Albanese Map of a Riemannian Manifold

2021-02-21 Fei Ye
𝔾iven a compact Riemannian manifold $M$ such that the first Betti number $b$ is nonzero, there exists a map $\operatorname{alb}: M\to \mathbb{R}^b/\mathbb{Z}^b$, called the Albanese map. The aim of post is to prove the smoothness of the Albanese map. Read more →

Coadjoint Orbits

2021-01-12 Fei Ye
ℂoadjoint orbits are examples of symplectic manifolds. This post aims at defining coadjoint orbits with a review on Lie group action. Read more →

A Brief Introduction to Cohomology of Lie Algebra

2021-01-07 Fei Ye
𝕃et $G$ be a compact simply connected Lie group and $\mathfrak{g}$ its Lie algebra. In this posts, we will discuss the motivation of Lie algebra cohomology of $\mathfrak{g}$ and its connection with the de Rham cohomology of $G$. Read more →

Zariski Decomposition on Algebraic Surfaces

2020-12-12 Fei Ye
ℤariski decompostion of pseudoeffective divisors on algebraic surface has many applications. In this post, we study Zariski decomposition on algebraic spaces of dimension 2 following an idea of Sakai. Read more →

Nakayama's Lemma and Some Applications

2020-12-02 Fei Ye
ℕakayama’s lemma is a powerful and useful tool in algebraic geometry. In this post, we will consider various versions of Nakayama’s lemma and some applications. Read more →

Seshadri Constants and Restricted Volumes

2020-11-22 Fei Ye
𝕋he purpose of this post is to show an application of the differentiation technique on multiplicity loci to Seshadri constants. Read more →

Hodge Index Theorem

2020-11-20 Fei Ye
ℍodge index theorem and its variations is a fundamental tool in study of smooth projective varieties. We aim at proving Hodge index theorem on higher dimensional varieties over an algebraically closed field of arbitrary characteristic. Read more →

Poisson Bivectors

2020-08-03 Fei Ye
𝔼quivalence between Poisson brackets and Poisson bivectors will be studied in this post. Read more →

Multiplicity Loci

2020-06-16 Fei Ye
𝔹undles of differential operators and their application to multiplicity loci will be discussed in this post. Read more →

Bertini's Theorem

2020-01-13 Fei Ye
𝕎e present a proof of Bertini’s theorem using differentiation in parameter directions to lower the multiplicities of divisors in families. Read more →

Bundles of Principal Parts

2019-08-25 Fei Ye
𝔻efinitions and properties of bundles of principal parts (also known as jet bundles) and sheaves of differential operators will be studied in this post. Read more →