A proof of a general slice-Bennequin inequality

In this blog post, I’ll provide a slick proof of a form of the slice-Bennequin inequality (as outlined by Kronheimer in a mathoverflow answer.) The main ingredient is the adjunction inequality for surfaces embedded in closed 4-manifolds. To obtain the slice-Bennequin inequality (which is a statement about surfaces embedded in 4-manifolds with boundary) we use […]

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 for the Gaussian curvature, is […]

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 ; also, the formula is related to alternating […]

The Picard number of a Kummer K3 surface

Let be a separably closed field of characteristic not , and an abelian surface. Then it is a basic fact (e.g. see Example 1.3 (iii) of Huybrechts’ “K3 Surfaces”) that one can make a K3 surface out of . The construction is as follows. Consider the involution given by The fixed locus of this involution […]

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 one has with a line […]

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 the intersection pairing . It […]