Varieties of general type with small volumes
Jungkai Alfred Chen, Ching-Jui Lai

TL;DR
This paper constructs examples of higher-dimensional algebraic varieties of general type with small volumes, extending known inequalities and providing new insights into their geometric properties.
Contribution
It generalizes Kobayashi's example to higher dimensions, demonstrating the existence of n-folds of general type with small volumes.
Findings
Examples of n-folds of general type with small volumes
Extension of Noether inequality to higher dimensions
New geometric properties of algebraic varieties
Abstract
Generalize Kobayashi's example for the Noether inequality in dimension three, we provide examples of n-folds of general type with small volumes.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometric and Algebraic Topology · Algebraic Geometry and Number Theory
Varieties of general type with small volumes
Jungkai Alfred Chen, Ching-Jui Lai
Abstract.
Generalize Kobayashi’s example for the Noether inequality in dimension three, we provide examples of -folds of general type with small volumes.
1. Introduction
From the point of view of classification theory, the following three type of algebraic varieties are considered to be the building blocks: varieties of Fano-type, varieties with , and varieties of general type. Unlike the other two types, varieties of general type can not be bounded. For example, curves of general type are those curves of genus , where can be arbitrarily large. Nevertheless, the classical theory of surfaces shows that there are some relations between some fundamental invariants. The Bogomolov-Miyaoka-Yau inequality shows that and Noether inequality gives .
It is thus natural and interesting to study the geography of higher dimensional varieties of general type, including the distribution of birational invariants and relations among invariants. There are some recent results in dimension three, due to Kobayashi, Meng Chen and many others. Let denotes the dimension of image of the canonical map . In [K], Kobayashi proved that for an -dimensional smooth projective variety of general type, provided . Hence in particular, for threefolds of general type with , it follows that . However, Kobayashi constructed examples of threefolds of general type with such that . A recent result of Meng Chen and the first-named authors prove that for any threefold of general type whose minimal model is Gorenstein, cf. [CC].
The purpose of this note is to explore the higher dimensional analogue of Noether inequality. We would like to remark the essential difference and difficulty in dimension three or higher is that its minimal model is no longer smooth. Therefore, the canonical volume can be a small positive rational number. This leads to the following two questions.
Question 1.1**.**
Is there a Noether-type inequality for varieties of general type of dimension ? More precisely, fix dimension , does there exists such that ?
This question is known only up to dimension , and partially known in dimension . One might want to ask the following weaker question, which is known up to dimension .
Question 1.2**.**
Fix dimension . Does there exists such that for -dimensional varieties of general type with Gorenstein minimal model?
Question 1.3**.**
Suppose that the answer to above questions are positive for all . What are and ?
Example 1.4**.**
Let be a general weighted hyperplane of degree . If is known that and . Let be a curve of genus and . Then and .
In the same spirit, let be an -dimensional variety with and small volume. Let . Since
[TABLE]
and , it is expected that there is an -dimensional variety with and is small enough such that . Hence it is expected that .
In this article, we produce series of examples of varieties of general type with small slope . Suppose that , recall that is the dimension of the image of canonical map .
Example A**.**
For any , there exists an -dimensional minimal smooth projective variety of general type and such that
[TABLE]
If , then we find a series of examples of -dimensional varieties of general type, where the slope goes to zero as increases (cf. Proposition 5.5).
Example B**.**
For each and , there exists an -dimensional smooth projective variety of general type such that and
[TABLE]
The following corollary answers Question 1.3 by fixing but letting increased.
Corollary 1.5**.**
Keep the notation as in Question 1.3. Then .
Note that we do not claim the constructed varieties in either Example A or B have Gorenstein minimal models. Therefore, we don’t know yet.
Our construction can be summarized in the following diagram:
[TABLE]
where , , all the horizontal maps are -bundles, and vertical maps are double covers (cf. Section 4 and 5).
This paper is organized as the following: In Section 2, we recall a proposition of Kobayashi, indicating the relation of the Noether inequality and varieties of minimal degree, for which our construction is based on. In Section 3, we take towers of -bundles over a variety of minimal degree and establish some basic properties. In Section 4 and 5, we construct Example A and B respectively. In Section 6, some related open questions are discussed.
Acknowledgement. We would like to thank Meng Chen for the helpful discussion. This work is done partially during the second author’s visit at Research Institute for Mathematical Sciences and he thanks for the warm hospitality of the institute. The second author is also supported by National Center of Theoretical Sciences in Taiwan and funded by MOST-104-2115-M-002-011-MY2.
