Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Projective embedding of log Riemann surfaces and K-stability   Given a smooth polarized Riemann surface (X, L) endowed with a hyperbolic
metric $\omega$ with cusp singularities along a divisor D, we show the L^2
projective embedding of (X, D) defined by L^k is asymptotically almost balanced
in a weighted sense. The proof depends on sufficiently precise understanding of
the behavior of the Bergman kernel in three regions, with the most crucial one
being the neck region around D. This is the first step towards understanding
the algebro-geometric stability of extremal K\"ahler metrics with
singularities.

11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
9
The categories of this paper are: nlin
24
The categories of this paper are: dg-ga
18
The categories of this paper are: econ
34
The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int