arXiv subjects

Tatsuo Suwa

Publications and source records attributed to Tatsuo Suwa.

10 recordsLinked to original sources

arXiv2026

On functoriality of Baum-Bott residues

We prove a functoriality formula for Baum--Bott residues under holomorphic pull-back. Let $f:V\to X$ be a holomorphic map generically transverse to a singular holomorphic foliation $\mathscr F$ on $X$. Under a compatibility condition on the tangent and normal sheaves, the localized Baum--Bott class of $f^*\mathscr F$ is the pull-back of that of $\mathscr F$, and the corresponding residue is obtained by localized intersection. We also discuss the universal relative class of Baum and Bott and its relation with the rationality conjecture. If the characteristic polynomial has rational coefficients, rational residues remain rational under the pull-backs considered here. For proper surjective generically finite maps of equal dimension, we obtain in addition a push-forward formula; hence rationality is equivalent before and after pull-back, in particular for bimeromorphic modifications satisfying the hypotheses of the theorem.

math.CV↗

arXiv2022

Sato hyperfunctions via relative Dolbeault cohomology

The relative Dolbeault cohomology which naturally comes up in the theory of Cech-Dolbeault cohomology turns out to be canonically isomorphic with the local (relative) cohomology of A. Grothendieck and M. Sato so that it provides a handy way of representing the latter. In this paper we use this cohomology to give simple explicit expressions of Sato hyperfunctions, some fundamental operations on them and related local duality theorems. This approach also yields a new insight into the theory of hyperfunctions and leads to a number of further results and applications. As one of such, we give an explicit embedding morphism of Schwartz distributions into the space of hyperfunctions.

math.CV↗

arXiv2020

Note on Dolbeault cohomology and Hodge structures up to bimeromorphisms

We construct a simply-connected compact complex non-Kähler manifold satisfying the $\partial\bar\partial$-Lemma, and endowed with a balanced metric. To this aim, we were initially aimed at investigating the stability of the property of satisfying the $\partial\bar\partial$-Lemma under modifications of compact complex manifolds and orbifolds. This question has been recently addressed and answered in \cite{rao-yang-yang, yang-yang, stelzig-blowup, stelzig-doublecomplex} with different techniques. Here, we provide a different approach using Čech cohomology theory to study the Dolbeault cohomology of the blow-up $\tilde X_Z$ of a compact complex manifold $X$ along a submanifold $Z$ admitting a holomorphically contractible neighbourhood.

math.DG↗

arXiv2019

Localization of Bott-Chern classes and Hermitian residues

We develop a theory of Cech-Bott-Chern cohomology and in this context we naturally come up with the relative Bott-Chern cohomology. In fact Bott-Chern cohomology has two relatives and they all arise from a single complex. Thus we study these three cohomologies in a unified way and obtain a long exact sequence involving the three. We then study the localization problem of characteristic classes in the relative Bott-Chern cohomology. For this we define the cup product and integration in our framework and we discuss local and global duality homomorphisms. After reviewing some materials on connections, we give a vanishing theorem relevant to our localization. With these, we prove a residue theorem for a vector bundle admitting a Hermitian connection compatible with an action of the non-singular part of a singular distribution. As a typical case, we discuss the action of a distribution on the normal bundle of an invariant submanifold (so-called the Camacho-Sad action) and give a specific example.

math.CV↗

arXiv2019

Relative Dolbeault cohomology

We review the notion of relative Dolbeault cohomology and prove that it is canonically isomorphic with the local (relative) cohomology of A. Grothendieck and M. Sato with coefficients in the sheaf of holomorphic forms. We deal with this cohomology from two viewpoints. One is the Cech theoretical approach, which is convenient to define such operations as the cup product and integration and leads to the study of local duality. Along the way we also establish some notable canonical isomorphisms among various cohomologies. The other is to regard it as the cohomology of a certain complex, which is interpreted as a notion dual to the mapping cone in the theory of derived categories. This approach shows that the cohomology goes well with derived functors. We also give some examples and indicate applications, including simple explicit expressions of Sato hyperfunctions, fundamental operations on them and related local duality theorems.

math.CV↗

arXiv2018

Representation of relative sheaf cohomology

We study the cohomology theory of sheaf complexes for open embeddings of topological spaces and related subjects. The theory is situated in the intersection of the general Cech theory and the theory of derived categories. That is to say, on the one hand the cohomology is described as the relative cohomology of the sections of the sheaf complex, which appears naturally in the theory of Cech cohomology of sheaf complexes. On the other hand it is interpreted as the cohomology of a complex dual to the mapping cone of a certain morphism of complexes in the theory of derived categories. We prove a "relative de Rham type theorem" from the above two viewpoints. It says that, in the case the complex is a soft or fine resolution of a certain sheaf, the cohomology is canonically isomorphic with the relative cohomology of the sheaf. Thus the former provides a handy way of representing the latter. Along the way we develop various theories and establishes canonical isomorphisms among the cohomologies that appear therein. The second viewpoint leads to a generalization of the theory to the case of cohomology of sheaf morphisms. Some special cases together with applications are also indicated.

math.AT↗

arXiv2016

Local and global coincidence homology classes

For two differentiable maps between two manifolds of possibly different dimensions, the local and global coincidence homology classes are introduced and studied by Bisi- Bracci-Izawa-Suwa (2016) in the framework of Cech-de Rham cohomology. We take up the problem from the combinatorial viewpoint and give some finer results, in particular for the local classes. As to the global class, we clarify the relation with the cohomology coincidence class as studied by Biasi-Libardi-Monis (2015). In fact they introduced such a class in the context of several maps and we also consider this case. In particular we define the local homology class and give some explicit expressions. These all together lead to a generalization of the classical Lefschetz coincidence point formula.

math.AT↗

arXiv2015

Localized intersection of currents and the Lefschetz coincidence point theorem

We introduce the notion of a Thom class of a current and define the localized intersection of currents. In particular we consider the situation where we have a smooth map of manifolds and study localized intersections of the source manifold and currents on the target manifold. We then obtain a residue theorem on the source manifold and give explicit formulas for the residues in some cases. These are applied to the problem of coincidence points of two maps. We define the global and local coincidence homology classes and indices. A representation of the Thom class of the graph as a Cech-de~Rham cocycle immediately gives us an explicit expression of the index at an isolated coincidence point, which in turn gives explicit coincidence classes in some non-isolated components. Combining these, we have a general coincidence point theorem including the one by S. Lefschetz.

math.CV↗

arXiv2010

Perturbation of Baum-Bott residues

We prove that Baum-Bott residues vary continuously under smooth deformations of holomorphic foliations. This provides an effective way to compute residues.

math.CV↗

arXiv2010

Localization of Atiyah classes

We construct Atiyah classes using debar-closed forms. Under this point of view and using the Cech-Dolbeault cohomology, we provide several types of results about vanishing and localization of Atiyah classes and applications.

math.CV↗