A note on resolution of rational and hypersurface singularities

It is well known that the exceptional set in a resolution of a rational surface singularity is a tree of rational curves. We generalize the combinatoric part of this statement to higher dimensions and show that the highest cohomologies of the dual complex associated to a resolution of an isolated rational singularity vanish. We also prove that the dual complex associated to a resolution of an isolated hypersurface singularity is simply connected. As a consequence, we show that the dual complex associated to a resolution of a 3-dimensional Gorenstein terminal singularity has the homotopy type of a point.


Introduction
Let o ∈ X be an isolated singularity of an algebraic variety (or an analytic space) X defined over a field of characteristic 0, dim X ≥ 2. Consider a good resolution f : Y → X (this means that the exceptional locus Z ⊂ Y of f is a divisor with simple normal crossings). Let Z = Z i , where Z i are irreducible. To the divisor Z we can associate the dual complex Γ(Z). It is a CW-complex whose cells are standard simplexes ∆ j i 0 ...i k corresponding to the irreducible components Z j i 0 ...i k of the intersections If X and Y are surfaces, then Γ(Z) is the usual resolution graph of f . Note that Γ(Z) is a simplicial complex iff all the intersections Z i 0 ∩ · · · ∩ Z i k are irreducible. This can be obtained for a suitable resolution. Also note that if dim X = n, then dim(Γ(Z)) ≤ n − 1.
The complex Γ(Z) was first studied by G. L. Gordon in connection to the monodromy in families (see [8]). We say that Γ(Z) is the dual complex associated to the resolution f . The main reason motivating the study of the dual complex is the fact that the homotopy type of Γ(Z) depends only on the singularity o ∈ X but not on the choice of a resolution f . This is a consequence of the Abramovich-Karu-Matsuki-W lodarczyk Weak Factorization Theorem in the Logarithmic Category (see [2]). Indeed, this theorem reduces the problem to the case of a blowup σ : (X ′ , Z ′ ) → (X, Z), where X and X ′ are smooth varieties with divisors Z and Z ′ with simple normal crossings, and the center of the blowup is admissible in some sense. It can be explicitly verified that Γ(Z) is homotopy equivalent to Γ(Z ′ ) (see [17]).
For example (it is taken from [8]), consider the singularity A good resolution can be obtained just by blowing up the origin. The reader can easily prove that the exceptional divisor Z consists of n hyperplanes in general position in P n−1 . We see that the complex Γ(Z) is the border of a standard n − 1-dimensional simplex and thus it has the homotopy type of the sphere S n−2 .
is a resolution of an isolated toric singularity (X, o), then the complex Γ(Z) has the homotopy type of a point ( [17]).
In this paper, we study the dual complex associated to a resolution in the case when X is a rational or a hypersurface singularity defined over the field C of complex numbers. We show that if f : Y → X is a good resolution of an isolated rational singularity o ∈ X, dim X = n, then H n−1 (Γ(Z), C) = 0 (see Theorem 2.2). The proof is a generalization of M. Artin's argument from [4] to the n-dimensional case. The main new ingredient is the lemma on the degeneracy of a spectral sequence associated to a divisor with simple normal crossings on a Kähler manifold (Lemma 2.4). If X is an isolated hypersurface singularity, dim X ≥ 3, then π(Γ(Z)) = 0 (see Theorem 3.1). This result is based on the well known fact that the link of an isolated hypersurface singularity of dimension n ≥ 3 is simply connected ( [15]). These results allow to prove that the homotopy type of the dual complex associated to a resolution of an isolated rational hypersurface 3-dimensional singularity is trivial (Corollary 3.3). As an application, we show that the dual complex associated to a resolution of a 3-dimensional Gorenstein terminal singularity has the homotopy type of a point (Corollary 3.4).
We prove our theorems for algebraic varieties but everything holds also for analytic spaces (with obvious changes).
The author is grateful to V. A. Iskovskikh, K. Matsuki, Yu. G. Prokhorov, J. Steenbrink and J. Whal for written and oral consultations that were very stimulating and useful. Also we thank to P.
Popescu-Pampu who noticed a mistake in the first version of the paper, to D. Arapura, P. Bakhtari, and J. W lodarczyk who corrected a mistake in the proof of Lemma 2.4, and to the referee of [18] who suggested several improvements to the text.

Rational singularities
Recall the Definition 2.1. An algebraic variety (or an analytic space) X has rational singularities if X is normal and for any resolution f : In the sequel, when we say that f is a good resolution we additionally assume that all the intersections Z i 0 ∩ · · · ∩ Z i k of prime components of the exceptional divisor Z = Z i of f are irreducible, thus Γ(Z) is a simplicial complex.
