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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.

14C05 Parametrization (Chow and Hilbert schemes)
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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