonical bundles twisted by pseudo-effective line bundles and multiplier ideal sheaves. As applications, we generalize Koll'ar's torsion freeness and Grauert-Riemenschneider's vanishing theorem. Moreover, we obtain a relative vanishing theorem of Kawamata-Viehweg-Nadel type and an extension theorem for holomorphic sections from fibers of morphisms to the ambient space. Our approach is based on transcendental methods and works for Kahler morphisms and singular hermitian metrics with non-algebraic singularities.

On Huber-type theorems in general dimensions

In this paper we present some extensions of the celebrated finite point conformal compactification theorem of Huber [20] for complete open surfaces to general dimensions based on the n-Laplace equation in conformal geometry. We are able to conclude a domain in the round sphere has to be the sphere deleted finitely many points if it can be endowed with a complete conformal metric with the negative part of the smallest Ricci curvature satisfying some integrable conditions. Our proof is based on the improvements of the Arsove-Huber-type theorem on n-superharmonic functions in our earlier work [26]. Moreover, using p-parabolicity, we push the injectivity theorem of Schoen-Yau to allow some negative curvature and therefore establish the finite point conformal compactification theorem for manifolds which have a conformal immersion into the round sphere. As a side product we establish the injectivity of conformal immersions from n-parabolicity alone, which is interesting by itself in conformal geometry.

Injectivity theorem for pseudo-effective line bundles and its applications

We formulate and establish a generalization of Kollár’s injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár’s torsion-freeness, Kollár’s vanishing theorem, and a generic vanishing theorem for pseudo-effective line bundles. Our approach is not Hodge theoretic but analytic, which enables us to treat singular Hermitian metrics with nonalgebraic singularities. For the proof of the main injectivity theorem, we use L 2 L^{2} -harmonic forms on noncompact Kähler manifolds. For applications, we prove a Bertini-type theorem on the restriction of multiplier ideal sheaves to general members of free linear systems.

Extension of cohomology classes and holomorphic sections defined on subvarieties

In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answer to a question posed by Cao-Demailly-Matsumura and unifies a few well-known injectivity theorems. The second result generalizes and optimizes a general L 2 L^2 extension theorem obtained by Demailly.

Several injectivity theorems on compact Kähler manifolds

In this short note, we use the Bochner technique and the Hodge theory in complex differential geometry to prove several injectivity results for the cohomology of holomorphic vector bundles on compact Kahler manifolds, which generalize Enoki’s original injectivity theorem.

Geometry of logarithmic forms and deformations of complex structures

We present a new method to solve certain ∂ ¯ \bar \partial -equations for logarithmic differential forms by using harmonic integral theory for currents on Kähler manifolds. The result can be considered as a ∂ ∂ ¯ \partial \bar \partial -lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness of logarithmic forms, give geometric and simpler proofs of Deligne’s degeneracy theorem for the logarithmic Hodge to de Rham spectral sequences at E 1 E_1 -level, as well as a certain injectivity theorem on compact Kähler manifolds. Furthermore, for a family of logarithmic deformations of complex structures on Kähler manifolds, we construct the extension for any logarithmic ( n , q ) (n,q) -form on the central fiber and thus deduce the local stability of log Calabi-Yau structure by extending an iteration method to the logarithmic forms. Finally we prove the unobstructedness of the deformations of a log Calabi-Yau pair and a pair on a Calabi-Yau manifold by the differential geometric method.

A transcendental approach to injectivity theorem for log canonical pairs

In this paper, we study transcendental aspects of the cohomology groups of adjoint bundles of log canonical pairs, aiming to establish an analytic theory for log canonical singularities. As a result, in the case of purely log terminal pairs, we give an analytic proof of the injectivity theorem originally proved by the Hodge theory. Our method is based on the theory of harmonic integrals and the L2-method for the dbar-equation, and it enables us to generalize the injectivity theorem to the complex analytic setting.

Kollár’s injectivity theorem for globally F-regular varieties

Kollár’s injectivity theorem for globally F-regular varieties

Injectivity theorems on compact complex manifolds

We use analytic methods in this paper to prove some new Enoki type injectivity theorems on compact complex manifolds which generalize more or less the original Enoki injectivity theorem.

An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities

The purpose of this paper is to establish an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, we obtain a Nadel type vanishing theorem. For the proof, we study the asymptotic behavior of the harmonic forms with respect to a family of regularized metrics, and give a method to obtain L 2 L^{2} -estimates of solutions of the ∂ ¯ \overline {\partial } -equation by using the de Rham-Weil isomorphism between the ∂ ¯ \overline {\partial } -cohomology and the C ˇ \rm {\check {C}} ech cohomology.

Vanishing and injectivity theorems for Hodge modules

We prove a surjectivity theorem for the Deligne canonical extension of a polarizable variation of Hodge structure with quasi-unipotent monodromy at infinity along the lines of Esnault-Viehweg. We deduce from it several injectivity theorems and vanishing theorems for pure Hodge modules. We also give an inductive proof of Kawamata-Viehweg vanishing for the lowest graded piece of the Hodge filtration of a pure Hodge module using mixed Hodge modules of nearby cycles.

Injectivity theorem for homotopy invariant presheafs with Witt-transfers

Injectivity theorem for homotopy invariant presheafs with Witt-transfers

Versions of injectivity and extension theorems

We give an analytic version of the injectivity theorem by using multiplier ideal sheaves, and prove some extension theorems for the adjoint bundle of dlt pairs. Moreover, by combining techniques of the minimal model program, we obtain some results for semi-ampleness related to the abundance conjecture in birational geometry and the Strominger-Yau-Zaslow conjecture for hyperKahler manifolds.

Some applications of the theory of harmonic integrals

Abstract In this survey, we present recent techniques on the theory of harmonic integrals to study the cohomology groups of the adjoint bundle with the multiplier ideal sheaf of singular metrics. As an application, we give an analytic version of the injectivity theorem.

A Nadel vanishing theorem for metrics with minimal singularities on big line bundles

The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and generalize Enoki's proof of Kollár's injectivity theorem. Moreover we investigate the asymptotic behavior of harmonic forms with respect to a family of regularized metrics.