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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 condition

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

For the reader who is more accustomed …

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 the fact that there exists a k-point; in general one …

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 = 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 N \in \mathcal{E}_{(r, 0)} one has N \cong F_r \otimes L with L a line bundle of degree 0. This …

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 “finite” mean? Well, it turns out that this depends on …

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 …

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.…