February 13-16, 2017

101 Faculty of Science, Kagoshima University

(日本語)

ORGANIZERS:

- Shun-ichi Kimura (Hiroshima University)
- Kiyoshi Takeuchi (Tsukuba Univerisity)
- Masaaki Murakami (Kagoshima University)
- Hiroyuki Nakaoka (Kagoshima University)
- Shoji Yokura (Kagoshima University)

SUPPORTED BY:

- Grants-in-Aid for Scientific Research (C) No.15K04825 (rep. Masaaki Murakami)
- Grants-in-Aid for Scientific Research (B) No.16H03936 (rep. Shoji Yokura)

February 13 (Monday)

- 12:30-13:00 : registration

- 13:00-13:50: Takeshi Saito (University of Tokyo)

"Characteristic cycle of an \(\ell\)-adic sheaf" (abstract) - 14:00-14:50 : Yuichi Ike (University of Tokyo)

"Compact exact Lagrangian intersection in cotangent bundles via sheaf quantizations" (abstract) - 15:10-16:00 : Tomohide Terasoma (University of Tokyo)

"Exotic S6 action on Selberg Hodge structure and Weil Hodge cycle" - 16:10-17:00 : Takahiro Saito (University of Tsukuba)

"On the monodromies and the limit mixed Hodge structures of families of algebraic varieties" (abstract)

February 14 (Tuesday)

- 10:00-10:50 : Hitoshi Moriyoshi (Nagoya University)

"The extension theory for C*-algebras and the Callias index theorem" (abstract) - 11:00-11:50 : Kazuaki Miyatani (Hiroshima University)

"p-adic hypergeometric D-modules and convolution" (abstract) - 13:30-14:20 : Chikashi Miyazaki (Kumamoto University)

"Cohomological Property of Vector Bundles on Multiprojective Space" (abstract) - 14:30-15:20 : Shin-ichi Matsumura (Tohoku University)

"Injectivity theorems with multiplier ideal sheaves for higher direct images" (abstract) - 15:40-16:30 : Hiroyuki Nakaoka (Kagoshima University)

"Cotorsion pairs on exact/triangulated categories" (abstract) - 16:40-17:30 : Jochen Brüning (Humboldt University)

"Global Analysis on Thom-Mather spaces" (abstract (pdf)) - 19:00- conference dinner

February 15 (Wednesday)

- 10:00-10:50 : Masaki Hanamura (Tohoku University)

"Integrals of logarithmic forms on semi-algebraic sets" (abstract) - 11:00-11:50 : Tetsuji Taniguchi (Hiroshima Institute of Technology)

"A representation of Hoffman graphs" (abstract) - 14:00-14:50 : Kenichiro Kimura (University of Tsukuba)

"Hodge realization of Bloch-Kriz mixed Tate motives via integral of logarithmic forms" (abstract) - 15:10-16:00 : Dai Tamaki (Shinshu University)

"Combinatorial Gradient Flows on Cell Complexes" (abstract) - 16:10-17:00 : Tatsuo Suwa (Hokkaido University)

"Relative Dolbeault cohomology and its applications" (abstract)

February 16 (Thursday)

Masanori Asakura "p-adic regulator via the hypergeometric functions"

Abstract: Otsubo and I introduced hypergeometric fibrations, which includes, for example, the Legendre family of elliptic curves. We constrcuted a certain element of K_2 of a hypergeometric fibration and showed that its Beilinson regulator is given by the hypergeometric function 3F2. In this talk I will give the p-adic counterpart along the discussion of the proof of Dwork's unit root formula.

Jochen Brüning "Global Analysis on Thom-Mather spaces"

Abstract: pdf

Masaki Hanamura "Integrals of logarithmic forms on semi-algebraic sets"

In this talk I will discuss the following aspects of our work (with K. Kimura and T. Terasoma):

(1) Study of integration of differential forms with logarithmic singularities on semi-algebraic sets. In particular a we formulate and show a simple sufficient criterion for absolute convergence.

(2) Intersection theory of semi-algebraic sets with a smooth divisor.

(3) Generalized Cauchy's residue formula. Cauchy's formula for integration will be generalized to a formula concerning integration on semi-algebraic sets of any dimension.

These are key issues to be settled when we construct a model of a mixed Hodge complex for complex \(n\)-spaces \(\mathbb{C}^n\); yet these results are of interest on their own, independent of Hodge theory or mixed motives.

