• Bridgeland Stability Conditions

    In classical algebraic geometry, the study of varieties often centers on their geometric aspects, such as analyzing embedded curves, hyperplane sections, etc. A different, more algebraic perspective, however, explores these varieties through their derived categories of coherent sheaves. This categorical approach has become increasingly popular among researchers in recent years.

  • Derived Torelli Theorem for K3 Surfaces

    We’ve been discussing derived categories and Fourier-Mukai transformations lately. This time, we’ll be focusing on $K3$ surfaces. Alongside two-dimensional compact complex tori, $K3$ surfaces represent the two-dimensional Calabi–Yau manifolds, which are also hyperkähler manifolds. These surfaces occupy a central role in the classification of algebraic surfaces, positioned between the positively curved Fano surfaces, which are relatively straightforward to classify, and the negatively curved surfaces of general type, which are largely resistant to classification.

  • K-theoretic and Cohomological Integral Transformations

    We’ve discussed Fourier-Mukai transformations between derived categories and as we are mainly using the derived categories as a tool to study our objects, the interplay between the derived category and cohomology is of interest.

  • The Derived Category of Coherent Sheaves and Fourier-Mukai Theory

    Last time we went through a brief overview of derived and triangulated categories. This time, our aim is to study the bounded derived category $D^b(X)$ of coherent sheaves on an algebraic variety $X$. We’ll also look at something called Fourier-Mukai theory or more concretely derived correspondences.

  • Derived and Triangulated Categories

    In the near future, I will write a short series on homological mirror symmetry and hopefully get a chance to talk about something called Bridgeland stability. This ties in somewhat beautifully with the previous discussions we’ve had here since there is a connection between the theory we are going to develop and the deformed Hermitian-Yang-Mills equation. Since the Bridgeland stability condition is stated in terms of $D^b(X)$, the derived category of coherent sheaves on $X$, we’ll need to go over some technical preliminaries first. I’ll try to give proper motivations for the series in a later post so this time around, unfortunately, we’ll be dealing with only technicalities.

  • Moduli Space of Hermite-Einstein Connections

    This is the first of three posts where we will delve into the moduli space of Hermite-Einstein connections. We will start by examining some of the properties of these connections. Next, we will prove that the moduli space of irreducible Hermite-Einstein connections constitutes an open subset of the moduli space of stable bundles. After that, we will demonstrate that the moduli space of Hermite-Einstein connections is a smooth manifold and explore an application of this theory in the context of surfaces. Finally, we will consider the Kobayashi–Hitchin correspondence.

  • Infinitesimal Deformations and The Kodaira-Spencer Map

    Infinitesimal deformation, broadly speaking, considers the infinitesimal variations in the structures of certain mathematical objects. Imagine an object in space, potentially rigid, whose shape changes due to the application of external forces. At each moment in time, $t$, the object assumes a slightly different shape.

  • Calabi-Yau Manifolds

    Today, we’ll be diving into the so-called Calabi-Yau manifolds. These are manifolds that play an important role in several areas of mathematical physics, most notably in string theory.

  • Kähler-Einstein Metrics and The Uniformization Theorem

    In Hermite-Einstein Metrics and Stability we talked about Hermite-Einstein metrics. These are metrics on a holomorphic vector bundle $E$ that satisfy an analogous condition with the Einstein condition on a Riemannian manifold. This time we will specialize in the case where $E$ is the holomorphic tangent bundle $TX$.

  • Instability in the Sense of Bogomolov

    This is going to be a quick one. We’ll cover the fact that an Hermite-Einstein vector bundle $E$ over a compact complex manifold $X$ is not instable in the sense of Bogomolov. The most important tool for proving this is a vanishing theorem for holomorphic sections in Hermite-Einstein bundles with negative proportionality factor.

  • Hermite-Einstein Metrics and Stability

    Last time we went over holomorphic connections and the Atiyah class on a holomorphic vector bundle. At the end of the post, we were looking at some of the properties of holomorphic vector bundles with vanishing Atiyah class and their correspondence with holomorphic bundles that admit flat connections. In one of the propositions, I mentioned something called a Hermite-Einstein metric, which we have not yet discussed. This, and the stability of vector bundles will be the topic for today.

  • Holomorphic Connections and The Atiyah Class

    Until now, our blog discussions have centered on real or complex vector bundles over real manifolds. However, today we’ll delve into connections on holomorphic vector bundles. I will assume a certain level of familiarity with complex geometry; however, if you possess a solid understanding of differential geometry, you should find this discussion accessible.

  • Flat Connections

    I’ve been currently reading about results considering certain characteristic classes on holomorphic vector bundles on Kähler manifolds and the notion of a flat connection plays an important role in the theory. This is by no means a surprise, if you think about it, the condition $d^2_\nabla = 0$ gives us a cochain complex and lets us wonder about all things cohomology. It turns out that the cohomology theory we obtain from this is very useful and appears, for example in the Riemann-Hilbert correspondence.

