Thom–Sebastiani Theorem

In complex analysis, a branch of mathematics, the Thom–Sebastiani Theorem states: given the germ $f:({ℂ}^{n_1+n_2},0)\to (ℂ,0)$ defined as $f(z_1,z_2)=f_1(z_1)+f_2(z_2)$ where $f_i$ are germs of holomorphic functions with isolated singularities, the vanishing cycle complex of $f$ is isomorphic to the tensor product of those of $f_1,f_2$. Moreover, the isomorphism respects the monodromy operators in the sense: $T_{f_1}\otimes T_{f_2}=T_f$.

The theorem was introduced by Thom and Sebastiani in 1971.

Observing that the analog fails in positive characteristic, Deligne suggested that, in positive characteristic, a tensor product should be replaced by a (certain) local convolution product.

References

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