2. polarized varieties of minimal degree
Given a non-degenerate projective variety of dimension . It is well-known that (cf. [EH]). If the equality holds, then is called a variety of minimal degree. Suppose that is a free and big line bundle on . Then defines a generically finite morphism. Let be its image in projective space. Then is a variety of degree . Recall that the -genus of a polarized variety introduced by T. Fujita is defined as
[TABLE]
It is clear that the image is a variety of minimal degree if and only if is a polarized variety with -genus . Moreover, polarized varieties with -genus zero are classified in [EH, F1, F2, F3].
The following result of Kobayashi reveals the connection between variety of minimal degree and varieties of general type with small slopes.
Proposition 2.1**.**
Let be an -dimensional minimal variety of general type with at most canonical singularities such that . Then . Equality holds only if is Gorenstein and admits a covering over a variety of minimal degree.
Two dimensional projective varieties of minimal degree are cone over rational normal curves, the embedding of Hirzebruch surface by a suitably chosen linear system, or the Veronese surface (cf. [Bea, Exercises IV.18.(4)]).
3. Towers of -bundles
The purpose of this section is to construct towers of -bundles and study some basic properties.
Given a lower triangular matrix with integer entries, we can construct a tower of -bundles successively as follows. Start with , and , then we construct as a -bundle over . Let be the distinguished section and fix . We then construct as a -bundle over . Inductively, we can construct all the way to get with distinguished section .
For , the distinguished section on is defined by the quotient of on , which satisfies on with normal bundle .
Let denote the induced map from to and denotes for simplicity. Also, for brevity, we abuse the notation of with and all other divisors or line bundles as well. It is easy to see that with as a set of generators. Hence, we can always write a line bundle on as and simply denote it as . An easy inductive computation shows that
[TABLE]
Example 3.1**.**
We consider to be the -dimensional tower of -bundle with the following building data of matrix,
[TABLE]
Then
[TABLE]
For , we also consider line bundles on or its pull-back on .
Proposition 3.2**.**
Keep the notation as in Example 3.1. The the line bundle satisfies the following properties:
. 2.
. 3.
If for all , then . 4.
If for all , then is free. 5.
* is free and big.* 6.
The polarized variety has -genus [math]. In other words, the image of is a variety of minimal degree. 7.
The morphism is a birational morphism contracting . 8.
* and for .*
Proof.
Write , then
[TABLE]
Since , by induction This proves .
We write again and by projection formula
[TABLE]
Since , one sees by induction.
We will prove by induction on dimension and on . The statement is standard on . Since ’s are rational, for all . Suppose now that with . We may and do assume that , otherwise it is reduced to lower dimensional case. Since , we can write with
[TABLE]
The standard restriction sequence to now reads:
[TABLE]
On , we consider . Through the isomorphism and , one sees that . It follows that by induction on dimension and on .
We now prove the freeness of , which is . Indeed, it suffices to prove the freeness of . By , one sees that . Pick a general which does not contain . We claim that is irreducible and is an isomorphism.
To this end, we write , where (resp. ) denotes the horizontal (resp. vertical) part with respect to . Since for a fiber of , the divisor is irreducible. By our choice of , is an effective divisor (possibly zero) on , hence so is .
Note that must be empty. Otherwise, is effective and non-empty. Since , for any ample divisor on , this leads to
[TABLE]
a contradiction. Similarly, is also empty. Therefore, is irreducible and smooth as it is then isomorphic to by .
Since , the section is disjoint from and . We have the following exact sequence:
[TABLE]
where the last map is the restriction to . By induction on dimension, we assume that is free on . Since , it follows that is free on . Let moves, then one sees that is free on away from .
Similarly, we consider the exact sequence by restricting to
[TABLE]
Since , it follows that is free on . This completes the proof of .
The statement of and follow from , and immediately. By and induction, defines a birational morphism on , which extends to a birational morphism on . As we saw in the proof of , restricts to a constant map on . This proves .
To compute that , let as before. Since , inductively we get
[TABLE]
and
[TABLE]
as for . ∎
4. Construction of Example A
Example 4.1**.**
We keep the notation as in Section 3. We take , which is -dimensional. In other words, is constructed by using the following building data of matrix
[TABLE]
Let be the distinguished section: and . By (3.1), one has
[TABLE]
Also, we define
[TABLE]
In what follows, we usually identify with and identify with on for any sheaf on .
Proposition 4.2**.**
Let be the variety as in Example 4.1 and write a line bundle as . We have the following similar properties as in Proposition 3.2.
, is not a base divisor of , and a general member is smooth and isomorphic to . 2.
If for all , then . 3.
If for all , then is free.
Proof.
The statement follows from the standard sequence by restricting to as in the proof of Proposition 3.2. The proof of and are almost the same as those in Proposition 3.2. For example, let . Since , one gets
[TABLE]
The restriction sequence to reads
[TABLE]
where on . One can then proceed by induction similarly.
