Normal bundles on the exceptional sets of simple small resolutions
Rong Du, Xinyi Fang

TL;DR
This paper investigates the structure of normal bundles on exceptional sets of certain singularities in higher dimensions, extending classical results to more complex cases with specific geometric conditions.
Contribution
It generalizes Nakayama, Ando, and Laufer's results to higher dimensions under specific conditions on the exceptional set and its normal bundle.
Findings
Extended Nakayama and Ando's results to higher dimensions.
Generalized Laufer's rationality and embedding dimension results.
Analyzed normal bundles with good filtrations on exceptional sets.
Abstract
We study the normal bundles of the exceptional sets of isolated simple small singularities in the higher dimension when the Picard group of the exceptional set is and the normal bundle of it has some good filtration. In particular, for the exceptional set is a projective space with the split normal bundle, we generalized Nakayama and Ando's results to higher dimension. Moreover, we also generalize Laufer's results of rationality and embedding dimension to higher dimension.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Meromorphic and Entire Functions · Commutative Algebra and Its Applications
Normal bundles on the exceptional sets of simple small resolutions
Rong Du*†*
School of Mathematical Sciences
Shanghai Key Laboratory of PMMP
East China Normal University
Rm. 312, Math. Bldg, No. 500, Dongchuan Road
Shanghai, 200241, P. R. China
and
Xinyi Fang
School of Mathematical Sciences
Shanghai Key Laboratory of PMMP
East China Normal University
No. 500, Dongchuan Road
Shanghai, 200241, P. R. China
Abstract.
We study the normal bundles of the exceptional sets of isolated simple small singularities in the higher dimension when the Picard group of the exceptional set is and the normal bundle of it has some good filtration. In particular, for the exceptional set is a projective space with the split normal bundle, we generalized Nakayama and Ando’s results to higher dimension. Moreover, we also generalize Laufer’s results of rationality and embedding dimension to higher dimension.
† The Research Sponsored by the National Natural Science Foundation of China (Grant No. 11471116, 11531007) and Science and Technology Commission of Shanghai Municipality (Grant No. 18dz2271000).
1. Introduction
Let be a complex manifold of dimension containing a compact smooth irreducible analytic submanifold of dimension . The submanifold is called exceptional or contractible, if there exists a birational proper morphism whose exceptional set is , where may be an algebraic space or an analytic space. If the subset of is of codimension greater than or equal to , the morphism is called a small contraction, and the pair is called the small singularity. Moreover, if dim , then the singularity is called isolated simple small singularity.
When is a smooth projective surface, an algebraic curves is exceptional if and only if its normal bundle is negative. When dim, Grauert has shown that if the normal bundle of algebraic set , , is negative, then is exceptional. But the reverse is not true. For example, we know many exceptional curves whose normal bundles are not negative (cf. [La1]). However, Nakayama and Ando have some very interesting works on the normal bundles of exceptional curves in the higher dimension (cf. [An1],[An2], [Na]), especially for the exceptional set is a rational curve.
Theorem 1.1**.**
(cf. [An2]) Let be a nonsingular projective variety of dimension over , and let . Assume that surjective morphism is a contraction map with the as the exceptional set. Let the normal bundle
[TABLE]
Then we have the following inequality holds.
[TABLE]
for any .
From another perspective, when is a smooth projective variety such that the canonical bundle is not numerically effective. According to the Mori’s theory, there exists a contraction morphism such that the fibers of are connected. The major goal of Mori’s theory is to construct a minimal model for each nonuniruled birational equivalence class of varieties. Studying the structure of small contraction maps, as well as finding the associated “surgery operation” (flip), is of great importance for the minimal model program. To this end, it is essential to analyze the exceptional loci and their normal sheaves ([Mo], [Ka]). When , Kawamata ([Ka]) proved that the exceptional locus of is a disjoint union of its irreducible components such that . In general, when , Zhang ([Zh]) proved that if each irreducible component of the exceptional locus of is a smooth subvariety of dimension , then . He also gave the result when and . Later, Su-Zhao studied small contractions of odd dimensional smooth complex projective varieties. More precisely, they showed that when , if each irreducible component of the exceptional locus of is a smooth subvariety of dimension , then is isomorphic to , quadratic hypersurface , or a linear -bundle over a smooth curve. Moreover, if dim , the third case never happens. Since dim dim by [Wi], we know that the structure of is very simple when the dimension of is the minimum for a small contraction . Therefore studying the projective space as the exceptional set is a basic and important research direction, especially for the normal bundle of the exceptional set. Kachi ([Kac]) study the normal bundle of as the exceptional set of some special flip contraction. However, as far as we know, such kind of result is very few for higher dimensional exceptional set . The simple reason is that the vector bundles on are not known very clearly. So, even if the normal bundle of the exceptional set splits, the explicit split type is still unknown.
