arXiv subjects

Tsz On Mario Chan

Publications and source records attributed to Tsz On Mario Chan.

13 recordsLinked to original sources

arXiv2026

Injectivity theorems for higher direct images under proper Kähler morphisms on snc spaces

Let $X$ be a complex manifold, and let $Y$ and $D$ be two reduced simple-normal-crossing (snc) divisors on $X$ with no common irreducible components. Given a proper locally Kähler morphism $π\colon X \to Δ$ from $X$ to a complex analytic space $Δ$, we prove Fujino's conjecture on the injectivity theorem in the relative setting in a generalized form. Specifically, we establish an injectivity result for the higher direct images under $π$ for the lc pairs $(X, D)$ as well as $(Y, D_Y)$, where $D_Y := D \cap Y$. As an application, this result immediately implies the injectivity theorem on holomorphically convex Kähler manifolds with reduced snc divisors. The main technique in the proof consists of the theory of harmonic integrals together with residue formulae associated with adjoint ideal sheaves, which are developed from our previous work for the absolute case (where $Δ$ is a point and $X$ is compact). Additionally, we make use of the Takegoshi harmonic forms to deal with the non-compactness of $X$.

math.CV↗

arXiv2026

An extension theorem in terms of adjoint ideal sheaves

In the context of the study of the Ohsawa--Takegoshi $L^2$ extension theorem, a "qualitative" extension theorem in terms of adjoint ideal sheaves (i.e. extension result without $L^2$ estimates but with some form of local $L^2$ condition ensured) is proved on compact Kähler manifolds via harmonic theory and residue computations. The arguments are adapted from those in the study of the injectivity theorem by Chan--Choi--Matsumura. The result guarantees, in particular, the existence of extensions of line-bundle-valued holomorphic top forms over the union of any log-canonical centres of the same codimension to top forms over the ambient space under suitable positivity assumptions.

math.CV↗

arXiv2024

A solution to Fujita's freeness conjecture via an extension theorem with analytic adjoint ideal sheaves

The effective freeness in Fujita's conjecture states that, for an ample line bundle $L$ on a complex projective manifold $X$, the adjoint bundle $K_X\otimes L^{\otimes m}$ is globally generated when $m \geq \dim_{\mathbb C} X + 1$. Following the approach of Angehrn and Siu, a solution is provided in this paper via the use of adjoint ideal sheaves, which provide a finer control of the non-integrable loci given by multiplier ideal sheaves, so that one can work directly with the lc singularities and the associated (minimal) lc centres as in the algebraic approaches of Kawamata and Helmke. The substitute for the Nadel or Kawamata-Viehweg vanishing theorem used in previous approaches is an extension theorem based on the techniques developed for the injectivity theorems.

math.AG↗

arXiv2024

An application of adjoint ideal sheaves to injectivity and extension theorems

This note reviews the authors' approach to Fujino's conjecture, i.e. the injectivity theorem for lc pairs on compact Kähler manifolds, via the use of adjoint ideal sheaves coupled with the associated residue computations in their previous work. Using only the techniques and results obtained from that study and under a slightly stronger positivity assumption, a "qualitative" extension result is obtained. Such extension result guarantees the existence of a global holomorphic extension $F$ of any holomorphic section $f$ on some $σ$-lc centres of the given lc pair. The extension $F$ can be shown to take values in the corresponding adjoint ideal sheaf, even though it comes without any $L^2$ estimate of $F$ in terms of $f$. Moreover, the proof invokes and implies no vanishing theorem in general.

math.CV↗

arXiv2024

An injectivity theorem on snc compact Kähler spaces: an application of the theory of harmonic integrals on log-canonical centers via adjoint ideal sheaves

Let $(X,D)$ be a log-canonical (lc) pair, in which $X$ is a compact Kähler manifold and $D$ is a reduced snc divisor, and let $F$ be a holomorphic line bundle on $X$ equipped with a smooth metric $h_F = e^{-φ_F}$. Via the use of the adjoint ideal sheaves (constructed from $φ_F$ and $D$) and the associated residue morphisms, sections of $K_D \otimes \left. F\right|_D$ on $D$ (as well as those of $K_X \otimes D \otimes F$ on $X$) can be related to the $F$-valued holomorphic top-forms on each lc center of $(X,D)$ by an inductive use of a certain residue exact sequence derived from the adjoint ideal sheaves. The theory of harmonic integrals is valid on each lc center (which is compact Kähler), so this provides a pathway to apply the techniques in harmonic theory to the possibly singular Kähler space $D$. To illustrate the use of such apparatus in problems concerning lc pairs, we prove a Kollár-type injectivity theorem for the cohomology on $D$ when $F$ is semi-positive. This in turn also solves the conjecture by Fujino on the injectivity theorem for the compact Kähler lc pair $(X,D)$, providing an alternative proof of a recent result by Cao and Păun.

math.CV↗

arXiv2023

A new definition of analytic adjoint ideal sheaves via the residue functions of log-canonical measures I