We leave the details to the readers. ∎
To construct our Example A, we need to consider the following line bundles:
[TABLE]
and the -line bundle
[TABLE]
The choices of these line bundles are inspired by the work of Kobayashi.
Lemma 4.3**.**
The linear system is free and we can choose a general member such that is a simple normal crossing divisor.
Proof.
Since , it is free by Proposition 4.2. Moreover, is free and the restriction map is surjective thanks to the vanishing of
[TABLE]
as proved in Proposition 4.2. The claim follows by Bertini’s Theorem. ∎
Example 4.4** (=Example A).**
We assume that . Consider the double cover branching along a general simple normal crossing divisor (cf. Lemma 4.3). It is clear that has singularities of type , which is Gorenstein and canonical. Note that
[TABLE]
is nef as is nef by Proposition 4.2. Blow up the singular locus of , which is crepant, we obtain a smooth minimal variety with for all .
Before moving on, we fix some notations. Let and be the ramification divisors on . Then and
[TABLE]
Proposition 4.5**.**
Keep the above notations. Then the following holds for the variety as constructed in Example 4.4.
The geometric genus
[TABLE] 2.
* is big and semiample.* 3.
The volume . 4.
One can identify the canonical image of with , which is -dimensional.
Proof.
Since the resolution is crepant, it is enough to work on . Let on for , then one has . In particular, is nef. By projection formula, one obtains
[TABLE]
Since , we have by induction. Since as is rational, , and , it follows again from projection formula that
[TABLE]
This proves .
We now prove . Since , it follows that is semiample by Proposition 4.2
It remains to show . Since is base point free, a general member is smooth and as for a fiber of . Since , it follows that is disjoint from . Therefore
[TABLE]
Now we can easily compute
[TABLE]
where we have used and in Proposition 3.2. Hence
[TABLE]
This proves assertion .
We now prove . By definition, if there is a constant such that
[TABLE]
then . We will show that . By projection formula, we have
[TABLE]
The proof can be completed by the following three steps.
Claim 1. If is even, then
[TABLE]
For any , we consider the exact sequence
[TABLE]
Since , , and , one sees that
[TABLE]
which can not be effective. Therefore,
[TABLE]
is an isomorphism. By the asymptotic Riemann-Roch formula for , the Claim follows.
Claim 2. .
From the definition of volume, we get
[TABLE]
where . Since is big, is free and big, and volume is homogeneous for big divisors by [L, Proposition 2.2.35], we get .
Claim 3. End of proof of .
Since it is easy to find an integer such that is effective, we obtain that
[TABLE]
and hence for ,
[TABLE]
Combined with Claim 1 and 2, we finish the proof of the statement .
For , it is easy to see that by the projection formula, both the maps
[TABLE]
are isomorphisms. Therefore, . It follows that the map defines by is the composition of the map induced by and . Since as long as , where is free and induces a birational morphism by Proposition 3.2., this shows that the image of is -dimensional and hence completes the proof. ∎
Remark 4.6*.*
One can actually prove that is base point free and contracts to , which can be identified with as .
5. Construction of Example B
The purpose of this section is to establish Example B. We start by considering a higher tower of -bundles.
Example 5.1**.**
Fix . For any , we construct -bundles inductively as follows: Start with and as in (4.1). Take and consider the line bundle on , where is the distinguished section with . By abusing the notation, we still write . We end up with a -dimensional tower of -bundles by considering the following building data of matrix,
[TABLE]
This matrix is the expansion of the matrix in Section 4 by the last rows. It is clear that we have as -cycle on for .
Note that the -dimensional tower of -bundle has canonical divisor of the following form
[TABLE]
We first establish some properties generalizing those in Section 4.
Lemma 5.2**.**
In the setup of the above notations, we have for all the following properties:
* is free and big;* 2.
; 3.
; 4.
.
Proof.
All the statements hold for from Proposition 3.2 and the proof of Proposition 4.2. We assume now .
Use and , one can show inductively that . It follows that and we now apply the argument as in Proposition 4.2. This proves freeness of and follows once we prove .
Since for , equality in follows from
[TABLE]
For , the proof is the same as Proposition 3.2.: there is a section disjoint from , then by restriction to
[TABLE]
The last statement is proved in the same way as . ∎
The next lemma is useful.
Lemma 5.3**.**
Denote by
[TABLE]
For any , if for some .
Proof.
It is easy to see that if , then . Otherwise, suppose that is an effective divisor of . By intersecting with a fibre of , one sees a contradiction immediately.
If now but , then
[TABLE]
where is either or (a multiple of) . This forces the last summand on to be negative and hence
[TABLE]
This implies if with arbitrary . The lemma now follows by induction via projection formula. ∎
On , we take the line bundles:
[TABLE]
It is easy to check that
[TABLE]
and
[TABLE]
Therefore, is free. Pick a general , it is clear that is a simple normal crossing divisor on .
