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 are examples of symplectic manifolds. This post aims at defining coadjoint orbits with a review on Lie group action. Read more →

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 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 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 →

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 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 →

𝔼quivalence between Poisson brackets and Poisson bivectors will be studied in this post. Read more →

𝔹undles of differential operators and their application to multiplicity loci will be discussed in this post. Read more →

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

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 →