Injectivity Theorem Research Articles

An $L^2$ injectivity theorem and its application

An $L^2$ injectivity theorem and its application

A class of generalized monotone operators

We consider the class of η-monotone operators which is intermediate between the class of Minty–Browder and the class of h-monotone operators. We provide sufficient conditions for η-monotonicity which generates a class of η-monotone operators. As a result we construct several η-monotone operators which fill the lack of examples in [11] of h-monotone operators. By using one of the main results in [11], applied to η-monotone operators, we prove a global injectivity theorem. Despite the infinite dimensional contexts in which the Minty–Browder monotonicity is generally used, the motivation behind the present work is rather geometric and the results proved here concern mostly the finite dimensional context.

An injectivity theorem

AbstractWe generalize the injectivity theorem of Esnault and Viehweg, and apply it to the structure of log canonical type divisors.

A Nadel vanishing theorem via injectivity theorems

The purpose of this paper is to establish Nadel type vanishing theorems 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 generalize Kollar’s injectivity theorem to an injectivity theorem for line bundles equipped with singular metrics, by making use of the theory of harmonic integrals. Moreover we give asymptotic cohomology vanishing theorems for high tensor powers of line bundles.

A transcendental approach to Kollár's injectivity theorem II

We treat a relative version of the main theorem in “A transcendental approach to Kollár's injectivity theorem” (Osaka J. Math., to appear). More explicitly, we give a curvature condition that implies Kollár type cohomology injectivity theorems in the relative setting. To carry out this generalization, we use the Ohsawa–Takegoshi twisted version of Nakano's identity.

Asymptotic behavior at infinity of the admissible growth of the quasiconformality coefficient and the injectivity of immersions of sub-Riemannian manifolds

We determine the asymptotic behavior of the admissible growth of the quasiconformality coefficient in a general global injectivity theorem for immersions of sub-Riemannian manifolds of conformally parabolic type. In the model case of a contact immersion of the Heisenberg group in itself, the asymptotic behavior of the admissible growth of the quasiconformality coefficient for which the mapping is still globally invertible was found by the author earlier.

A TRANSCENDENTAL APPROACH TO KOLLÁR'S INJECTIVITY THEOREM

We treat Kollar's injectivity theorem from the analytic (or differential geometric) viewpoint. More precisely, we give a curvature condition which implies Kollar type cohomology injectivity theorems. Our main theorem is formulated for a compact Kahler manifold, but the proof uses the space of harmonic forms on a Zariski open set with a suitable complete Kahler metric. We need neither covering tricks, desingularizations, nor Leray's spectral sequence.

Equivariant cohomology for Hamiltonian torus actions on symplectic orbifolds

We study Hamiltonian R-actions on symplectic orbifolds [M/S], where R and S are tori. We prove an injectivity theorem and generalize Tolman and Weitsman’s proof of the GKM theorem [TW] in this setting. The main example is the symplectic reduction X//S of a Hamiltonian T-manifold X by a subtorus S ⊂ T. This includes the class of symplectic toric orbifolds. We define the equivariant Chen–Ruan cohomology ring and use the above results to establish a combinatorial method of computing this equivariant Chen–Ruan cohomology in terms of orbifold fixed point data.

L2‐signatures, homology localization, and amenable groups

AbstractAimed at geometric applications, we prove the homology cobordism invariance of the L2‐Betti numbers and L2‐signature defects associated to the class of amenable groups lying in Strebel's class D(R), which includes some interesting infinite/finite non‐torsion‐free groups. This result includes the only prior known condition, that Γ is a poly‐torsion‐free abelian group (or a finite p‐group). We define a new commutator series that refines Harvey's torsion‐free derived series of groups, using the localizations of groups and rings of Bousfield, Vogel, and Cohn. The series, called the local derived series, has versions for homology with arbitrary coefficients and satisfies functoriality and an injectivity theorem. We combine these two new tools to give some applications to distinct homology cobordism types within the same simple homotopy type in higher dimensions, to concordance of knots in three manifolds, and to spherical space forms in dimension 3. © 2012 Wiley Periodicals, Inc.

