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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
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Is the dimension of the pieces of a mixed Hodge structure constant under smooth deformations?
In the case of a family of compact complex manifolds we have the following:
Theorem. Let $f:X→B$ be a family of complex manifolds and assume that $X_0$ is Kähler for some $0\in B$. Then for $b$ in a n …