The following theorem can be considered as a generalization of the classical fact that the exceptional locus in a resolution of a rational surface singularity is a tree of rational curves ( [4]).
Theorem 2.2. Let o ∈ X be an isolated rational singularity of a variety (or an analytic space) X of dimension n ≥ 2, and let f : Y → X be a good resolution with the exceptional divisor Z. Then the highest (complex) cohomologies of the complex Γ(Z) vanish: Z i be the decomposition of the divisor Z to its prime components Z i . We can assume that X is projective (since the given singularity is isolated) and f is obtained by a sequence of smooth blowups (Hironaka's resolution [13]). Thus all Z i and Y are Kähler manifolds.
The sheaves R i f * O Y are concentrated at the point o. Via Grothendieck's theorem on formal functions (see [12], (4.2.1), and [10], Ch. 4, Theorem 4.5 for the analytic case) the completion of the stalk of the where (r) = (r 1 , . . . , r N ) and there is a natural surjective map g of sheaves on Z: Since dimension of Z is n − 1, the map g induces a surjective map of cohomologies . Recall that the given singularity o ∈ X is rational, and thus the projective limit (1) is 0. Therefore the cohomology group H n−1 (Z, O Z ) vanishes too (because the projective system in (1) is surjective). Now it follows from the lemma 2.3 below that H n−1 (Γ(Z), C) = 0.
Then the k-th cohomologies with coefficients in C of the complex Γ(Z) vanish too: Proof. Let us introduce some notation.
..ip of differential forms of bidegree (0, q) on Z p and their direct images K p = ϕ p * O Z p and K p,q = ϕ p * A p,q on Z. The sequence of sheaves {K p } p≥1 forms a complex via the combinatoric differentials δ p : is a section of the sheaf K p over an open set U ⊆ Z, then Note that there is also a natural injection of . . is exact. This is easy to check by considering the stalks; in particular, the exactness at K 0 is a consequence of the following fact which holds locally in a sufficiently small neighborhood of every point of Z: if {f i } is a collection of regular functions on Z i such that their restrictions onto intersections Z i ∩ Z j coincide, then there exists a regular function f on Z such that f | Z i = f i for all i (it is important here that the divisor Z has normal crossings). Therefore the complexes . . are quasiisomorphic. It is clear that the hypercohomologies of the first complex are isomorphic to the cohomologies of Z with coefficients in the structure sheaf: . Now let us calculate the hypercohomologies of the complex K * by using the acyclic resolutions where∂ is the Dolbeaux differential.
There is a filtration F p K n = p ′ +q=n p ′ ≥p K p ′ ,q on the complex K * . It is known (see [11], Ch. 3, section 5) that the spectral sequence E r associated to the filtration F p K n converges to the cohomologies H * (K * , d) and E p,q 0 ≃ K p,q ; E p,q 1 ≃ H q ∂ (K p, * ) ; E p,q 2 ≃ H p δ (H q ∂ (K * , * )) . In particular, E p,0 is isomorphic to the cochain complex that one uses to calculate cohomologies of Γ(Z) (here we denote by the same letter δ the map between cohomologies induced by the combinatoric differential). It follows that E p,0 2 ≃ H p (Γ(Z), C) . We shall show that the spectral sequence E r degenerates in E 2 . The method of the proof is based on the standard technique of the theory of the mixed Hodge structures. We learned it from [14], Chapter 4, §2. Compare also [9]. Since the result about E r can be of a particular interest, we state it as a separate lemma.
Lemma 2.4. Let Z = Z i be a reduced divisor with simple normal crossings on a compact Kähler manifold Y , and let E r be the associated spectral sequence as described above. Then d r = 0 for all r ≥ 2, i. e., this spectral sequence degenerates in E 2 .
Remark. Our proof of this lemma contained in [18] and in the previous version of this preprint is incorrect. The mistake was that the restriction of a harmonic differential form onto a subvariety is generally not harmonic even on compact Kähler varieties. The mistake was fixed by D. Arapura where∂δ + δ∂ = 0. First we take cohomologies in the vertical rows and obtain the sequences . . . . . .
Here δ is the induced map between cohomologies. Then we calculate δ-cohomologies and obtain the differential (H q−1 ∂ (K * , * ) that acts as described below.
