Reference/Known results on the singular behaviour of the fibres of a holomorphic map between compact Kähler manifolds

I have been interested in the following situation of late: Let $ X$ and $ Y$ be compact Kähler manifolds with $ \dim_{\mathbb{C}}(Y) < \dim_{\mathbb{C}}(X)$ and let $ f : X \to Y$ be a surjective holomorphic map with connected fibres. Let $ S = \{ s_1, …, s_k \}$ denote the critical values of $ f$ , which is a subvariety of $ Y$ .

I cannot find a detailed account of how bad the singular behaviour of the fibres of $ f$ can be. For example, do the fibres contain $ (-1)$ curves (i.e., curves with self-intersection number $ -1$ ) or $ (-2)$ curves?

If anyone can provide references where I can get a better understanding of this, that would be tremendously appreciated.

(Singular) metric associated to the higher cohomology

Suppose $ X$ is a smooth complex variety and $ L$ is a line bundle with a metric $ h_L$ , then a section $ s \in H^0(X, L)$ gives another metric $ \tilde h_L:= e^{-\phi}h_L$ where $ \phi=\log \|s\|^2_{h_L}$ .

If $ u \in H^q(X,L)$ is a section (or just $ u\in H^q(X,K_X)$ ), is there a way to construct a metric on $ L$ with some relation to $ u$ ?

Why does convention say DB table names should be singular but RESTful resources plural?

It’s a pretty established convention that database table names, in SQL at least, should be singular. SELECT * FROM user; See this question and discussion.

It’s also a pretty established convention that RESTful API resource names should be plural. GET /users/123 and POST /users See this one.

In the simplest database-backed API, the name of the resource in the URL would be the table, and the data elements in the URL and request/response bodies would map directly to the columns in the DB. Conceptually, I don’t see a difference between operating on the data through this theoretical API versus operating on it directly through SQL. And because of that, the difference in naming conventions between user and users doesn’t make sense to me.

How can the difference in pluralization be justified when, conceptually, the REST API and the SQL are doing the same thing?

Filtrations of spectra related to cellular ones and singular homology

I would like to study filtrations of spectra (i.e., objects of the “topological” stable homotopy category $ SH$ ; a filtration of a spectrum $ E$ is a sequence of compatible maps $ E_{\le i}\to E$ ) whose levels lie “between” subsequent levels of cellular filtrations (and cones of comparison maps are shifted Moore spectra). An interesting particular case is the filtration that gives the “canonical” filtration on singular homology (note that this is not the Postnikov $ t$ -structure filtration, since the latter is “much further from the cellular one”); the levels of this filtration for finite spectra are finite as well (and “quotients” are the shifted Moore spectra corresponding to $ H_*(E)$ ).

Did anybody study anything similar previously? Can you suggest me any possible applications for filtrations of this sort?

Singular integral of the composition of the Hilbert transform and fractional Laplacian

Given $ 0<s<1$ , we can define the Fractional Laplacian by

$ $ \Lambda^{-s}f(x):=(-\Delta)^{-s/2}(x)=\int_{-\infty}^{+\infty}|x-y|^{-1+s}f(y)dy$ $ or by means of Fourier transform as $ $ \widehat{\Lambda^{-s}f}(\xi)=c_s|\xi|^{-s}\widehat{f}(\xi)\;\mbox{for all}\;\xi\neq0.$ $

The Hilbert transform is defined by $ $ Hf(x)=p.v. \int_{-\infty}^{+\infty}\frac{f(y)}{x-y}dy$ $ or by means of Fourier transform as $ $ \widehat{Hf}(\xi)=-isgn(\xi)\widehat{f}(\xi)\;\mbox{for all}\;\xi\neq0.$ $

Therefore, we can write \begin{equation}\label{eq1}\widehat{\Lambda^{-s}Hf}(\xi)=C_ssgn(\xi)|\xi|^{-s}\widehat{f}(\xi)\;\mbox{for all}\;\xi\neq0. \end{equation} My question is how can i define $ \Lambda^{-s}Hf$ by singular integrals.

Singular part of the Chen-Frid pairing for divergence-measure vector fields

In their fundamental paper about vector fields with measure divergence, Chen and Frid prove the following Theorem (see Thm. 3.2 of the quoted paper):

Theorem. Let $ F \colon \mathbb R^N \to \mathbb R^N$ be a bounded, measurable vector field. Assume that its distributional divergence $ \text{div} F$ is (a distribution represented by a) Radon measure with finite total variation.

Let $ g \in \text{BV}(\mathbb R^N)\cap L^\infty(\mathbb R^N)$ . Then the distribution $ $ \mu := \text{div}(gF) – \tilde{g}\text{div} F $ $ ($ \tilde g$ is the precise representative of the function $ g$ ) is a Radon measure absolutely continuous with respect to $ \vert Dg\vert$ , whose absolutely continuous part with respect to the Lebesgue measure coincides with $ F\cdot (∇g)_{ac}$ a.e. .

This interesting theorem is proved in the aforementioned paper. For the research I am currently carrying on, I am wondering if something more is known on the measure $ \mu$ . In particular, has anybody characterized this measure $ \mu$ (i.e. its jump part/the Cantor part)?

Besides some results spread in the literature, I would be interested in guessing what is the correct formula (especially for the Cantor part). I do not have any idea about it… Any ideas? Thanks.