Posts
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.
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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.
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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 $G$.
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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.
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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.
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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.
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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.
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Poisson Bivectors
2020-08-03
Fei Ye
𝔼quivalence between Poisson brackets and Poisson bivectors will be studied in this post.
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Multiplicity Loci
2020-06-16
Fei Ye
𝔹undles of differential operators and their application to multiplicity loci will be discussed in this post.
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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.
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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.
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