Workshop on Complex geometry in Osaka 2022

Date: 2022 September 20th (Tue) - 22th (Thu)

Venue: Osaka University, Toyonaka

Graduate School of Science E404 (大阪大学理学部E404)

Speakers, Talks:

Jonas Stelzig (Universität München) 3 talks

Results surrounding the topology and cohomologies of (almost) complex manifolds.

Talk 1: linear combinations of cohomological invariants:

Talk 2: Analogues of the ddc-property beyond the Kähler realm.

Talk 3: Maximally non-integrable almost complex structures: An h-principle and cohomological properties

Ryushi Goto (Osaka University)

Generalized contact structures and generalized Sasakian structures

Keizo Hasegawa (Osaka University)

On Sasaki and CR Lie groups and some other related topics

Nobuhiro Honda (Tokyo Institute of Technology)

A new family of space-like Zoll 3-manifolds

Isamu Iwanari (Tohoku University):

Motivic rational homotopy types and motivic Galois groups

Taro Sano (Kobe University)

Title: Construction of non-Kähler Calabi-Yau manifolds by log deformations

Shun Wakatsuki(Shinshu University)

BV exactness and the S^1-equivariant cohomology of free loop spaces

Natsuo Miyatake (Kyushu University)

Takashi Ohno (Osaka University)

Deformations of Higgs Bundle

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Program

Time table

9/20

13:00—14:30 Stelzig

15:00— 16:00 Honda

16:30–-17:30 Hasegawa

9/21

10:00—11:30 Stelzig

13:00— 14:00 Iwanari

14:30–15:30 Wakatsuki

16:00---16:30 Ohno

16:50— 17:20 Miyatake

9/22

10:00—11:30 Stelzig

13:00— 14:00 Sano

14:30–15:30 Goto

9/20

13:00—14:30

Jonas Stelzig (Universität München)

Talk 1: linear combinations of cohomological invariants

In the 50s, Hirzebruch asked which linear combinations of Hodge and Chern numbers are topological invariants of compact complex manifolds. Building on ideas of Schreieder and Kotschick, who solved the Kähler case, I will present a general answer to this question (and some related ones). Furthermore, I will outline a program how to tackle similar questions when incorporating more cohomological invariants, eg the dimensions of the Bott Chern cohomology groups. This will naturally lead to an algebraic study of the structure of bicomplexes, as well as a number of challenging geometric construction problems.

15:00— 16:00

Nobuhiro Honda (Tokyo Institute of Technology)

A new family of space-like Zoll 3-manifolds

de Sitter space is a Lorentzian analogue of the unit sphere in the Euclidean space, and it has a positive constant curvature. At least in the 3-dimensional case, de Sitter space has another common property with the sphere that its all maximal space-like geodesics are closed. In this talk, we show that the same 3-dimensional manifold admits a family of indefinite Einstein-Weyl structures whose all maximal space-like geodesics are closed but which are different from the above standard one. Our construction uses so-called the minitwistor correspondence that realizes the Einstein-Weyl space as the space of particular rational curves on a complex surface. This is joint work with F. Nakata at Fukushima.

16:30–-17:30

Keizo Hasegawa (Osaka University)

On Sasaki and CR Lie groups and some other related topics. We first review basic results on Sasaki and Vaisman structures on homogeneous spaces (Lie groups) and some well-known non-Kaehler complex manifolds such as Hopf surfaces, Inoue surfaces and Kodaira surfaces. In particular, we discuss locally conformally Kaehler structures on Hopf manifolds

and their small deformations. Sasaki structures are closely related to Vaisman

structures; and may be considered as strongly pseudo-convex CR structures. We see their relations for the cases of Lie groups, along with some explicit classifications of such Lie groups.

9/21

10:00—11:30

Jonas Stelzig (Universität München)

Talk 2: Analogues of the ddc-property beyond the Kähler realm.

The ddc (or \partial\bar\partial) property is a powerful tool in the study of compact complex Kähler manifolds (and closely related ones). It admits many different characterisations, is stable under various geometric constructions and has topological implications. Building on some results introduced in the second half of the previous talk, I will introduce two variants of this property which share similar properties, but hold on natural classes of not-necessarily-Kähler manifolds (eg complex parallelizable solvmanifolds, Vaisman manifolds,...). Survey of joint works with H. Kasuya, D. Popovici, L. Ugarte and S. Wilson.

