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Hodge theory and the Hilbert scheme. (English) Zbl 0804.14004
Let $$X$$ be an algebraic manifold and $$Y \subset X$$ a submanifold with normal bundle $$N$$.
Grothendieck’s estimate states that any component $$H$$ of the Hilbert scheme $$\text{Hilb}_ X$$ containing $$\{Y\}$$ satisfies $$\dim H \geq h^ 0 N - h^ 1N$$. In 1972, Bloch improved this result and he proved that if the semiregularity map $$\pi : H^ 1N \to H^{p + 1} (\Omega_ X^{p - 1})$$, $$p = \text{codim} (Y,X)$$, is injective then $$\dim H = h^ 0N$$ and hence $$H$$ is smooth at $$\{Y\}$$.
In this paper the author improves Bloch’s estimate and he shows that $\dim H \geq h^ 0 N - h^ 1N + \dim (\text{im} (\pi)).$ He also finds interesting applications of this result.

##### MSC:
 14C05 Parametrization (Chow and Hilbert schemes) 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
##### Keywords:
normal bundle; Hilbert scheme
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