On subtransversality (Q1095471)

Let X and Y be smooth manifolds, and A and B closed submanifolds of X and Y. Let \(g: X\to Y\) be a smooth map such that g(A)\(\subset B\); then \(C_ a^{\infty}(X)\cdot g^*I(B)_{g(a)}\subset I(A)_ a\), where \(C_ a^{\infty}(X)\) is the ring of germs of smooth functions at \(a\in A\) and \(I(A)_ a\) the ideal of germs vanishing on A. We say that g is subtransverse to B at a if the conductor ideal \(c_ g(I(A)_ a,I(B)_{g(a)})\subset C_ a^{\infty}(X)\) is regular of codimension equal to \(co\dim_{g(a)}B\), and strongly subtransverse to B at a if \(c_ g(I(A)_ a,I(B)_{g(a)})+I(A)_ a\) is regular of codimension equal to \(co\dim_ aA+co\dim_{g(a)}B\). Finally we say that g is (strongly) \(\sigma\)-subtransverse to B at a if \(g\circ \sigma\) is (strongly) subtransverse to B at any point of \(\sigma^{-1}(a)\), \(\sigma\) being the projection of the blowing up of X along A. These notions are due to A. Andreotti. The authors show that (strong) \(\sigma\)-subtransversality has a simple geometric meaning in terms of ordinary transversality in tangent and normal bundles. The connection is made via a ``completed'' bijet bundle.