## The torsion component of the Picard scheme

This post is a continuation of Sean Cotner’s most recent post [see An example of a non-reduced Picard scheme]. Since writing that post, Bogdan Zavyalov shared some notes of his proving the following strengthened version of the results described there.

Main Theorem. Let be a noetherian local ring and …

## Chern connections in the context of CR-geometry

This post is a result of several discussions with Rodion Déev.

#### 1. Preliminaries: Chern’s theorem

Fix a complex manifold and a complex vector bundle over . Recall that a structure of holomorphic bundle on is given by an operator

which satisfies -Leibniz identity

and the integrability condition

For the …

## An example of a non-reduced Picard scheme

Let be a smooth projective connected scheme over an algebraically closed field (experts will notice that several of these hypotheses can be weakened in what follows). Attached to is the Picard scheme , a locally finite type -scheme defined functorially as sending a -scheme to the group . (This uses …

## An explicit construction of indecomposable vector bundles over an elliptic curve

In the celebrated paper “Vector bundles over an elliptic curve,” M. Atiyah classifies indecomposable vector bundles, namely he provides a bijection between indecomposable bundles of arbitrary rank and degree (denoted by ) and (where ). The latter is described explicitly: there is a distinguished element such that for any other …

## A proper scheme with infinite-dimensional fppf cohomology

In algebraic geometry, very often one encounters theorems of the following flavor:

Theorem: Let be a proper morphism of spaces. Then for every sheaf on that is finite, so is its pushforward .

Notice how I was being deliberately vague in the theorem above. What are and ? What does …

## Demystification of the Willmore integrand

The Willmore energy for a surface in Euclidean 3-space is defined as , where is the mean curvature of and its area form. It’s known to be invariant under the conformal transformations (whereas the mean curvature itself is not). White, and later Bryant noticed that the 2-form , where stands …

## Looking for a better counterexample

This post is doubling my old question asked on Mathoverflow.

Let be a smooth projective complex variety of dimension , and let be an ample irreducible divisor. Recall that a cohomology class is called -primitive, if . Here, is the first Chern class of the line bundle associated to . …

## The étale cohomology of curves over finite fields

When I was a graduate student, Zev Rosengarten (a former student of Brian Conrad) and I used to eat dinner at Stanford’s Arrillaga dining hall a lot. We’d talk about math for hours, but one thing that will forever be ingrained in my mind is how Zev was able to …

## A shortcut in Kapovich’s proof of Haupt’s theorem

The Teichmüller space of genus curves carries the Hodge bundle , the total space of which maps into the first cohomology space via the period map (i. e., a holomorphic 1-form maps into its cohomology class). Haupt’s (or Haupt–Kapovich) theorem describes the image in terms of the integral structure on and …