  • The Lefschetz Operator

    Soon, I’ll be diving into Kähler manifolds and Hodge theory on these manifolds. This post is going to act as a warm-up in linear algebra for what’s to come in the next posts.

  • Line Bundles and Divisors

    This is going to be one on line bundles and divisors. We’ll begin with the simplest line bundle one can consider which we’ll denote by $\mathcal{O}(-1)$ for reasons that become apparent soon.

  • Sheaf Cohomology

    We’ll be continuing our journey on sheaves and their cohomology today. Last time we ended up in a situation where we considered a short exact sequence of sheaves

    \[0 \longrightarrow \mathcal{F} \longrightarrow \mathcal{G} \longrightarrow \mathcal{H} \longrightarrow 0,\]

    on a topological space $X$. This gave us a short exact sequence of sections

    \[0 \longrightarrow \mathcal{F}(U) \longrightarrow \mathcal{G}(U) \longrightarrow \mathcal{H}(U),\]

    for every open set $U \subset X$.

  • Chern Classes

    The final topic in our discussions considering characteristic classes will be that of Chern classes. In the foreseeable future, I might write about characteristic classes from a different viewpoint. The way we have been doing this for now is based on the Chern-Weil Theory, but there is an alternative, more topological approach using something called classifying spaces.

  • Pontryagin and Euler Classes

    Last time we finished with a quite general description of Characteristic Classes. In particular, we gave a construction for the Chern–Weil homomorphism $c_E : \mathrm{Inv}(\mathfrak{gl}(n,\Bbb R)) \to H^\ast(M)$. The goal today is to look at different types of characteristic classes. We’ll begin with Pontryagin classes and look at Euler classes afterward. The next post will be about complex vector bundles and Chern classes.

  • Characteristic Classes

    The previous post on Connection & Curvature Forms recalled some basic notions on Cartan formalism and how these forms transform under a change of frame. This time, we are going to pick up from that and see where it leads us.

  • Connection and Curvature Forms

    In the previous post, we looked at invariant polynomials. The topic for this one is going to be something called connection $1$-forms and curvature $2$-forms.

  • Coefficients and Characteristic Polynomials

    Last time in our discussion regarding invariant polynomials, I mentioned that the coefficients $f_k(A)$ of the characteristic polynomial of a matrix $A$ are sums of the principal $k \times k$ minors modulo signs. This post will be a short one, giving some justification for this claim.

  • Invariant Polynomials

    I’m finally done with the talk regarding the Bochner Technique. I was planning on writing about the actual result we proved, but it turned out to be such an analytic mess that I don’t think any of the readers here would find reading about it enjoyable. Instead, I’ll begin with a journey toward something called Characteristic Classes.

  • Curvature

    Today will be all about curvature. Specifically, we will be looking at the Riemann curvature tensor and something called Ricci curvature.

  • Gradient, Divergence and the Laplacian

    I’m currently tasked to give a short presentation about the Bochner Technique and I thought that it would probably be beneficial to track my thoughts prior to the presentation. As you might know, the Bochner Technique is a way to relate the Laplacian to the curvature tensor. Before this, however, we need to understand what the gradient, divergence, and Laplacian are on a Riemannian manifold. This is going to be the topic for this post.

  • Sheaves

    I was planning on writing about derived functors and sheaf cohomology, but I figured that since I had no posts regarding sheaves nor category theory I’ll begin with talking about sheaves and go along from there. This posts assumes some elementary knowledge about category theory.

  • Connections

    The notion of a connection seems to be a source of confusion for many students seeing it for the first time. Questions such as “What is the geometric intuition?” are usually present in discussions with students going through connections and I don’t blame them, the definition is rather abstract.

  • Tensor Contraction

    Today, we’re keeping it brief and diving into tensor contraction, particularly its application in Riemannian geometry. Our motivating example will be the Ricci tensor, which, as we’ll see, is actually a contraction of the curvature tensor.

  • Tensor Characterization Lemma

    Some of my classmates found it challenging to grasp the concept that the torsion tensor, defined as $T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]$ is a $(1,2)$-tensor field. Therefore, the aim of this post is to elucidate this concept. I apologize in advance for the notational overload.

  • Sections, Tensor Products and Duals

    This will be a quickie regarding some of the properties of the global sections of a vector bundle $\pi : E \to M$ on a smooth manifold $M$.

  • Local Operators

    In the following we consider vector bundles $E \to M$ and $F \to M$ over a smooth manifold $M$ and instead of focusing on the bundles we’ll digress on something called local operators which we will need in a upcoming post. Most of this is just paraphrasing results from the book by Loring W. Tu on Differential Geometry which I gladly recommend.