The purpose of this paper is to study the normal bundle of the exceptional set of isolated simple small singularities in the higher dimension case when the Picard group of the exceptional set is and the normal bundle of it has some good filtration. The typical example is that the exceptional set is projective space with splitting normal bundle. So we generalized Nakayama ([Na]) and Ando’s ([An2]) results to higher dimension.
Definition 1.2**.**
Let be a holomorphic vector bundle on a variety of rank and Pic. If there exists a filtration
[TABLE]
with all are invertible sheaves and , where , we call has a good filtration.
Let be a complex manifold with a compact smooth irreducible analytic submanifold with Pic . Suppose dim, dim, and .
Our main result is as follows.
Theorem 1.3**.**
Let be a complex manifold of dimension and be an exceptional set of dimension of an isolated simple small singularity such that Pic . Let be the ideal sheaf of . If the conormal bundle has a good filtration, then
[TABLE]
and
[TABLE]
where
[TABLE]
and
[TABLE]
for any .
Remark 1.4**.**
There are many new restrictions of those ’s indeed if the dimension of is odd. For example, fix the same assumption as Theorem 1.3 and let dim, then
[TABLE]
and
[TABLE]
where
[TABLE]
for any .
Corollary 1.5**.**
Let be a complex manifold of dimension and be the exceptional set of an isolated simple small singularity, . If the normal bundle , then we have a system of inequalities
[TABLE]
where
[TABLE]
and or .
If we let , then we can get Nakayama and Ando’s result, Theorem 1.1.
Corollary 1.6**.**
Fix the same assumption as Theorem 1.3 and suppose dimension of is odd of codimension in , then and .
In [La1], Laufer gave a sufficient condition for the rationality of an isolated singularity when the exceptional set is . Moreover, he also calculated the Hilbert function of the singularity and gave the embedding dimension of it especially. We generalize Laufer’s results to higher dimension as follows.
Theorem 1.7**.**
Let be an exceptional set in the n-dimensional manifold . Suppose the normal bundle and for . Let is the contraction morphism, then is a rational singularity. Let be the maximal ideal of at . Let be the Hibert function for at , then
[TABLE]
In particular, at the embedding dimension of is .
2. Normal bundles of the exceptional sets
Let be a complex manifold of dimension and be an exceptional set of of dimension with Pic . Let be the ideal sheaf of and be the conormal bundle of in .
Definition 2.1**.**
For two ideals and with and Supp , there exists a filtration with every is a locally free and is a zero-dimensional sheaf. Then we define
[TABLE]
and
[TABLE]
Clearly, these are independent of the choice of a filtration by the properties of Chern classes.
Definition 2.2**.**
Two coherent sheaves and over a smooth manifold of dimension is called to be numerical equivalent (denoted , if length length and for all .
If the conormal bundle has a good filtration, then we have corresponding filtration with all are invertible sheaves and .
For any , let . Put
[TABLE]
By the similar argument by Ando in [An1], Proposition 2.5, for any nonnegative integer integer , we have the following result. We mimic Ando’s proof as follows in order to keep this paper self-contained.
Proposition 2.3**.**
[TABLE]
[TABLE]
[TABLE]
Proof.
On a small neighborhood of , we can find a set of functions on corresponding to the above filtration with and the class of modulo is a local base of the invertible sheaf . Clearly
[TABLE]
[TABLE]
Next we only prove the second assertion. Put , and . We formally put . First we prove . It is enough to show this on . Let be a monomial in of degree , be a monomial in of degree . Since and are generated by the monomials in the form , it is enough to show that if , then . Note that if and only if and . Assume but , then , but , so we have and . Since , we have . But leads to , which is impossible. Thus we have .
Since , we have . Put . By the isomorphism theorem , . Thus we have
[TABLE]
On the other hand
[TABLE]
Thus we have
[TABLE]
By the definition of numerical equivalence, we have
[TABLE]
Because , . We have
[TABLE]
For the first assertion, put , and . By repeating the above process, we are done. For the last assertion , put and we are done. ∎
The following combinatoric results will be used in the proof of the main theorem.
Theorem 2.4**.**
(Euler’s Finite Difference Theorem, cf. [Go-Qu], Section 10, Page 45)
[TABLE]
Theorem 2.5**.**
(Variations of Theorem 2.4, cf. [Go-Qu], Section 10, Page 45)
[TABLE]
Lemma 2.6**.**
For any positive integer ,
[TABLE]
where
[TABLE]
Proof.
[TABLE]
where ∎
Proof of Theorem 1.3:
Proof.
