## 제 11회 고등과학원 기하학 겨울학교 |

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Lecture series by 이재혁(이화여대), 최인송(건국대)

Lecture series by 이재혁(이화여대), 최인송(건국대)

Complex Manifolds I

In this talk, we have a brief introduction to complex manifolds and review elementary properties on them. We also discuss complex manifolds determined by particular structures such as Kähler manifolds and Calabi-Yau manifolds.

Complex Manifolds II

In this talk, we consider examples of complex manifolds and their properties. We also present typical examples of computation and argument on complex manifolds.

Complex Manifolds III

We introduce the language of line bundles and divisors of complex manifolds. Also we explain the Cech cohomology to compute the number of independent sections of a line bundle.

Complex Manifolds IV

We study the line bundles on projective space. Using this, we prove Chow's theorem, which states that a compact complex manifold embedded in projective space must be cut out by algebraic equations.

**김영욱(고려대)**

**Title:**기하학의 역사

**김동수 (전남대)**

**Title:**Geodesic chord properties and some problems

**Abstract:**Hyperspheres, hypercylinders and hyperplanes are the only isoparametric hypersurfaces in the Euclidean n-space Eⁿ. They are also characterized as hypersurfaces such that the chord joining any two points on them meets the hypersurfaces at the two endpoints with equal angles.

Generalizing this property, we study hypersurfaces in non-flat space form M or the product M x R satisfying the so-called geodesic chord property. As a result, we completely classify such hypersurfaces. Finally, we suggest some open problems.

**고성은 (건국대)**

**Title:**S³ 살펴보기

**조철현 (서울대)**

**Title:**Introduction to Fukaya category of Lefschetz fibration

**Abstract:**Given a Lefschetz fibration, its vanishing cycles form an A-infinity category which was defined by Paul Seidel. We give an elementary introduction.

**한강진 (DGIST)**

**Title:**Geometry of tensors: tensor ranks and equations of secant varieties

**이정찬 (부산대)**

**Title:**Curvature relations and bi-harmonicity of anti-invariant submersions

**Abstract:**We study the geometry of anti-invariant Riemannian submersions from a Kähler manifold onto a Riemannian manifold. We first determine the base space when the total space of an anti-invariant Riemannian submersion is Einstein. We also investigate curvature relations between the total manifold and the base manifold of an anti-invariant Riemannian submersions from almost Hermitian manifolds and obtain certain properties of distributions which are arose from the definition of Riemannian submersions. We find biharmonic conditions of anti-invariant Riemannian submersions from a Kähler manifold onto a Riemannian manifold.

References:

Jungchan Lee, JeongHyeong Park, Bayram Sahin and Dae-Yup Song, Einstein conditions for the base manifold of anti-invariant submersions and Clairaut submersions, Taiwanese J. Math., 19(4)(2015), 1145-1160.

B. Sahin, Anti-invariant Riemannian submersions from almost Hermitian manifolds. Cent. Eur. J. Math., 8(3)(2010), 437-447.

**최영준 (KIAS)**

**Title:**The fiberwise Kähler-Einstein metric on a family of complex manifolds

**Abstract:**Let p : X -> Y be a surjective holomorphic submersion between complex manifolds such that every fiber , is a Kähler manifold.

This is called a family of Kähler manifolds. If every fiber admits a Kähler-Einstein metric, which is a Kähler metric whose Ricci curvature tensor is proportional to the Kähler metric, then this family of Kähler-Einstein metrics induces the fiberwise Kähler-Einstein metric. Since the Kähler-Einstein metric is determined uniquely by the complex structure, the fiberwise Kähler-Einstein metric gives many information of the family of Kähler manifolds.

In the first lecture, we discuss the basic notions of Kähler-Einstein metrics and families of Kähler manifolds. In the second lecture, we discuss the positivity of the fiberwise Kähler-Einstein metrics.

**권도현 (NIMS)**

**Stability of Normalized Mean Curvature Flow**

Title:

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**Abstract:**For several types of mean curvature flow, evolution of convex sets are well understood by Huisken, Andrews, and Bellettini. For non-convex sets, Angenent proved that simply connected curves in two dimensions shrinks to a point by intersection comparison principle. In this talk, we discuss about geometric stability and long-time behavior of a normalized mean curvature flow, where the unique smooth stationary solution is a ball of prescribed volume. The flow preserves the p-reflection property, which is a quantitative smalleness condition between the set and the nearest ball. It is shown that the evolving set becomes smooth in finite time and uniformly converges to a ball in Hausdorff topology exponentially fast. This is joint work with Inwon Kim.

**어윤희 (NIMS)**

**Integrability of orthogonal almost complex structures on spheres.**

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**Abstract:**In this talk, we first introduce some de definitions and properties of almost complex structure. Then we describe the integrability of orthogonal almost complex structures on spheres and the product of spheres.

**박경동 (서울대**)

**Title:**Cartan geometries and cone structures on complex manifolds

**Abstract:**During the 19th century, two generalizations of classical Euclidean geometry appeared by Gauss, Bolyi, Lobachevski, and Riemann. The first was the discovery of non-Euclidean geometry such as hyperbolic geometry, spherical geometry, projective geometry and so on, which were unified as various examples of homogeneous spaces G/H of Lie groups by Felix Klein. The second was the study of a curved manifold modeled on Euclidean space, which we now call Riemannian geometry. Around a hundred years ago, Elie Cartan studied a common generalization of these theories. Cartan geometry may be regarded as a curved geometry modeled on a homogeneous space G/H. To be precise, Cartan geometry is a pair of a principal bundle and a Cartan connection that generalizes the Maurer-Cartan form on a Lie group.

In this talk, I will explain the basic concepts and results on Cartan geometry. After relating these to geometric G-structures and cone structures on a complex manifold, I would like to state that the cone structure given by the variety of minimal rational tangents on a horospherical manifolds of the Hermitian symmetric type is locally flat.