When is a complete toric variety a Poincare duality space? Is there an "if and only if" condition? And is this condition local? Given an analytically-locally-toric compactification of a smooth variety, can the condition be described purely in terms of the local cones?
$\begingroup$
$\endgroup$
4
-
5$\begingroup$ Not sure from the question if you want integrally or rationally? An easy condition is that if for every cone in the fan the rays are linearly independent (I think such cones are called simplicial), then the toric variety has quotient singularities and Poincare duality holds rationally. $\endgroup$– Evgeny ShinderCommented Feb 26, 2021 at 23:14
-
$\begingroup$ Thanks for the question, I only wanted PD rationally. $\endgroup$– Philip EngelCommented Feb 27, 2021 at 13:52
-
$\begingroup$ The example I was thinking about was not simplicial, but is very close: The number of generators of any maximal cone exceeds the dimension by exactly 1. $\endgroup$– Philip EngelCommented Feb 27, 2021 at 14:02
-
1$\begingroup$ Philip Engel: most likely PD won't hold in this case. E.g. for threefolds you have 4 rays in maximal cones, and I think $H_2(X)$ and $H_4(X)$ will typically have different dimensions. One way to approach this is to subdivide the cones to make an orbifold resolution $X' \to X$ and compare homology of $X$ and of $X'$. $\endgroup$– Evgeny ShinderCommented Feb 27, 2021 at 14:54
Add a comment
|