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 …

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

1.Preliminaries: Chern theorem

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

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 …

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 …