Wonderful resolutions and categorical crepant resolutions of singularities

# Wonderful resolutions and categorical crepant resolutions of singularities

Roland Abuaf 111Institut Fourier, 100 rue des maths, 38402, Saint Martin d’Hères, France. E-mail :roland.abuaf@ujf-grenoble.fr
###### Abstract

Let be an algebraic variety with Gorenstein singularities. We define the notion of a wonderful resolution of singularities of by analogy with the theory of wonderful compactifications of semi-simple linear algebraic groups. We prove that if has rational singularities and has a wonderful resolution of singularities, then admits a categorical crepant resolution of singularities. As an immediate corollary, we get that all determinantal varieties defined by the minors of a generic square/symmetric/skew-symmetric matrix admit categorical crepant resolution of singularities.

## 1 Introduction

Let be an algebraic variety over . Hironaka proved in [hiro] that one can find a proper birational morphism , with smooth. Such a is called a resolution of singularities of . Unfortunately, given an algebraic variety , there is, in general, no minimal resolution of singularities of . In case is Gorenstein, a crepant resolution of (that is a resolution such that ) is often considered to be minimal. The conjecture of Bondal-Orlov (see [BO]) gives a precise meaning to that notion of minimality:

###### Conjecture 1.0.1

Let be an algebraic variety with canonical Gorenstein singularities. Assume that has a crepant resolution of singularities . Then, for any other resolution of singularities , there exists a fully faithful embedding:

 Db(~X)↪Db(Y).

Varieties admitting a crepant resolution of singularities are quite rare. For instance, non-smooth Gorenstein -factorial terminal singularities (e.g. a cone over , for even , see [kuz], section ) never admit crepant resolution of singularities. Thus, it seems natural to look for minimal resolutions among categorical ones. Kuznetsov has given the following definition ([kuz]):

###### Definition 1.0.2

Let be an algebraic variety with Gorenstein and rational singularities. A categorical resolution of singularities of is a triangulated category with a functor such that:

• there exists a resolution of singularities such that is admissible and ,

• we have and for all :

 RπT∗Lπ∗TF≃F,

where is the left adjoint to .

If for all , there is a quasi-isomorphism:

 Lπ∗TF≃Lπ!TF,

where is the right adjoint of , we say that is weakly crepant.

Finally, if has a structure of module category over and the identity is a relative Serre functor for with respect to , then is said to be strongly crepant.

Obviously, if is a strongly crepant resolution of , then it is also a weakly crepant resolution of . The converse is false, as shown is section of [kuz]. If is a crepant resolution of , the one easily shows that is a strongly crepant categorical resolution. The main result of [kuz] is the:

###### Theorem 1.0.3

Let be an algebraic variety with Gorenstein rational singularities. Let be a resolution of singularities with a positive integer such that , where is the scheme-theoretic exceptional divisor of . Assume moreover that we have a semi-orthogonal decomposition:

 Db(E)=⟨OE(mE)⊗Bm,…,OE(E)⊗B1,B0⟩,

with:

 Lπ∗Db(π(E))⊂Bm⊂⋯⊂B1⊂B0,

then admits a categorical weakly crepant resolution of singularities.

Assume moreover that , then admits a categorical strongly crepant resolution of singularities.

As a consequence, Kuznetsov obtains (see [kuz], sections and ) the:

###### Corollary 1.0.4

The following varieties admit a categorical strongly crepant resolution of singularities:

• a cone over (odd ),

• a cone over (any ),

• the Pfaffian variety : (odd ).

The following varieties admit categorical weakly crepant resolution of singularities:

• a cone over a smooth Fano variety in its anti-canonical embedding,

• the Pfaffian variety (even ).

Of course, one would like to generalize Kuznetsov’s result, to apply it to higher corank determinantal varieties for instance. Using Kodaira relative vanishing theorem and some adjunction formulae, it is not difficult to prove the following (this is the case of Proposition LABEL:keypropbis):

###### Proposition 1.0.5

Let be an algebraic variety with Gorenstein rational singularities. Let be a resolution of singularities such that the exceptional divisor of is irreducible, smooth and flat over . Then there exists a positive integer such that:

 K~X=KX⊗O~X(mE),

and we have a semi-orthogonal decomposition:

 Db(E)=⟨OE(mE)⊗Bm,…,OE(E)⊗B1,B0⟩,

with .

As a consequence of this proposition, we get a first mild generalization of the first part of Kuznetsov’s theorem:

###### Theorem 1.0.6

Let be an algebraic variety with Gorenstein rational singularities. Let be a resolution of singularities such that the exceptional divisor of is irreducible, smooth and flat over . Then admits a categorical weakly crepant resolution of singularities.

Note that all examples in Corollary 1.0.4 satisfy the hypotheses of Theorem 1.0.6.