A new definition of analytic adjoint ideal sheaves for quasi-plurisubharmonic (quasi-psh) functions with only neat analytic singularities is studied and shown to admit some residue short exact sequences which are obtained by restricting sections of the newly defined adjoint ideal sheaves to some unions of $σ$-log-canonical ($σ$-lc) centres. The newly defined adjoint ideal sheaves induce naturally some residue $L^2$ norms on the unions of $σ$-lc centres which are invariant under log-resolutions. They can also describe unions of $σ$-lc centres without the need of log-resolutions even if the quasi-psh functions in question are not in a simple-normal-crossing configuration. This is hinting their potential use in discussing the $σ$-lc centres even when the quasi-psh functions in question have more general singularities. Furthermore, their relations between the algebraic adjoint ideal sheaves of Ein--Lazarsfeld as well as those of Hacon--McKernan are described in order to illustrate their role as a (potentially finer) measurement of singularities in the minimal model program. In the course of the study, a local $L^2$ extension theorem is proven, which shows that holomorphic sections on any unions of $σ$-lc centres can be extended holomorphically to some neighbourhood of the unions of $σ$-lc centres with some $L^2$ estimates. The proof does not rely on the techniques in the Ohsawa--Takegoshi-type $L^2$ extension theorems.

math.CV↗

arXiv2023

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, which is also an improvement of Matsumura's result, is given.

math.CV↗

arXiv2023

Residue functions and Extension problems

The "qualitative" extension theorem of Demailly guarantees existence of holomorphic extensions of holomorphic sections on some subvariety under certain positive-curvature assumption, but that comes without any estimate of the extensions, especially when the singular locus of the subvariety is non-empty and the holomorphic section to be extended does not vanish identically there. Residue functions are analytic functions which connect the $L^2$ norms on the subvarieties (or their singular loci) to $L^2$ norms with specific weights on the ambient space. Motivated by the conjectural "dlt extension", this note discusses the possibility of retrieving the $L^2$ estimates for the extensions in the general situation via the use of the residue functions. It is also shown in this note that the $1$-lc-measure defined via the residue function of index $1$ is indeed equal to the Ohsawa measure in the Ohsawa--Takegoshi $L^2$ extension theorem.

math.CV↗

arXiv2022

On an $L^2$ extension theorem from log-canonical centres with log-canonical measures

With a view to prove an Ohsawa-Takegoshi type $L^2$ extension theorem with $L^2$ estimates given with respect to the log-canonical (lc) measures, a sequence of measures each supported on lc centres of specific codimension defined via multiplier ideal sheaves, this article is aiming at providing evidence and possible means to prove the $L^2$ estimates on compact Kähler manifolds $X$. A holomorphic family of $L^2$ norms on the ambient space $X$ is introduced which is shown to "deform holomorphically" to an $L^2$ norm with respect to an lc-measure. Moreover, the latter norm is shown to be invariant under a certain normalisation which leads to a "non-universal" $L^2$ estimate on compact $X$. Explicit examples on $\mathbb{P}^3$ with detailed computation are presented to verify the expected $L^2$ estimates for extensions from lc centres of various codimensions and to provide hint for the proof of the estimates in general.

math.CV↗

arXiv2021

Extension from lc centres and the extension theorem of Demailly but with estimates

This paper generalises the result of Jean-Pierre Demailly on his Ohsawa--Takegoshi-type $L^2$ extension theorem, which guarantees holomorphic extensions for some sections $f$ on analytic subspaces $Y$ defined by multiplier ideal sheaves of plurisubharmonic functions. The result in this paper provides $L^2$ estimates for the extended sections even if those $f$ do not vanish on the singular sets of the reduced loci of the subspaces $Y$, at least for the case when the ambient manifold $X$ is compact and the relevant metrics have only neat analytic singularities. This is achieved by replacing the generalised Ohsawa measure by a measure supported on log-canonical centres.

math.CV↗

arXiv2021

Extension with log-canonical measures and an improvement to the plt extension of Demailly--Hacon--Paun

With a view to proving the conjecture of "dlt extension" related to the abundance conjecture, a sequence of potential candidates for replacing the Ohsawa measure in the Ohsawa-Takegoshi $L^2$ extension theorem, called the "lc-measures", which hopefully could provide the $L^2$ estimate of a holomorphic extension of any suitable holomorphic section on a subvariety with singular locus, are introduced in the first half of the paper. Based on the version of $L^2$ extension theorem proved by Demailly, a proof is provided to show that the lc-measure can replace the Ohsawa measure in the case where the classical Ohsawa-Takegoshi $L^2$ extension works, with some improvements on the assumptions on the metrics involved. The second half of the paper provides a simplified proof of the result of Demailly-Hacon-Paun on the "plt extension" with the superfluous assumption "$\operatorname{supp} D \subset \operatorname{supp}(S+B)$" in their result removed. Most arguments in the proof are readily adopted to the "dlt extension" once the $L^2$ estimates with respect to the lc-measures of holomorphic extensions of sections on subvarieties with singular locus are ready.

math.CV↗

arXiv2013

The Index Theorem for Quasi-Tori

The Index theorem for holomorphic line bundles on complex tori asserts that some cohomology groups of a line bundle vanish according to the signature of the associated hermitian form. In this article, this theorem is generalized to quasi-tori, i.e. connected complex abelian Lie groups which are not necessarily compact. In view of the Remmert-Morimoto decomposition of quasi-tori as well as the Künneth formula, it suffices to consider only Cousin-quasi-tori, i.e. quasi-tori which have no non-constant holomorphic functions. The Index theorem is generalized to holomorphic line bundles, both linearizable and non-linearizable, on Cousin-quasi-tori using $L^2$-methods coupled with the Kazama-Dolbeault isomorphism and Bochner-Kodaira formulas.

math.AG↗

arXiv2012

Kulikov surfaces form a connected component of the moduli space

We show that the Kulikov surfaces form a connected component of the moduli space of surfaces of general type with p_g=0 and K^2=6. We also give a new description for the surfaces, extending ideas of Inoue. Finally we calculate the bicanonical degree of a Kulikov surface, and prove that they verify the Bloch conjecture.

math.AG↗