Safe Haskell | None |
---|---|
Language | Haskell2010 |
Compute the non-equivariant CSM in P^n
recursively
Synopsis
- delta_star :: Partition -> ZMod US -> ZMod HS
- pi_star :: Int -> ZMod HS -> ZMod G
- tangentChernClass :: Int -> ZMod US
- smallestOrbitCSM :: Int -> ZMod G
- upperCSM :: Partition -> ZMod HS
- lowerCSM :: Partition -> ZMod G
- openCSM :: Partition -> ZMod G
- closedCSM :: Partition -> ZMod G
- highestCoeff_ :: ZMod G -> Integer
- lowestCoeff_ :: ZMod G -> Integer
- highestCoeff :: ZMod G -> (G, Integer)
- lowestCoeff :: ZMod G -> (G, Integer)
Pushforwards
delta_star :: Partition -> ZMod US -> ZMod HS Source #
A group generator on the left is a subset (=product) of U-s, which we map to a linear combinaton of H-s. This is then extended additively to the cohomology ring.
The pushforward map pi_*
along pi
.
A (cohomology) group generator above is a subset (=product) of H-s, which we map to a group generator below. This defines the map on the cohomology ring by additive extension.
Easy things
tangentChernClass :: Int -> ZMod US Source #
The total Chern class of the tangent bundle of Q^d = P^1 x P^1 x ... x P^1
This is just the product of (1+2u_i)
-s for i=[1..d]
smallestOrbitCSM :: Int -> ZMod G Source #
The CSM of the smallest orbit: 1 point with multiplicity n
,
which is just the rational normal curve in P^n
.
CSM calculation
upperCSM :: Partition -> ZMod HS Source #
We know that:
csm(im(Delta) = Delta_* c(TQ^d) c(TQ^d) = (1+2*u1) (1+2*u2) ... (1+2*ud)
From these, we can compute csm(im(Delta_nu))
recursively
lowerCSM :: Partition -> ZMod G Source #
A formula for pi_*(csm(im(delta)))
. This should satisfy
lowerCSM part = pi_star n (upperCSM part)
closedCSM :: Partition -> ZMod G Source #
To get the CSM of the closed strata, we just sum over the open strata contained in the closure.