Semipositive Curvature Research Articles

Singular hermitian metrics and the decomposition theorem of Catanese, Fujita, and Kawamata

We prove that a torsion-free sheaf F \mathcal {F} endowed with a singular hermitian metric with semi-positive curvature and satisfying the minimal extension property admits a direct-sum decomposition F ≃ U ⊕ A \mathcal {F}\simeq \mathcal {U}\oplus \mathcal {A} where U \mathcal {U} is a hermitian flat bundle and A \mathcal {A} is a generically ample sheaf. The result applies to the case of direct images of relative pluricanonical bundles f ∗ ω X / Y ⊗ m f_* \omega _{X/Y}^{\otimes m} under a surjective morphism f : X → Y f\colon X \to Y of smooth projective varieties with m ≥ 2 m\geq 2 . This extends previous results of Fujita, Catanese–Kawamata, and Iwai.

Singular Griffiths semi-positivity of higher direct images

We show that the higher direct image sheaves of the relative canonical bundle tensorized with a Nakano semi-positive vector bundle admit singular Hermitian metrics with Griffiths semi-positive curvature and the minimal extension property.

On the complement of a hypersurface with flat normal bundle which corresponds to a semipositive line bundle

We investigate the complex analytic structure of the complement of a non-singular hypersurface with unitary flat normal bundle when the corresponding line bundle admits a Hermitian metric with semipositive curvature.

Plurisubharmonic functions on a neighborhood of a torus leaf of a certain class of foliations

Abstract Let C be a smooth elliptic curve embedded in a smooth complex surface X such that C is a leaf of a suitable holomorphic foliation of X. We investigate the complex analytic properties of a neighborhood of C under some assumptions on the complex dynamical properties of the holonomy function. As an application, we give an example of ( C , X ) {(C,X)} in which the line bundle [ C ] {[C]} is formally flat along C, however it does not admit a C ∞ {C^{\infty}} Hermitian metric with semi-positive curvature. We also exhibit a family of embeddings of a fixed elliptic curve for which the positivity of normal bundles does not behave in a simple way.

HIGHER CODIMENSIONAL UEDA THEORY FOR A COMPACT SUBMANIFOLD WITH UNITARY FLAT NORMAL BUNDLE

Let $Y$ be a compact complex manifold embedded in a complex manifold with unitary flat normal bundle. Our interest is in a sort of the linearizability problem of a neighborhood of $Y$. As a higher codimensional generalization of Ueda’s result, we give a sufficient condition for the existence of a nonsingular holomorphic foliation on a neighborhood of $Y$ which includes $Y$ as a leaf with unitary-linear holonomy. We apply this result to the existence problem of a smooth Hermitian metric with semipositive curvature on a nef line bundle.

Ueda theory for compact curves with nodes

Let $C$ be a compact complex curve included in a non-singular complex surface such that the normal bundle is topologically trivial. Ueda studied complex analytic properties of a neighborhood of $C$ when $C$ is non-singular or is a rational curve with a node. We propose an analogue of Ueda's theory for the case where $C$ admits nodes. As an application, we study singular Hermitian metrics with semi-positive curvature on the anti-canonical bundle of the blow-up of the projective plane at nine points in arbitrary position.

On the minimality of canonically attached singular Hermitian metrics on certain nef line bundles

We apply Ueda theory to a study of singular Hermitian metrics of a (strictly) nef line bundle $L$. Especially we study minimal singular metrics of $L$, metrics of $L$ with the mildest singularities among singular Hermitian metrics of $L$ whose local weights are plurisubharmonic. In some situations, we determine a minimal singular metric of $L$. As an application, we give new examples of (strictly) nef line bundles which admit no smooth Hermitian metric with semi-positive curvature.

Toward a higher codimensional Ueda theory

Ueda’s theory is a theory on a flatness criterion around a smooth hypersurface of a certain type of topologically trivial holomorphic line bundles. We propose a codimension two analogue of Ueda’s theory. As an application, we give a sufficient condition for the anti-canonical bundle of the blow-up of the three dimensional projective space at 8 points to be non semi-ample however admit a smooth Hermitian metric with semi-positive curvature.

Dynamical construction of Kähler-Einstein metrics

AbstractIn this article, we give a new construction of a Kähler-Einstein metric on a smooth projective variety with ample canonical bundle. As a consequence, for a dominant projective morphism f: X → S with connected fibers such that a general fiber has an ample canonical bundle, and for a positive integer m, we construct a canonical singular Hermitian metric hE,m on with semipositive curvature in the sense of Nakano.

Dynamical construction of Kähler-Einstein metrics

AbstractIn this article, we give a new construction of a Kähler-Einstein metric on a smooth projective variety with ample canonical bundle. As a consequence, for a dominant projective morphismf:X→Swith connected fibers such that a general fiber has an ample canonical bundle, and for a positive integerm, we construct a canonical singular Hermitian metrichE,monwith semipositive curvature in the sense of Nakano.

Hyperkähler Syz Conjecture and Semipositive Line Bundles

Let M be a compact, holomorphic symplectic Kahler manifold, and L a non-trivial line bundle admitting a metric of semipositive curvature. We show that some power of L is effective. This result is related to the hyperkahler SYZ conjecture, which states that such a manifold admits a holomorphic Lagrangian fibration, if L is not big.

Extension of twisted Hodge metrics for Kähler morphisms

Let f : X --> Y be a holomorphic map of complex manifolds, which is proper, Kahler, and surjective with connected fibers, and which is smooth over Y-Z the complement of an analytic subset Z. Let E be a Nakano semi-positive vector bundle on X, and consider direct image sheaves F = R^qf_*(K_{X/Y} \otimes E) for q \geq 0. In our previous paper, we discussed the Nakano semi-positivity of F with respect to the so-called Hodge metric, when the map f is smooth. In this paper, we discuss the extension of the induced metric on the tautological line bundle O(1) on the projective space bundle P(F) ``over Y-Z'' as a singular Hermitian metric with semi-positive curvature ``over Y''. As a particular consequence, if Y is projective, R^qf_*(K_{X/Y} \otimes E) is weakly positive over Y-Z in the sense of Viehweg.

Hodge metrics and positivity of direct images

Building on Fujita-Griffiths method of computing metrics on Hodge bundles, we show that the direct image of an adjoint semi-ample line bundle by a projective submersion has a continuous metric with Griffiths semi-positive curvature. This shows that for every holomorphic semi-ample vector bundle $E$ on a complex manifold, and every positive integer $k$, the vector bundle $S^kE\otimes\det E$ has a continuous metric with Griffiths semi-positive curvature. If $E$ is ample on a projective manifold, the metric can be made smooth and Griffiths positive.

Properties of compact complex manifolds carrying closed positive currents

We show that a compact complex manifold is Moishezon if and only if it carries a strictly positive, integral (1, 1)-current. We then study holomorphic line bundles carrying singular hermitian metrics with semi-positive curvature currents, and we give some cases in which these line bundles are big. We use these cases to provide sufficient conditions for a compact complex manifold to be Moishezon in terms of the existence of certain semi-positive, integral (1,1)-currents. We also show that the intersection number of two closed semi-positive currents of complementary degrees on a compact complex manifold is positive when the intersection of their singular supports is contained in a Stein domain.

Oeljeklaus–Toma Manifolds Admitting No Complex Subvarieties

The Oeljeklaus-Toma (OT-) manifolds are complex manifolds constructed by Oeljeklaus and Toma from certain number fields, and generalizing the Inoue surfacesSm. On each OT-manifold we construct a holomorphic line bundle with semipositive curvature form !0 and trivial Chern class. Using this form, we prove that the OT-manifolds