The 6th HU and SNU Symposium on Mathematics - Geometry and Topology -as a part of The 11th HU-SNU Joint Symposium

Symplectic $4$-manifolds with $b_2^+=1$ versus complex surfaces with $p_g=0$ (Please note that a title is a little bit changed): One of the fundamental problems in the study of $4$-manifolds is to find a new family of simply connected smooth (symplectic, complex) $4$-manifolds. Though many interesting $4$-manifolds have been constructed using techniques such as fiber sum, rational blow-down, knot surgery, Luttinger surgery and so on, it is still very hard to find a new family of $4$-manifolds with small Euler characteristic. Since I discovered a new simply connected symplectic $4$-manifold with $b_2^+ =1$ and $c_1^2=2$ in 2004 using a rational blow-down surgery, many new simply connected $4$-manifolds with small Euler characteristic have been constructed and now it is one of most active research areas in $4$-manifolds theory to find a new family of $4$-manifolds with $b_2^+ =1$ (equivalently $p_g=0$ in complex category). The aim of this talk is to review recent development in the construction of $4$-manifolds with small Euler characteristic. In particular, I'd like to survey the existence problems of simply connected symplectic $4$-manifolds with $b_2^+=1$ and complex surfaces of general type with $p_g=0$.

Analytic continuation of holomorphic maps respecting varieties of minimal rational tangents: In this talk, we will discuss how to extend a local holomorphic embedding respecting the variety of minimal rational tangents. As an application, we will show that, given an equivariant embedding of a homogeneous variety X into a homogeneous variety Y, a local biholomorphism f from X into Y becomes the given embedding up to the group action, provided that f respects the varieties of minimal rational tangents. This is a joint work with N. Mok.

On Floer cohomology in toric manifolds: We review developments of Floer homology theory due to Fukaya Oh, Ohta and Ono and explain recent advances in the computations of toric manifold cases.

Potentials of homotopic cyclic A-infinity algebras: For an A-infinity algebra with cyclic inner product, a potential recording the structure constants can be defined. We explain how to construct a potential directly in the case that cyclic inner product is given in an A-infinite quasi-isomorphic algebra. (joint work with Cheol-Hyun Cho).

A simply connected surface of general type with $p_g=0$ and $K^2=4$: We construct a simply connected minimal complex surface of general type with $p_g=0$ and $K^2=4$ by using a rational blow-down surgery and $\mathbb{Q}$-Gorenstein smoothing theory.

A logarithmic module of a generalized multiarrangement: A multiarrangement is a pair of an arrangement of hyperplanes and positive integer-valued multiplicities. We can associate the logarithmic vector field to a multiarrangement, which has been playing an important role in the arrangement theory. Recently, several researches on it make us feel that negative multiplicities on hyperplanes and the associated logarithmic module should be also important if we could formulate them correctly. In this talk, we will give a definition of a generalized multiarrangement including negative multiplicities, and the associated logarithmic module. As an application, a duality of Coxeter multiarrangements will be given.

On a Hopf hypersurface of a complex space form: A Hopf hypersurface is defined to be a real hypersurface of a complex space form whose structure vector field is a principal curvature vector field. We study some conditions on the holomorphic distribution on real hypersurfaces which contain the definition of a Hopf hypersurface. On the other hand, we study real hypersurfaces whose structure vector field is an eigenvector field of the Ricci operator.

Analogies to Gauss' results on the arithmetic-geometric mean and the hypergeometric function: Let m(a, b) be an arithmetic-geometric mean of two positive numbers a and b. That is, it is a common limit of sequences {a_n} and {b_n} defined by a_1 = a, b_1 = b, a_{n+1} = (a_n+b_n)/2, b_{n+1} = (a_n*b_n)^(1/2). Gauss showed the equation a/m(a,b) = F( 1/2, 1/2, 1 ;1-(b/a)^2) in 1799, where F(p, q, r; z) denotes the hypergeometric function with parameters p, q, and r. We can prove this equation simply. The key of the proof is, to use transformation formulas for the hypergeometric function skillfully. It is well known that there are several types of transformation formulas of the hypergeometric function. In this talk, we use those in the Goursat's paper in 1881, and give extensions of arithmetic-geometric mean and introduce analogies to Gauss' theorem.

A positive characteristic analogue of the log canonical singularity: The log canonical singularity is a class of singularities, which appeared in birational geometry. Since this singularity is defined via resolution of singularities, the base field is always the complex number field. On the other hand, for a field of positive characteristic, an another type of singularity is defined via the Frobenius morphism. It is called the F-pure singularity. After reduction to mod p >0, F-pure singularities imply log canonical singularities. However these two type of singularities have different behavior. In this talk, the speaker introduces the notion of two type of singularities and gives examples which illustrate their differences.

A certain handle decomposition and Engel structure: An Engel structure is a distribution of tangent subspace of dimension $2$ on a $4$-dimensional manifold which is maximally non-integrable. Engel structure has no local invariant, like contact structure. Contact structures on $3$-dimensional manifolds have an important relation to $3$-dimensional topology. However, we know little about parallelizable $4$-dimensional manifolds, only where Engel structures exist. I will talk on some geometry of closed parallelizable $4$-dimensional manifolds from the view point of Engel geometry.

The 6th HU and SNU Symposium on Mathematics
- Geometry and Topology -
as a part of The 11th HU-SNU Joint Symposium