By Proposition 2.3, we have
[TABLE]
where is the exponential Chern character of . By Lemma 2.6, we know that for any,
[TABLE]
is a polynomial in a single indeterminate of degree . Therefore
[TABLE]
where
[TABLE]
is a polynomial in a single indeterminate of degree n. By Lemma 2.6, for any , we know the leading coefficient of
[TABLE]
is .
Thus the leading coefficient of the above polynomial is
[TABLE]
up to a positive constant.
Therefore by generalized Grothendieck-Hirzebruch-Riemann-Roch theorem, on the one hand, we can express the Euler characteristic as a polynomial in a single indeterminate , the leading coefficient of it is
[TABLE]
up to a positive constant.
On the other hand,
[TABLE]
However, because Z is the exceptional set,
[TABLE]
by the holomporphic functions theorem (cf. [Knu], Chapter 5, Theorem 3.1), where is the contraction morphism. Therefore,
[TABLE]
Thus there exists some constant which is independent of such that dim for all .
Thus
[TABLE]
Moreover, for any , put , we can consider the sheaf , then by the similar argument as above, we only need to focus on the leading coefficient of the polynomial . Similarly, we have
[TABLE]
where
[TABLE]
∎
If the exceptional set is and the normal bundle splits, then can have several good filtrations. So we have many inequalities.
Corollary 2.7**.**
Let be a complex manifold of dimension and be the exceptional set of an isolated simple small singularity, . If the normal bundle , then we have a system of inequalities
[TABLE]
where
[TABLE]
and or .
Remark 2.8**.**
Set , we can get Nakayama and Ando’s result, Theorem 1.1.
Proof of Corollary 1.6:
Proof.
By Theorem 1.3, we know that
[TABLE]
and
[TABLE]
where dim.
Since
[TABLE]
and
[TABLE]
we have
[TABLE]
Similarly,
[TABLE]
∎
3. Isolated small singularities
Fixed the same notations as the formal sections. When dim, Grauert has shown that if the normal bundle of algebraic set , , is negative, then is exceptional. But the reverse is not true. Even the first Chern class of the normal bundle is not negative. However, if we consider locally a Stein neighborhood of an isolated Gorenstein singularity such that the canonical divisor of is negative, then . In fact, since a small resolution is crepant and the canonical divisor is trivial, by adjunction formula, we have
[TABLE]
So which is negative.
If the exceptional set of a simple small singularity is , Laufer studied the embedding dimension of such singularity ([La1]). If the exceptional set is , we can have similar result.
Theorem 3.1**.**
Let be an exceptional set in the n-dimensional manifold . Suppose the normal bundle and for . Let is the contraction morphism, then is a rational singularity. Let be the maximal ideal of at . Let be the Hibert function for at , then
[TABLE]
In particular,at the embedding dimension of is .
Proof.
: is the defining idea of in , we formlly put . Consider the exact sheaf sequence
[TABLE]
Since for , for all and . Hence is onto for all and . By [Gra], Satz 4.2, for all and . In particular, for all . Hence is a rational singularity.
Then, as in [La2], and , hence
[TABLE]
In particular, at the embedding dimension of is . ∎
Example 3.1**.**
Let , and be with coordinates , and . We construct and by the following transition functions
[TABLE]
is contracted by
[TABLE]
It is easy to check . So is a rational singularity with embedding dimension . Here is the contraction morphism of (see [An1] Section 3).
Acknowledgements
Both authors would like to thank for the reviewers for pointing out some typos in the original version.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[An 1] T. Ando: On the normal bundle of an exceptional curve in a higher dimensional algebraic manifold , Math. Ann. 306 (1996), 625-645.
- 2[An 2] T. Ando: On the normal bundle of ℙ 1 superscript ℙ 1 \mathbb{P}^{1} in the higher dimensional projective variety , Amer. J. Math. 113(1991), 949-961.
- 3[Go-Qu] H. Gould and J. Quaintance: Combinatorial Identities: Table I: Intermediate Techniques for Summing Finite Series , https://www.math.wvu.edu/ gould/Vol.4.PDF.
- 4[Gra] H. Grauert: U ¨ ¨ 𝑈 \ddot{U} ber Modifikationen und exzeptionelle analytische Mengen , Math. Ann., 146(1962), 331-368.
- 5[Na] N. Nakayama: On smooth exceptional curves in threefolds , J. Fac. Sci. Univ. Tokyo Sect. IA. Math. 37(1990), 511-525.
- 6[Ka] Y. Kawamata: Small contractions of four dimensional algebraic manifolds , Math. Ann., 284 (1989), 595-600.
- 7[Kac] Y. Kachi: Flips from 4-folds with isolated complete intersection singularities , Amer. J. of Math., 120(1998), 43-102.
- 8[Knu] D. Knutson: Algebraic space , L.N.M. 203, Springer, 1971.