13:00— 14:00

Isamu Iwanari (Tohoku University):

Motivic rational homotopy types and motivic Galois groups

This talk is about motivic rational homotopy types. This notion gives a motivic generalization of rational homotopy theory. The main invariant is a motivic version of Sullivan’s polynomial de Rham algebras. Another main notion is cotangent motives that provide motives of rational homotopy groups.I plan to discuss motivic rational homotopy types from several perspective: examples, realizations, anabelian perspective, and motivic Galois actions via higher Tannaka duality theory.

14:30–15:30

Shun Wakatsuki(Shinshu University)

BV exactness and the S^1-equivariant cohomology of free loop spaces

It is difficult to compute the S^1-equivariant cohomology of free loop spaces, even with rational coefficients. In this talk, we introduce a new notion "BV exactness", which enables us to compute it only from cohomological information. Moreover, BV exactness is closely related to other notions such as formality, positive weights on Sullivan models and the periodicity operator on cyclic homology. If time permits, we also discuss "r-BV exactness", which is a generalization of BV exactness from the viewpoint of a spectral sequence.

This is a joint work with Katsuhiko Kuribayashi, Takahito Naito, and Toshihiro Yamaguchi.

16:00---16:30

Takashi Ohno (Osaka University)

Deformations of Higgs Bundle

The purpose of this talk is to introduce the DGLA which governs the deformation of Higgs bundle. Since We construct the DGLA via differential geometric approach, the differential of this DGLA has a explicit form which we couldn’t obtain previously. Using the differential we obtained here, we can construct the Kuranishi Space of Higgs bundle.

16:50— 17:20

Natsuo Miyatake (Kyushu University)

Restriction of Donaldson's functional to diagonal metrics on Higgs bundles with not-holomorphic Higgs fields

Let $X$ be a compact connected Riemann surface. In this talk, we consider a Higgs bundle $(E,\Phi)\rightarrow X$ with a smooth, but not holomorphic Higgs field $\Phi$. We first discuss the meaning of the solution of the Hermitian-Einstein equation of a Higgs bundle with a not-holomorphic Higgs field. Secondly, we suppose that the holomorphic vector bundle $E$ decomposes into a direct sum of holomorphic line bundles, and we give some necessary and sufficient conditions for Donaldson's functional to attain a minimum over diagonal metrics concerning the decomposition. From this, under a strong assumption, we see that Donaldson's functional has a critical point even if the Higgs field is not holomorphic. Finally, we discuss how to extend the theorem to not-holomorphic splittings of the vector bundle.

9/22

10:00—11:30

Jonas Stelzig (Universität München)

Talk 3: Maximally non-integrable almost complex structures: An h-principle and cohomological properties

We study almost complex structures with lower bounds on the pointwise rank of the Nijenhuis tensor. If the lower bound is maximal, such structures appear naturally in the study of almost complex Dolbeault cohomology a la Cirici -Wilson, which was proposed as a possible solution to another Question of Hirzebruch from the 50s, concerning a metric independent definition of the Hodge numbers on almost complex manifolds. They also appear in many geometric situations, eg on nearly Kähler six-manifolds. Somewhat surprisingly, there are topological obstructions to their existence in low dimensions. Nevertheless, we prove an h-principle, showing that these obstructions are essentially the only ones (except possibly in real dimension 8). We furthermore show that the Hodge numbers defined by Cirici and Wilson are often infinite, even on compact manifolds. (Joint work with R. Coelho and G. Placini)

13:00— 14:00

Taro Sano (Kobe University)

Construction of non-Kähler Calabi-Yau manifolds by log deformations

Projective Calabi--Yau (CY) manifolds play an important role in the classification of algebraic varieties. Reid suggested connecting projective CY manifolds through non-Kähler CY manifolds via conifold transitions and Friedman exhibited infinitely many topological types of non-Kähler CY 3-folds with b_2=0. In this talk, I'll explain construction of non-Kähler CY manifolds with arbitrarily large b_2 by log deformation theory of　normal crossing varieties.

14:30–15:30

Ryushi Goto (Osaka University)

Generalized contact structures and generalized Sasakian structures

In the talk we discuss a new approach to generalized geometry on odd dimensional manifolds. We introduce generalized contact structures and generalized Sasakian structures in terms with Dirac structures on an odd dimensional manifold $M$, which are also given by admissible pure spinors of the Clifford algebra of $M$. One of our approach is based on the one to one correspondence between generalized geometry on $M$ and conical generalized geometry of the cone $C(M)$ of $M$. From another view point of cylindrical structures, we also discuss generalized contact structures and generalized Sasakian structures on odd dimensional compact Lie groups.

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Organizer

Hisashi Kasuya (Osaka University)

Supported by

JSPS KAKENHI 19H01787

Grant-in-Aid for Scientific Research (B)