Example 5.4** (=Example B).**
Now take the double cover ramified along the divisor . Locally has singularity type at triple intersection of and at double locus of , where the first one is of -type terminal singularities and the second one is of -type canonical singularities.
Notice that is not nef: Let be a curve on contained in . Then
[TABLE]
where is the pushforward of on . Hence if is supported on the fibres of .
Now take to be a resolution of singularities. Notice that for all .
We will compare with
[TABLE]
Observe that is big and free by Lemma 5.1.
Proposition 5.5**.**
Keep the above notation, then the following holds for .
The geometric genus is give by
[TABLE] 2.
The volume is
[TABLE] 3.
The image of the canonical map of can be identified with , which is -dimensional.
Proof.
It is enough to work on , for which we denote by in the following computation since there should be no confusion. Similarly, we denote by , , and the corresponding divisors on .
Statement can be proved via projection formula and Lemma 5.3:
[TABLE]
Hence part also follows from Proposition 4.5.
Similar to the proof of Proposition 4.5, we have
[TABLE]
Claim : for even.
Proof of the Claim.
For and or we consider
[TABLE]
One has
[TABLE]
Note that by Lemma 5.3, the line bundle
[TABLE]
has vanishing if any of , holds. Hence in the following computation, the summation up to is equal to the summation only up to . More precisely,
[TABLE]
Which verifies the Claim. ∎
Now by the same reasoning as in Proposition 4.5, we get . From and Lemma 5.1, it follows that
[TABLE]
This completes the proof. ∎
From the above proposition, we have
[TABLE]
The variety is the required variety in Example B.
6. Related Problems
Our examples show that the geography, or the relation between canonical volume and genus, depends on , the dimension of variety and on , the dimension of image of the canonical map as well. It is expected that one can say a lot more if . For example, it is reasonable to ask the following several questions.
Question 6.1**.**
Suppose that is a -fold minimal model with Gorenstein singularities. Suppose that . Is it true that either or is fibered by curves of genus ?
By a similar argument to Kobayashi’s proof, one can prove the following result, partially answers the above question.
Proposition 6.2**.**
Let be a smooth minimal -fold of general type. Suppose that . Then More precisely, either or is fibered by curves of genus .
Proof.
We may write , where is the mobile part and is the base divisor. Let be a resolution of so that the proper transform is free. That is, and where . Let be a general fibre curve of the morphism induced on .
Since consists of a sequence of blowup along smooth centers, we have and hence , where .
Notice that has image of dimension and hence . Therefore, we have the following
[TABLE]
Since , it follows that . Now
[TABLE]
Hence and therefore by (6.1).
Suppose that . Then clearly, and hence . By (6.2), and hence . Together with (6.1) , one sees that either or is fibered by curves of genus . ∎
Question 6.3**.**
Suppose that is a -fold minimal model of general type with at worst Gorenstein singularities. Suppose furthermore that . Does the relation
[TABLE]
holds?
Question 6.4**.**
Suppose that is a -fold minimal model of general type with at worst Gorenstein singularities. Suppose furthermore that and
[TABLE]
Is it true that is a double covering over a variety of minimal degree, similar to example ?
It is known in dimension three that the above equality characterizes the variety essentially as Kobayashi’s example if (cf. [CH]). The same proof partially works in the setting of Question 6.4 with the extra assumption .
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[Bea] Beauville, Arnaud, Complex algebraic surfaces. Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. Second edition. London Mathematical Society Student Texts, 34. Cambridge University Press, Cambridge, 1996.
- 2[CC] Chen, Jungkai A.; Chen, Meng, The Noether inequality for Gorenstein minimal 3-folds. Comm. Anal. Geom. 23 (2015), no. 1, 1–9.
- 3[CH] Chen, Yifan; Hu, Yong, On canonically polarized Gorenstein 3-folds satisfying the Noether equality . ar Xiv:1411.2200
- 4[EH] Eisenbud, David; Harris, Joe, On varieties of minimal degree (a centennial account). Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 3–13, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987.
- 5[F 1] Fujita, Takao, On the structure of polarized varieties with Δ Δ \Delta -genera zero. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 103-115.
- 6[F 2] Fujita, Takao, On polarized varieties of small Δ Δ \Delta -genera. Tohoku Math. J. (2) 34 (1982), no. 3, 319-341.
- 7[F 3] Fujita, Takao, Classification of projective varieties of Δ Δ \Delta -genus one. Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 3, 113-116.
- 8[K] Kobayashi, Masanori, On Noether’s inequality for threefolds. J. Math. Soc. Japan 44 (1992), no. 1, 145-156.
