On the existence of analytic contractions

Algebraic approximations of compact Kähler manifolds of algebraic codimension 1

“…In general, based on Ancona, Tomassini, and Bingener's work on formal modifications [4], we will observe that if X ′ is a bimeromorphic model of a given compact complex variety X and Y ⊂ X ′ a subvariety containing the locus that are modified under the bimeromorphic map X ′ X, then a deformation Π : X ′ → ∆ of X ′ induces a deformation of X X ′ provided Π induces a trivial deformation of the completion Ŷ of X ′ along Y (1) . In particular if X is a compact Kähler manifold, then an algebraic approximation of X ′ preserving Ŷ induces an algebraic approximation of X.…”

“…Let Y min ⊂ X be the minimal subvariety which is modified under the induced bimeromorphic map X X and let Y := f −1 ( f (Y min )). Up to replacing f ′ with a more carefully constructed bimeromorphic model (see Step 2,3,and 4 in the proof of Lemma 5.3), we may assume that the projection Y → f (Y) has a multi-section. It is for the pair (X, Y) that we will prove that the tautological family Π associated to f : X → B contains a subfamily which is an algebraic approximation of X preserving the completion Ŷ of X along Y.…”

Cohomology Vanishing Theorems on Weakly 1-Complete Manifolds

“…Theorem 6.5 has been considerably generalized by Fujiki [9] and Bingener [7] (see also Ancona-Tomassini [33]). …”

“…Let ae Hom(V+,Vf) = F$® K* be a hermitian metric of F 4 and let i^,..., y B e F$ be a basis such that " p % cr= 2 i;,-®*',-. We define the hermitian metrics of A K*® A F* associated to (7 Here, < , > denotes the inner product with respect to a and h. Proof. Similar as above.…”

Flops

“…Proof. Let I c be the biggest ideal sheaf on X that agrees with (Θ(-D), Θ(-E))dΘ x generically along C. It is easy to check that I c and the map Spec Θ X \I C -• Spec C satisfy the contractibility criterion of Artin [Al.6.2] (see [Bi,6.1] for the analytic case).…”