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In complex analysis, a branch of mathematics, the Thom–Sebastiani Theorem states: given the germ ${\displaystyle f:(\mathbb {C} ^{n_{1}+n_{2}},0)\to (\mathbb {C} ,0)}$ defined as ${\displaystyle f(z_{1},z_{2})=f_{1}(z_{1})+f_{2}(z_{2})}$ where ${\displaystyle f_{i}}$ are germs of holomorphic functions with isolated singularities, the vanishing cycle complex of ${\displaystyle f}$ is isomorphic to the tensor product of those of ${\displaystyle f_{1},f_{2}}$. Moreover, the isomorphism respects the monodromy operators in the sense: ${\displaystyle T_{f_{1}}\otimes T_{f_{2}}=T_{f}}$.

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

1. Fu, Lei (30 December 2013). "A Thom-Sebastiani Theorem in Characteristic p". arXiv:1105.5210. {{cite journal}}: Cite journal requires |journal= (help)
2. ^ Illusie 2016, § 0.
3. Sebastiani, M.; Thom, R. (1971). "Un résultat sur la monodromie". Inventiones Mathematicae. 13 (1–2): 90–96. Bibcode:1971InMat..13...90S. doi:10.1007/BF01390095. S2CID 121578342.

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

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