Yuichi Ike "Compact exact Lagrangian intersection in cotangent bundles via sheaf quantizations"

I will talk about recent sheaf-theoretic approaches to symplectic geometry, especially non-displaceability of Lagrangian submanifolds. Tamarkin introduced a new category defined as a quotient of the derived category of sheaves and proved the non-displaceability theorem by means of the category. The theorem asserts that if there exist sheaves associated with given two Lagrangians and the Hom space between them is non-zero, then the Lagrangians are non-displaceable by Hamiltonian isotopies. In this talk, I will show that the cardinality of the intersection of two compact Lagrangians in a cotangent bundle is bounded from below by the dimension of Hom space of the Guillermou's sheaf quantizations of the Lagrangians in Tamarkin's category.

Kenichiro Kimura "Hodge realization of Bloch-Kriz mixed Tate motives via integral of logarithmic forms"

Bloch and Kriz constructed a candidate of the category of mixed Tate motives (MTM). They define MTM as the category of graded comodules over a certain graded Hopf algebra constructed from algebraic cycles. The motivation of our work with M. Hanamura and T. Terasom is to understand the Hodge realization functor of MTM in terms of integral of logarithmic differential forms. I will outline the construction of the Hodge realization and explain the case of polylogarithms in some detail.

Toshiki Mabuchi "The extremal Kähler version of the Yau-Tian-Donaldson Conjecture"

For Kähler-Einstein metrics, the Yau-Tian-Donaldson conjecture on polarized algebraic manifolds was recently solved affirmatively by Chen-Donaldson-Sun and Tian. However, for general polarizations, or more generally for extremal Kähler metrics, the generalized versions of the above conjecture are still open. In this talk, I'll discuss the extremal Kähler version of the above conjecture focusing on the existence problem.

Shin-ichi Matsumura "Injectivity theorems with multiplier ideal sheaves for higher direct images"

The injectivity theorem is a celebrated generalization of the Kodaira vanishing theorem to "semi-positive" line bundles in algebraic geometry. In this talk, after we review algebraic geometric methods based on the Hodge theory and analytic methods based on the theory of harmonic integrals, I would like to introduce several results on injectivity theorem formulated for "singular semi-positive" line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. Our method depends on the \(L^2\)-method for the dbar-equation and the theory of harmonic integrals on noncompact Kähler manifolds. If time permits, I explain a further generalization to singular varieties and topics related to extension problems of holomorphic functions. A part of this talk is based on a joint work with Osamu Fujino (Osaka University).

Kazuaki Miyatani "p-adic hypergeometric D-modules and convolution"

The notion of Frobenius structure on p-adic differential equations is a fundamental object which connects the theory of line ar differential equations and the number theory. It is, however, difficult in general to construct a Frobenius structure on a given p-adic differential equation. In this talk, we discuss the existence of Frobenius structures on the p-adic hypergeometric equations. An essential ingredie nt is an extensive use of multiplicative convolution of arithmetic D-modules.

Chikashi Miyazaki "Cohomological Property of Vector Bundles on Multiprojective Space"

My talk is based on a joint work with Francesco Malaspina. I will talk about the cohomological property of vector bundle towards Horrocks-type criteria and characterize the tensor product of pullbacks of exterior products of differential sheaves on multiprojective space by using a notion of Castelnuovo-Mumford regularity and Koszul complexes. Also, I will give a cohomological criterion for the Buchsbaum property of the Segre products of these bundles.

Hitoshi Moriyoshi "The extension theory for C*-algebras and the Callias index theorem"

Let A be a C*-algebra and K denote the algebra of compact operators on the Hilbert space. A short exact sequences of C*-algebras: 0->K->E->A->0 is called an extension of A by K. It is known that such extensions gi ve rise to the K-homology group of A, which is denoted by Ext(A). For instance, the K-homology group is isomorphic to the i ntegers for the group C*-algebra of the real numbers and a generator is given by the Hopf-Wiener extension. It is a Toeplitz tyep extension by definition, but also constructed in a geometric way by exploiting a bordism with cylindrical end, namely, a complete manifold with half-cylinder attached to the boundary. On such a manifold we can develop the Callias index theorem, an index theorem for the Dirac operator with a suitable potent ial. I shall talk about the relation between the Callias index and the connection map in the 6-term exact sequence of K-theory defined from a bordism extension.

Hiroyuki Nakaoka "Cotorsion pairs on exact/triangulated categories"

The notion of a cotorsion pair goes back to Salce. It can be defined in an abelian category, or more generally in an exact category, and have some relation with model structures as shown by Hovey and Gillespie. Its counterpart in a triangulated category is essentially the same as that of a torsion pair in the sense of Iyama and Yoshino, which contains \(t\)-structures, cluster tilting subcategories and co-\(t\)-structures as examples. I will talk about cotorsion pairs and related structures, in a way applicable both to exact categories and triangulated categories.

