Thom–Sebastiani Theorem

In complex analysis, a branch of mathematics, the Thom–Sebastiani Theorem states: given the germ $f:(\mathbb {C} ^{n_{1}+n_{2}},0)\to (\mathbb {C} ,0)$ 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

• Illusie, Luc (24 April 2016). "Around the Thom-Sebastiani theorem". arXiv:1604.07004. {{cite journal}}: Cite journal requires |journal= (help)