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 \(S\) 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 \(X\) and a complex vector bundle \(E\) over \(X\). Recall that a structure of holomorphic bundle on \(E\) is given by an operator

\[\overline{\partial}_{\mathcal{E}} \colon \Gamma (E) \to \Gamma(E \otimes \Lambda^{0,1}X)\]

which satisfies \(\overline{\partial}\)-Leibniz identity

\[\overline{\partial}_{\mathcal{E}}(fs) = f\overline{\partial}_{\mathcal{E}}(s) + \overline{\partial}(f) \otimes s\]

and the integrability condition

\[\overline{\partial}_{\mathcal{E}}^2 = 0.\]

For the …

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 \(\alpha_{(r, d)}\) between indecomposable bundles of arbitrary rank \(r\) and degree \(d\) (denoted by \(\mathcal{E}_{(r, d)}\)) and \(\mathcal{E}_{(h, 0)}\) (where \(h = \text{gcd}(d, r)\)). The latter is described explicitly: there is a distinguished element \(F_r \in \mathcal{E}_{(r, 0)}\) 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 \(f : X \to S\) be a proper morphism of spaces. Then for every sheaf \(\mathcal{F}\) on \(X\) that is finite, so is its pushforward \(Rf_\ast \mathcal{F}\).

Notice how I was being deliberately vague in the theorem above. What are \(X\) and \(Y\)? What does …

Simple Proof of Tokuyama’s Formula

Tokuyama’s Formula is a combinatorial result that expresses the product of the deformed Weyl denominator and the Schur polynomial as a sum over strict Gelfand-Tsetlin patterns. This result implies Gelfand’s parametrization of the Schur polynomial, Weyl’s Character Formula, and Stanley’s formula on Hall-Littlewood polynomials — all for \(\text{GL}_n\); …

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

The Teichmüller space \(T_g\) of genus \(g\) curves carries the Hodge bundle \(\Omega T_g\), the total space of which maps into the first cohomology space \(V = H^1(S_g,\mathbb{C})\) via the period map (i. e., a holomorphic 1-form maps into its cohomology class). Haupt’s (or Haupt–Kapovich) theorem describes the image \(per(\Omega T_g) \subset V\) in terms of the integral structure on \(V = H^1(S, \mathbb{Z}) \otimes \mathbb{C}\) and …