Rationality conjecture for residues of 2-flags of holomorphic foliations

Let F=(F1,F2)\mathcal{F}=(\mathcal{F}_{1},\mathcal{F}_{2}) be a 2-flag of holomorphic foliations on a complex manifold MM. Let SS be a compact connected component of the singular set of the flag, and let φ=(φ1,φ2)\varphi=(\varphi_{1},\varphi_{2}), where each φi\varphi_i is a homogeneous symmetric polynomial of degree did_i satisfying the referenced condition.

Rationality conjecture. If the polynomials φi\varphi_i have rational coefficients, then

Resφ1,φ2(F,NF,S)H2n2(d1+d2)(S;Q).\operatorname{Res}_{\varphi_{1},\varphi_{2}}(\mathcal{F},\mathcal{N}_{\mathcal{F}},S)\in H_{2n-2(d_{1}+d_{2})}(S;\mathbb{Q}).

The claim asserts that residues associated with rational characteristic polynomials define homology classes with rational coefficients. The supplied text does not indicate whether this result is proved or remains open.

Sources & referencesView supporting material

Primary source

Jean-Paul Brasselet, Maurício Corrêa and Fernando Lourenço, “Residues for flags of holomorphic foliations”, arXiv:1602.09136 (2017).

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