Thom polynomials of maps and bundle sections Characteristic classes in singularity theory Andrzej Weber joint with Malgorzata Mikosz and Piotr Pragacz AndrzejWeber Characteristicclassesinsingularitytheory:Thompolynomials of maps and bundle sections Characteristic classes Thom polynomials in singularity theory Andrzej Weber joint with Malgorzata Mikosz and Piotr Pragacz AndrzejWeber Characteristicclassesinsingularitytheory:Thompolynomials and bundle sections Characteristic classes Thom polynomials in singularity theory of maps Andrzej Weber joint with Malgorzata Mikosz and Piotr Pragacz AndrzejWeber Characteristicclassesinsingularitytheory:Thompolynomials Characteristic classes Thom polynomials in singularity theory of maps and bundle sections Andrzej Weber joint with Malgorzata Mikosz and Piotr Pragacz AndrzejWeber Characteristicclassesinsingularitytheory:Thompolynomials → Eg. vector field v on a manifold M P → Poincar´e–Hopf theorem: i (v) = χ(M) x ⇒ local contribution to a global invariant ⇐ non vanishing of the global invariant forces the existence of singularities Singularities Characteristic Classes ! Local Theory Global Theory AndrzejWeber Characteristicclassesinsingularitytheory:Thompolynomials ⇒ local contribution to a global invariant ⇐ non vanishing of the global invariant forces the existence of singularities Singularities Characteristic Classes ! Local Theory Global Theory → Eg. vector field v on a manifold M P → Poincar´e–Hopf theorem: i (v) = χ(M) x AndrzejWeber Characteristicclassesinsingularitytheory:Thompolynomials ⇐ non vanishing of the global invariant forces the existence of singularities Singularities Characteristic Classes ! Local Theory Global Theory → Eg. vector field v on a manifold M P → Poincar´e–Hopf theorem: i (v) = χ(M) x ⇒ local contribution to a global invariant AndrzejWeber Characteristicclassesinsingularitytheory:Thompolynomials Singularities Characteristic Classes ! Local Theory Global Theory → Eg. vector field v on a manifold M P → Poincar´e–Hopf theorem: i (v) = χ(M) x ⇒ local contribution to a global invariant ⇐ non vanishing of the global invariant forces the existence of singularities AndrzejWeber Characteristicclassesinsingularitytheory:Thompolynomials If c (ξ) 6= 0 then H 6= ∅ 1 Is H singular? Generically smooth (e.g. for ξ very ample) Singularities Σ ⊂ H have to occur in one-dimensional families ξ → M line bundle, s ∈ Γ(ξ) section H = {s = 0} hyperplane [H] = c (ξ) ∈ H2(M) 1 AndrzejWeber Characteristicclassesinsingularitytheory:Thompolynomials Is H singular? Generically smooth (e.g. for ξ very ample) Singularities Σ ⊂ H have to occur in one-dimensional families ξ → M line bundle, s ∈ Γ(ξ) section H = {s = 0} hyperplane [H] = c (ξ) ∈ H2(M) 1 If c (ξ) 6= 0 then H 6= ∅ 1 AndrzejWeber Characteristicclassesinsingularitytheory:Thompolynomials