Portal:Mathematics/Selected article/38

2013-10-23T14:20:19Z

106.192.45.162:

In [[complex geometry]], a '''polar homology''' is a group which captures holomorphic invariants  of a [[complex manifold]] in a similar way to usual [[Homology (mathematics)|homology]] of a [[manifold (mathematics)|manifold]] in [[differential topology]]. Polar homology was defined by B. Khesin and A. Rosly in 1999.

==Definition==
Let ''M'' be a [[complex projective manifold]]. The space <math>C_k</math> of polar ''k''-chains is a vector space over <math>{\Bbb C}</math> defined as a quotient <math>A_k/R_k</math>, with <math>A_k</math> and <math>R_k</math> vector spaces defined below.

===Defining <math>A_k</math>===
The space <math>A_k</math> is freely generated by the triples <math>(X, f, \alpha)</math>, where ''X'' is a smooth, ''k''-dimensional complex manifold, <math>f:\; X \mapsto M</math> a holomorphic map, and <math>\alpha</math> is a rational ''k''-form on ''X'', with first order poles on a [[normal crossing divisor|divisor with normal crossing]].

===Defining <math>R_k</math>===
The space <math>R_k</math> is generated by the following relations.

#<math>\lambda (X, f, \alpha)=(X, f, \lambda\alpha)</math>
#<math>(X,f,\alpha)=0</math> if <math>\dim f(X) < k</math>.
#<math>\ \sum_i(X_i,f_i,\alpha_i)=0</math> provided that
::<math>\sum_if_{i*}\alpha_i\equiv 0,</math>

:where

:<math>dim \;f_i(X_i)=k</math> for all <math>i</math> and the push-forwards <math>f_{i*}\alpha_i</math> are considered on the smooth part of <math>\cup_i f_i(X_i)</math>.

===Defining the boundary operator ===

The boundary operator <math>\partial:\; C_k \mapsto C_{k-1}</math> is defined by

:<math>\partial(X,f,\alpha)=2\pi \sqrt{-1}\sum_i(V_i, f_i, res_{V_i}\,\alpha)</math>,

where <math>V_i</math> are components of the polar divisor of <math>\alpha</math>, ''res'' is the [[Poincaré residue]], and <math>f_i=f|_{V_i}</math> are restrictions of the map ''f'' to each component of the divisor.

Khesin and Rosly proved that this boundary operator is well defined, and satisfies <math>\partial^2=0</math>. They defined the '''polar cohomology''' as the quotient <math> \operatorname{ker}\; \partial / \operatorname{im} \; \partial</math>.

== Notes ==

* B. Khesin, A. Rosly, ''[http://arxiv.org/abs/math/0102152 Polar Homology and Holomorphic Bundles]'' Phil. Trans. Roy. Soc. Lond. A359 (2001) 1413-1428

[[Category:Complex manifolds]]
[[Category:Several complex variables]]
[[Category:Homology theory]]

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