By John Michael Schlessinger
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Extra resources for Infinitesimal deformations of singularities.
8) where as usual g(C) denotes the genus of C. (ii). Viewed as a complex manifold, X is Kobayashi hyperbolic if there are no non-constant entire holomorphic mappings g : C −→ X. Algebraic hyperbolicity was introduced and studied by Demailly in . 8) to hold for the normalizations of embedded curves, but it is easily seen using Riemann–Hurwitz that this is equivalent to the condition stated above. Similarly, the absence of entire holomorphic mappings is usually not taken as the definition of hyperbolicity, but for compact targets it is equivalent to the standard definition thanks to a theorem of Brody.
13. (Formal functions along ample subvarieties). Assume that d = dim X ≥ 1 and that N = NX/M is ample, and consider the formal completion M = M/X of M along X. (i). e. H 0 M , OM/X = C. (ii). Given a locally free sheaf E on M , denote by E its completion along X. Then H 0 M , E is finite dimensional. Recall that if U ⊇ X is any connected neighborhood of X in M , then the natural map H 0 U, E|U −→ H 0 M , E is injective. The same is true if U is a connected neighborhood in the classical topology and the group on the left is replaced by the space H 0 Uan , Ean of holomorphic sections of E on U .
Finally, in the spirit of the Lefschetz hyperplane theorem, one can attempt to compare the topology of X and M . Assume now that M is projective. Napier and Ramachandran  used L2 -methods to prove that if N = NX/M is ample, then the image of the map π1 (X) −→ π1 (M ) on fundamental groups has finite index in π1 (M ). 19. (Analogue of theorem of Napier–Ramachandran). Let M be a connected complex projective manifold, and let X ⊆ M 34 Chapter 6. Ample and Nef Vector Bundles be a smooth irreducible subvariety with N = NX/M ample.
Infinitesimal deformations of singularities. by John Michael Schlessinger