Thuses

Brieskorn resolutions via algebraic spaces

I’d like to discuss simultaneous resolutions of surfaces from a moduli-theoretic perspective, following Michael Artin’s paper on Brieskorn resolutions.

Artin’s approach to moduli begins with the most desirable aspect of a moduli space, its universal property. That is to say, define a functor the space should represent and then check …

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 example of a non-reduced Picard scheme

Let $X$ be a smooth projective connected scheme over an algebraically closed field $k$ (experts will notice that several of these hypotheses can be weakened in what follows). Attached to $X$ is the Picard scheme $\mathrm{Pic}_{X/k}$ , a locally finite type $k$ -scheme defined functorially as sending a $k$ -scheme $T$ to the group $\mathrm{Pic}(X_T)/\mathrm{Pic}(T)$ . (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 $\alpha_{(r, d)}$ 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 …

Demystification of the Willmore integrand

The Willmore energy for a surface $S$ in Euclidean 3-space is defined as $\tilde{W}(S) = \int_S \mu^2\omega_S$ , where $\mu$ is the mean curvature of $S$ and $\omega_S$ 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 $\Omega_S = (\mu^2-K)\omega_S$ , where $K$ stands …

Looking for a better counterexample

This post is doubling my old question asked on Mathoverflow.

Let $X$ be a smooth projective complex variety of dimension $d$ , and let $Y\subset X$ be an ample irreducible divisor. Recall that a cohomology class $b\in H^k(X,\mathbf{Q})$ is called $Y$ -primitive, if $b\smile c_1(Y)^{d-k+1} =0$ . Here, $c_1(Y)\in H^{2}(X,\mathbf{Q})$ is the first Chern class of the line bundle associated to $Y$ . …

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 $\text{GL}_n$ ; …

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 …