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 for the Gaussian curvature, is invariant under conformal transformations; its …

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. We denote by H^k_{\mathrm{prim}}(X,\mathbf{Q}) \subset H^k(X,\mathbf{Q}) the subset of primitive elements.

Let denote …

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; also, the formula is related to alternating sign matrices. …

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 do all these complicated spectral sequence arguments off the top …

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 the intersection pairing \omega \colon \Lambda^2V \to \mathbb{C}. It might be stated as follows.…

Fun example. Empty colimit does not commute with empty limit

One of the important properties of filtered colimits is that they commute with finite limits in the category of sets.

Theorem: Let F \colon \mathcal{C}\times \mathcal{D} \to \mathbf{Sets} be a functor, where \mathcal{C} is a filtered small category and \mathcal{D} is a finite category. Then the natural mapping

    \[\mathrm{colim}_{\mathcal{C}} \lim_{\mathcal{D}} F (c, d) \to \lim_{\mathcal{D}} \mathrm{colim}_{\mathcal{C}} F(c, d)\]

is an isomorphism.

This statement is, for example, useful to check that a continuous morphism of sites \mathcal{D} \to \mathcal{C} commuting …

Non-noetherian local ring with finitely generated maximal ideal

Some time ago I found the following interesting lemma on the stackproject:

Theorem: (https://stacks.math.columbia.edu/tag/05GH) Let I be a finitely generated ideal in a ring R. Then the I-adic completion \widehat{R} is noetherian if so is R/I.

Corollary: Let R be a complete local ring with a finitely generated maximal ideal \mathfrak{m}. Then R is noetherian.

This corollary turns out to be actually quite useful …