Constructible functions

Last updates: 17 February 2011

Constructible functions

Let $X$ be an algebraic variety over $ℂ$ with the Zariski topology.

• A locally closed subset is the intersection of an open set and a closed set.
• A constructible subset is a subset $A\subseteq X$ which is a finite union of locally closed subsets.
• The vector space of constructible functions on $X$ is the span of the characteristic functions of constructible subsets of $X$,  $ℳ=\mathrm{span-}\left\{{\chi }_{A}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}A\phantom{\rule{.5em}{0ex}}\text{is constructible}\right\}$. (constfcn)
Let
 $𝒵=\mathrm{span-}\left\{\phantom{\rule{.5em}{0ex}}\text{irreducible components}\phantom{\rule{.5em}{0ex}}Z\phantom{\rule{.5em}{0ex}}\text{of}\phantom{\rule{.5em}{0ex}}X\right\}$
and define ${f}_{Z}$ by requiring that ${f}_{Z}$ is equal to 1 on a dense open subset of $Z$ and equal to 0 on a dense open subset of any other irreducible component $Z\prime$.

Notes and References

This summary of the theory of constructible functions is part of joint work with A. Ghitza and S. Kannan on the relationship between MV-cycles and the Borel-Weil-Bott theorem. This presentation follows [GLS, Section 4.1].

References

[GLS] C. Geiss, B. Leclerc and J. Schröer, Semicanonical bases and preprojective algebras, Ann. Sc. École Norm. Sup. 38 (2005), 193-253. (2003), 567-588, arXiv:math/0402448, MR2144987.