Poincaré invariants

Abstract

We construct an obstruction theory for relative Hilbert schemes in the sense of [BF] and compute it explicitly for relative Hilbert schemes of divisors on smooth projective varieties. In the special case of curves on a surface V , our obstruction theory determines a virtual fundamental class [[Hilb m V ]] ∈ A m(m−k)

Degeneracy loci, virtual cycles and nested Hilbert schemes, I

“…So we may assume both β, β ∨ are effective and vd β ≥ 0. By [DKO,Corollary 3.15] this implies vd β = 0 and h 1 (O S ) = 1. So, just as in (6.4), O (1) is trivial on the 0-dimensional [S β ] vir and we may replace CO [n 1 ,n 2 ] β in (6.4) by Rπ * L−RH om π (I 1 , I 2 ⊗L) for one L ∈ Pic β (S).…”

“…So we assume that AJ * [S β ] vir = 0. By [DKO,Definition 3.19] this means that β is a basic class. And by [DKO,Proposition 3.20] S is of simple type, which implies that vd β = 0.…”

“…By Serre duality down π and the identity c i (E) = (−1) i c i (E ∨ ), this is (−1) n 1 +n 2 c n 1 +n 2 Rπ * L β − RH om π (I 1 , (6.9) We have noted that h 1 (O S ) = 1, so χ(O S ) = 0. Thus, identifying Pic β (S) ∼ = Pic β ∨ (S) via L → K S ⊗ L −1 , [DKO,Theorem 3.16] gives…”

Localized Donaldson-Thomas theory of surfaces

“…As a result S [n] β is equipped with a virtual fundamental class denoted by [S [n] β ] vir . This allows us to define new invariants for S recovering in particular Poincaré invariants of [DKO07], and (after reduction) stable pair invariants of [KT14].…”

On the Fer expansion: Applications in solid-state nuclear magnetic resonance and physics

“…Poincaré's integral invariant is the most important invariant in Hamiltonian dynamics. For any phase space set, the addition of the regions of all of its orthogonal projections onto all the non-intersection canonically conjugate planes is invariant under Hamiltonian evolution [271]. Channel and Scovel [232] have also presented some numerical hydrodynamics examples to show that symplectic integrator algorithm is extremely stable.…”