Now, one remembers the strong version of Hironaka’s theorem ([hiro]) : any variety can be desingularized by a sequence of blow-ups such that every exceptional divisor is flat over its center of blowing-up and the total exceptional divisor of the resolution has (with its reduced structure) simple normal crossings. So, one could hope to get a very far-reaching generalization of Kuznetsov’s result : any Gorenstein variety with rational singularities admits a categorical weakly crepant resolution of singularities.

Unfortunately, things are not so simple. Indeed, in order to construct a categorical crepant resolution starting from a sequence of blow-ups which desingularizes our variety, one needs strong compatibility conditions between the semi-orthogonal decompositions of the derived categories of the various exceptional divisors. Those compatibility conditions can be formulated at the categorical level (see Proposition LABEL:keypropbis). But one would rather like to know geometric situations where these compatibility conditions are satisfied. I have thus formalized the notion of wonderful resolution of singularities (see definition 2.1.2 of the present paper), which applies to a sequence of blow-ups:

 Xn→⋯→X1→X,

giving a resolution of singularities of . It is remarkable that this definition, which I made up to describe geometrically some compatibility conditions among the derived categories of the exceptional divisors of a resolution of singularities, happens to be the one which perfectly identifies the resolution process of the boundary divisor for the most basic wonderful compactifications of semi-simple linear algebraic groups (see [procesi-decon] for details). With this notion in hand, quite technical but predictable computations yields the:

###### Theorem 1.0.7 (Main Theorem)

Let be an algebraic variety with Gorenstein rational singularities. Assume that has a wonderful resolution of singularities. Then admits a categorical weakly crepant resolution of singularities.

At first glance, the notion of wonderful resolution of singularities seems to be quite restrictive and one could naively guess that there are too few examples of such resolutions. However, some reformulations of the work of Vainsencher [vain] and Thaddeus [thad] show that it is not the case. Indeed, we have the:

All determinantal varieties (square as well as symmetric and skew-symmetric) admit wonderful resolution of singularities.

As a consequence, we get the:

###### Corollary 1.0.9

All Gorenstein determinantal varieties (square as well as symmetric and skew-symmetric) admit categorical weakly crepant resolutions of singularities.

Let us now briefly indicate the plan of the paper. In section , we give the definition of a wonderful resolution of singularities and study its basic cohomological properties. We also exhibit some examples of varieties which have a wonderful resolution of singularities. In section , we prove the main theorem. This is the technical core of the paper. In section , we discuss some minimality properties for categorical crepant resolutions of singularities and some existence problems related to prehomogeneous spaces.

Aknowledgements : I would like to thank Sasha Kuznetsov for many interesting discussions on categorical resolutions of singularities and Christian Lehn for many helpful comments on the first drafts of this paper. I would also like to thank Laurent Manivel for his constant support and insightful criticism during the preparation of this work.

## 2 Wonderful resolutions of singularities

We work over the field of complex numbers. An algebraic variety is a reduced algebraic scheme of finite type over (in particular it may be reducible). For any proper morphism of schemes of finite type over , we denote by the total derived functor , by the total derived functor and by the right adjoint functor to . In case we need to use specific homology sheaves of these functors, we will denote them by and .

### 2.1 Wonderful resolutions

Let be a closed irreducible subvariety of . We say that is a normally flat center in X if the natural map:

 E→Y

is flat, where is the exceptional divisor of the blow up of along . Hironaka proved in [hiro] that any algebraic variety can be desingularized by a finite sequence of blow-ups along smooth normally flat centers.

###### Example 2.1.1

Let be a vector space of dimension at least and let be the nd Pfaffian variety in . Then is singular exactly along . Here is a smooth normally flat center for X. Indeed, if is the exceptional divisor of the blow-up of along , then is the flag variety and the natural map onto is given by the second projection. Obviously it is flat. See [kuz], section 8 for more details.

Given a semi-simple affine algebraic group, one often wants to find a good equivariant compactification of . Equivariant compactifications of for which the boundary divisors have simple normal crossings are called wonderful in the literature (see [hurug], chapter for instance). One notices that the most basic wonderful compactifications we know are obtained by the following procedure. Take be a naive 222For instance, let be a linear representation of and consider the closure of in . equivariant compactification of . Then find an embedded resolution of the boundary divisor in such that it’s smooth model has simple normal crossings with the exceptional divisors of the modification of . This embedded resolution is obtained by a succession of blow-ups along smooth centers which satisfy nice intersection properties. The following definition captures the most essential features of this sequence of blow-ups.

###### Definition 2.1.2 (Wonderful resolutions)

Let be an algebraic variety with Gorenstein singularities. For all , we define a -step wonderful resolution of singularities in the following recursive way:

• A -step wonderful resolution is a single blow up:

 π:~X→X,

over a smooth normally flat center , such that and the exceptional divisor are smooth.