Our aim is to show that d r = 0, r ≥ 2. The differential d 2 is trivial if the representative a ∈ K p,q can be chosen in such a way that δ(a) is exactly 0 but not only 0 modulo∂K p+1,q−1 . But this is true because there are harmonic differential forms in the classā ∈ H q ∂ (K p, * ) and we can take a to be a harmonic form of pure type (0, q). Notice that since we work on a compact Kähler variety the form a is not only∂-closed but also ∂ +∂-closed, where ∂ +∂ is the complex de Rham differential. The form δa is defined by means of restrictions onto subvarieties and linear operations. We can not claim that it is also harmonic, but it remains ∂ +∂-closed and of pure type (0, q). But it is 0 mod (∂K p+1,q−1 ), i. e., ∂-exact. It follows from the ∂∂-lemma ( [11], Ch. 1, section 2) that δa is also ∂-exact and thus is exactly 0. Further, this δa = 0 can be lifted to a ′ = 0 in K p+1,q−1 , thus δ(a ′ ) = 0 and so forth. This shows that d r = 0 also for all r ≥ 3. Now let us come back to the proof of Lemma 2.3. We have If H p (Γ(Z)) is not trivial, then also H p (Z, O Z ) is not trivial.

Hypersurface singularities
Theorem 3.1. Let o ∈ X be an isolated hypersurface singularity of an algebraic variety (or an analytic space) X of dimension at least 3 defined over the field C of complex numbers. If f : Y → X is a good resolution of o ∈ X, Z its exceptional divisor, then the fundamental group of Γ(Z) is trivial: Proof. Let n be dimension of X. We can assume that X is a hypersurface in C n+1 and the singular point o coinsides with the origin. Consider the link M of singularity o ∈ X, i. e., the intersection of X with a sphere S 2n+1 of sufficiently small radius around the origin. The link M is an (n − 2)-connected smooth manifold ( [15], Corollary 2.9, Theorem 5.2), in particular, M is simply connected.
We can also consider M as the border of a tubular neighborhood of the exceptional divisor Z ⊂ Y . It is known (see [3]), that there is a surjective map ϕ : M → Z whose fibers are tori. It follows that the induced map ϕ * : π(M) → π(Z) is also surjective and hence π(Z) = 0.
It remains to show that π(Z) = 0 implies π(Γ(Z)) = 0. It is enough to construct a surjective map ψ : Z → Γ(Z) with connected fibers. The following lemma is, essancially, a partial case of the general construction of a map from a topological space Z = ∪Z i to the nerve Γ(Z) corresponding to the covering {Z i } (see [5], p. 355). As in section 2, we assume that the intersections Z i 0 ...ip are irredusible so that Γ(Z) is a simplicial complex.
Lemma 3.2. Let Z be a divisor with simple normal crossings on an algebraic veriety or an analytic space X, and let Γ(Z) be the corresponding dual complex. Then there exists a map ψ : Z → Γ(Z) which is (i) simplicial in some triangulations of Z and Γ(Z), (ii) surjective, and (iii) has connected fibers.
Proof. First, let us take a triangulation Σ ′ of Z such that all the intersections Z i 0 ...ip are subcomplexes. Next we make the barycentric subdivision Σ of Σ ′ and the barycentric subdivision of the complex Γ(Z). Now let v be a vertex of Σ belonging to the subcomplex Z i 0 ...ip but not to any smaller subcomplex Then let ψ(v) = the center of the simplex ∆ i 0 ...ip .
This determins the map ψ completely as a simplicial map (depending on the triangulation Σ ′ ). It is clear from our construction that ψ is surjective. We claim that the map ψ has connected fibers. Indeed, first observe that ψ can be represented as a composition of topological contractions of connected subcomplexes By an open neighborhood of a subcomplex K we mean the union of interior points of all simplicial stars of Σ with centers on K. All complexes in (2) are connected because the codimension of Further, the contraction of a subcomplex K from (2) can be factored into one-by-one contraction of maximal simplexes of K. The preimage of every connected set under such contraction is connected since the preimage of every simplex is a simplex. Therefore the map ψ has all the needed properties.
Some important types of singularities are rational hypersurface. Combining Theorems 2.2 and 3.1, we can obtain some precise results in the 3-dimensional case. Corollary 3.3. Let o ∈ X be an isolated rational hypersurface singularity of dimension 3. If f : Y → X is a good resolution with the exceptional divisor Z, then the dual complex Γ(Z) associated to the resolution f has the homotopy type of a point.
For instance, 3-dimensional Gorenstein terminal singularities are exactly isolated compound Du Val points (up to an analytic equivalence, for details see [16]). Here it is sufficient to us that compound Du Val points are hypersurface singularities. On the other hand, terminal singularities (and, moreover, canonical) are rational (see [7]). Combining these results with Corollary 3.3, we get Corollary 3.4. Let o ∈ X be a 3-dimensional Gorenstein terminal singularity and let f : Y → X be a good resolution with the exceptional divisor Z. Then the dual complex Γ(Z) of f has the homotopy type of a point.