On the combinatorial anabelian geometry of nodally nondegenerate outer representations

Let $\Sg$ be a nonempty set of prime numbers. In the present paper, we continue the study, initiated in a previous paper by the second author, of the combinatorial anabelian geometry of semi-graphs of anabelioids of pro-$\Sg$ PSC-type, i.e., roughly speaking, semi-graphs of anabelioids associated to pointed stable curves. Our first main result is a partial generalization of one of the main combinatorial anabelian results of this previous paper to the case of nodally nondegenerate outer representations, i.e., roughly speaking, a sort of abstract combinatorial group-theoretic generalization of the scheme-theoretic notion of a family of pointed stable curves over the spectrum of a discrete valuation ring. We then apply this result to obtain a generalization, to the case of proper hyperbolic curves, of a certain injectivity result, obtained in another paper by the second author, concerning outer automorphisms of the pro-$\Sg$ fundamental group of a configuration space associated to a hyperbolic curve, as the dimension of this configuration space is lowered from two to one. This injectivity allows one to generalize a certain well-known injectivity theorem of Matsumoto to the case of proper hyperbolic curves

Twisted Homology Cobordism Invariants of Knots in Aspherical Manifolds

We study concordance of homotopically essential, null-homologous knots in three-manifolds M with poly-torsion-free-abelian fundamental group. We fix a knot J and, using L2-signature techniques, construct a family of concordance invariants of knots homotopic to J. We then construct an infinite family of non-concordant knots that are characteristic to J. Our invariants are ρ-invariants of certain three-manifolds associated to these knots, where the three-manifolds depend on the fixed knot J. In order to obtain concordance invariants, we define a series that admits an injectivity theorem (that is, a theorem reminiscent of Stallings’ theorem). We define this series by constructing a localization on the category of groups over π1(M), following Levine’s construction of the algebraic closure for groups [10].

On the combinatorial cuspidalization of hyperbolic curves

In this paper, we continue our study of the pro-$\Sigma$ fundamental groups of configuration spaces associated to a hyperbolic curve, where $\Sigma$ is either the set of all prime numbers or a set consisting of a single prime number, begun in an earlier paper. Our main result may be regarded either as a combinatorial, partially bijective generalization of an injectivity theorem due to Matsumoto or as a generalization to arbitrary hyperbolic curves of injectivity and bijectivity results for genus zero curves due to Nakamura and Harbater--Schneps. More precisely, we show that if one restricts one's attention to outer automorphisms of such a pro-$\Sigma$ fundamental group of the configuration space associated to a(n) affine (respectively, proper) hyperbolic curve which are compatible with certain ``fiber subgroups'' (i.e., groups that arise as kernels of the various natural projections of a configuration space to lower-dimensional configuration spaces) as well as with certain cuspidal inertia subgroups, then, as one lowers the dimension of the configuration space under consideration from $n+1$ to $n \ge 1$ (respectively, $n \ge 2$), there is a natural injection between the resulting groups of such outer automorphisms, which is a bijection if $n \ge 4$. The key tool in the proof is a combinatorial version of the Grothendieck conjecture proven in an earlier paper by the author, which we apply to construct certain canonical sections.

On preimages of a class of generalized monotone operators

In this paper we first provide a geometric interpretation of the Minty–Browder monotonicity which allows us to extend this concept to the so called h -monotonicity, still formulated in an analytic way. A topological concept of monotonicity is also known in the literature: it requires the connectedness of all preimages of the operator involved. This fact is important since combined with the local injectivity, it ensures global injectivity. When a linear structure is present on the source space, one can ask for the preimages to even be convex. In an earlier paper, the authors have shown that Minty–Browder monotone operators defined on convex open sets do have convex preimages, obtaining as a by-product global injectivity theorems. In this paper we study the preimages of h -monotone operators, by showing that they are not divisible by closed connected hypersurfaces, and investigate them from the dimensional point of view. As a consequence we deduce that h -monotone local homeomorphisms are actually global homeomorphisms, as the proved properties of their preimages combined with local injectivity still produce global injectivity.

Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform

Using a vanishing theorem for microlocally real analytic distributions and a theorem on flatness of a distribution vanishing on infinitely many hyperplanes we give a new proof of an injectivity theorem of Bélisle, Massé, and Ransford for the ray transform on $\R^n$. By means of an example we show that this result is sharp. An extension is given where real analyticity is replaced by quasianalyticity.

A local uniqueness theorem for weighted Radon transforms

We consider a weighted Radon transform in the plane, $R_m(\xi, \eta) = \int_{\R} f(x, \xi x + \eta) m(x,\xi,\eta) dx$, where $m(x,\xi,\eta)$ is a smooth, positive function. Using an extension of an argument of Strichartz we prove a local injectivity theorem for $R_m$ for essentially the same class of $m(x,\xi,\eta)$ that was considered by Gindikin in his article in this issue.

Global division of cohomology classes via injectivity

We note that the vanishing and injectivity theorems of Koll\'ar and Esnault-Viehweg can be used to give a quick algebraic proof of a strengthening of the Ein-Lazarsfeld Skoda-type division theorem for global sections of adjoint line bundles vanishing along suitable multiplier ideal sheaves, and to extend it to higher cohomology classes as well. For global sections, this is also a slightly more general statement of the algebraic translation of an analytic result of Siu. Along the way we write down an injectivity statement for multiplier ideals, and its standard consequences.

On the applicability to semirings of two theorems from the theory of rings and modules

Problems concerning the extension of the Baer criterion for injectivity and embedding theorem of an arbitrary module over a ring into an injective module to the case of semirings are treated. It is proved that a semiring S satisfies the Baer criterion and every S-semimodule can be embedded in an injective semimodule if and only if S is a ring.

Injectivity theorems and algebraic closures of groups with coefficients

ABSTRACTRecently, Cochran and Harvey defined torsion-free derived series of groups and proved an injectivity theorem on the associated torsion-free quotients. We show that there is a universal construction which extends such an injectivity theorem to an isomorphism theorem. Our result relates injectivity theorems to a certain homology localization of groups. In order to give a concrete combinatorial description and existence proof of the necessary homology localization, we introduce a new version of algebraic closures of groups with coefficients by considering certain types of equations.

QUASI-LOCAL CONJUGACY THEOREMS IN BANACH SPACES

Let f : U(x0) ⊂ E → F be a C1 map and f′(x0) be the Frechet derivative of f at x0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surjectivity theorem, local injectivity theorem, and the local conjugacy theorem are well known. Those theorems are established by using the properties: f′(x0) is double splitting and $R(f^{\prime}(x))\cap N(T_{0}^{+})=\{0\}$ near x0. However, in infinite dimensional Banach spaces, f′(x0) is not always double splitting (i.e., the generalized inverse of f′(x0) does not always exist), but its bounded outer inverse of f′(x0) always exists. Only using the C1 map f and the outer inverse ${T_{0}^{\#}}$ of f′(x0), the authors obtain two quasi-local conjugacy theorems, which imply the local conjugacy theorem if x0 is a locally fine point of f. Hence the quasi-local conjugacy theorems generalize the local conjugacy theorem in Banach spaces.

On a Curvature Condition that Implies a Cohomology Injectivity Theorem of Kollár–Skoda Type

The curvature condition for the singular Hermitian metric in a generalized L^2 extension theorem on complex manifolds implies also a cohomology injectivity theorem in certain circumstances. This shows that the curvature condition is still available for the extension theorem in more general situations than before.