• For , a -step wonderful resolution of X is a sequence of blow-ups:

 Xn\lx@stackrelπn→Xn−1…X1\lx@stackrelπ1→X0=X

over smooth normally flat centers such that:

1. all the are Gorenstein,

2. the map is a -step wonderful resolution of ,

3. the intersection of with is proper and smooth (where is the total transform of , the exceptional divisor of , with respect to , for ),

4. the map is a -step wonderful resolution of singularities.

As far as I know, the term wonderful resolution first appeared in [fu-chap] where it was used to describe the resolution of indeterminacies of a stratified Mukai flop.

###### Example 2.1.3 (Determinantal varieties)

Let a vector space of dimension . Let and let be the subvariety of defined by the vanishing of the minors of size . It is well known that is Gorenstein with rational singularities (see [weyman], corollary ). We define to be the blow up:

 π1:X1→X,

of along , where is the subvariety of defined by the minors of size . For , we define recursively to be the blow-up:

 πk:Xk→Xk−1,

of along , where is the strict transform through of the subvariety of defined by the vanishing of the minors of rank . By theorem of [vain], we know that is smooth and that all for are smooth. Moreover, theorem of [vain] shows that the are normally flat centers and that item , , and in the definition of a wonderful resolution are satisfied for the following resolution of :

 Xr−1\lx@stackrelπr−1→…X1\lx@stackrelπ1→X

Thus, has Gorenstein rational singularities and admits a wonderful resolution of singularities.

Let (resp. denotes the determinantal variety defined by the vanishing of the minors of size of the generic symmetric (resp. skew-symmetric) matrix with linear entries. If is even (resp. no conditions), we know by [weyman], corollary (resp. proposition ) that (resp. ) is Gorenstein with rational singularities. Moreover, the appendices and of [thad] show that and also admit wonderful resolutions of singularities.

###### Example 2.1.4 (Secant variety of OP2)

Let be the Cayley plane into his highest weight embedding, where is the maximal parabolic associated to the root of the root system. The Cayley plane can also be recovered as the scheme defined by the minors of the generic hermitian octonionic matrix:

 M=⎛⎜⎝a1a2a3¯¯¯¯¯a2a4a5¯¯¯¯¯a3¯¯¯¯¯a5a6⎞⎟⎠,withai∈Oand such thatt¯¯¯¯¯¯M=M,

where is the octonionic conjugate of .

Let be the secant variety of inside . It can be seen as the scheme defined by the determinant of the above matrix . The variety is a cubic hypersurface (so it is Gorenstein) which is singular exactly along . Let

 π:˜S(X)→S(X)

be the blow up of along . It is smooth and the exceptional divisor is isomorphic to (where is the parabolic associated to the roots and ). It is a fibration into smooth -dimensional quadrics over . As a consequence, the map is a wonderful resolution of singularities. We refer to [zak] ch. III and [manifaen] for more details on the beautiful geometric and categorical features of the Cayley plane.

### 2.2 Wonderful resolutions and singularities of the intermediate divisors

The above examples also suggest that the definition of a wonderful resolution imposes strong conditions on the singularities of the exceptional divisor . Indeed, the following three propositions show that they must be similar to the singularities of .

###### Proposition 2.2.1

Let be an algebraic variety with Gorenstein and rational singularities. Let be a wonderful resolution of singularities of . Then, for all , the varieties and the exceptional divisors have Gorenstein rational singularities.

Proof :
The fact that the are Gorenstein is in the definition of a wonderful resolution of singularities. Let be the blow-up along with exceptional divisor and let be the irreducible components of . We also denote by the total transform of under . Since the meet transversally the for all , the coefficients appearing in front of the in the expression of are the same as the coefficients in front of the in the expression of . Now is Gorenstein with rational singularities, hence it has canonical singularities by [kollar] corollary . Thus, the coefficients in front of the in the expression of are positive, and so are the coefficient in front of the in the expression of . As a consequence, we have . Since we also have , we find that has rational singularities. An obvious induction shows that all have rational singularities.

The divisors are obviously Gorenstein, as they are Cartier divisors inside Gorenstein varieties. The main point of the proposition is thus to show that the have rational singularities for any .

Item of definition 2.1.2 implies that for any the map

 πk|E(k)1:E(k)1→E(k−1)1

is the blow-up along (we recall that is the total transform of through ). Thus, we deduce from item of definition 2.1.2 that for any , the map

 Πk=πk+1…πn|E(n)1:E(n)1→E(k)1

is a resolution of singularities. Hence, to prove that has rational singularities we only have to compute .

For all , we have a fibered diagram:

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