This is partly based on a joint work with Yann Palu.

Takahiro Saito "On the monodromies and the limit mixed Hodge structures of families of algebraic varieties"

We study the monodromies and the limit mixed Hodge structures of families of complete intersection varieties over a punctured disk in the complex plane. For this purpose, we express their motivic nearby fibers in terms of the geometric data of some Newton polyhedra. In particular, the limit mixed Hodge numbers and some part of the Jordan normal forms of the monodromies of such a family will be described very explicitly. This is a joint work with Kiyoshi Takeuchi.

Takeshi Saito "Characteristic cycle of an \(\ell\)-adic sheaf"

For an \(\ell\)-adic sheaf on a smooth variety over a perfect field, its characteristic cycle is defined as a Z-linear combination of irreducible components of the singular support, defined by Beilinson as a closed conical subset of the cotangent bundle. It gives an analogue of that defined by Kashiwara- Schapira in a transcendental setting. We discuss its properties, including the index formula and the relation with the MacPherson Chern class.

Tatsuo Suwa "Relative Dolbeault cohomology and its applications"

The Čech-de Rham cohomology is effectively used in various problems related to localization of characteristic classes. Likewise we may develop the Čech-Dolbeault cohomology theory and on the way we naturally come up with the relative Dolbeault cohomology. This 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 expressing the latter. In this talk I will present this cohomology theory and give, as applications, simple expressions of the Sato hyperfunctions, some fundamental operations on them and related local duality theorems. The talk includes a joint work with N. Honda and T. Izawa.

Dai Tamaki "Combinatorial Gradient Flows on Cell Complexes"

One of the most famous discretization of Morse theory is Forman's work in middle 90's, in which the notion of discrete Morse function on a cell complex was introduced. He also introduced an analogue of gradient flows, which turned out to be quite useful for computing homology and cohomology in various contexts. Unfortunately, however, Forman's gradient flows are not sufficient to recoverthe homotopy type of the original cell complex. In this talk, we introduce thenotion of flow paths, a simple extension of Forman's gradient flows, with which a systematic method of recovering the homotopy type from a discrete Morse function is obtained. More precisely, we construct a poset-enriched category C(f) from a discrete Morse function \(f\colon F(X) \to \mathbb{R}\) on a regular cell complex \(X\) whose classifying space BC(f), in an appropriate sense, is homotopy equivalent to \(X\). If time permits, we also discuss a possible extension to "discrete Morse theory with cycles".

This is a report of a joint work with Vidit Nanda (Oxford), Kohei Tanaka (Shinshu), and Hidetaka Tokuno (Shinshu).

Tetsuji Taniguchi "A representation of Hoffman graphs"

Hoffman graphs were introduced by Woo and Neumaier [3] to extend the results of Hoffman [1]. Hoffman proved what we would call Hoffman's limit theorem which asserts that, in the language of Hoffman graphs, the smallest eigenvalue of a fat Hoffman graph is a limit of the smallest eigenvalues of a sequence of ordinary graphs with increasing minimum degree.

Woo and Neumaier [3] gave a complete list of fat indecomposable Hoffman graphs with smallest eigenvalue at least \(-1-\sqrt{2}\). Moreover they [3, Open Problem 4] raised the problem of classifying fat Hoffman graphs with smallest eigenvalue at least \(-3\).

In [2], we showed that the special graph \(\mathcal{S}^-(\mathfrak{H})\) of a such a Hoffman graph \(\mathfrak{H}\) is isomorphic to one of the Dynkin graphs \(\mathsf{A}_n\), \(\mathsf{D}_n\), or extended Dynkin graphs \(\tilde{\mathsf{A}}_n\) or \(\tilde{\mathsf{D}}_n\). Also we showed that, even when the Hoffman graph \(\mathfrak{H}\) does not admit an integral representation, its special graph \(\mathcal{S}(\mathfrak{H})\) can be represented by one of the exceptional root lattices \({\mathsf{E}}_n\) (\(n=6,7,8\)).

In this talk, we introduce a Hoffman graph and its representation. Moreover we talk about some results and problems.

[2] H. J. Jang, J. Koolen, A. Munemasa, and T. Taniguchi, On fat Hoffman graphs with smallest eigenvalue at least \(-3\), Ars Math. Contemp., 7 (2014), 105--121.

[3] R. Woo and A. Neumaier, On graphs whose smallest eigenvalue is at least \(-1-\sqrt{2}\), Linear Algebra Appl. 226-228 (1995), 577--591.

Tomohide Terasoma "Exotic S6 action on Selberg Hodge structure and Weil Hodge cycle"