The dual complex of Calabi–Yau pairs
Abstract.
A log Calabi–Yau pair consists of a proper variety and a divisor on it such that is numerically trivial. A folklore conjecture predicts that the dual complex of is homeomorphic to the quotient of a sphere by a finite group. The main result of the paper shows that the fundamental group of the dual complex of is a quotient of the fundamental group of the smooth locus of , hence its profinite completion is finite. This leads to a positive answer in dimension . We also study the dual complex of degenerations of Calabi–Yau varieties. The key technical result we prove is that, after a volume preserving birational equivalence, the transform of supports an ample divisor.
January 27, 2021
A log Calabi–Yau pair, abbreviated as logCY, is a pair consisting of a proper variety and an effective divisor such that is log canonical and is linearly equivalent to 0. Any Calabi–Yau variety can be naturally identified with the log Calabi–Yau pair . At the other extreme, if is a Fano variety and is an effective divisor then is also logCY (provided that it is log canonical).
Here we are interested in these Fano–type logCYs, especially when is “large.” Being Fano is not preserved under birational equivalence, thus it is better to define “large” without reference to Fano varieties. There are several natural candidates for this notion; we were guided by the concepts of large complex structure limit and maximal unipotent degeneration used in Mirror Symmetry.
Definition 1.
Let be a log canonical pair of dimension and a crepant log resolution. That is, , , is smooth and is a simple normal crossing divisor.
Note that is usually not effective but, since is log canonical, all divisors appear in with coefficient . Let denote the union of all irreducible components of whose coefficient equals .
The combinatorics of is encoded in its dual complex, denoted by ; see Definition 12. By [dFKX12] is independent of the choice of , upto PLhomeomorphism. We call this PLhomeomorphism type the dual complex of and denote it by .
Note that since, on a variety of dimension , at most irreducible components of a simple normal crossing divisor meet at a point. We say that has maximal intersection if equality holds.
By [Kol14], every finite simplicial complex of dimension appears as for some dimensional simple normal crossing pair . Thus it is interesting to understand which algebraic restrictions on have meaningful topological consequences for . The aim of this paper is to study the dual complex of logCY pairs. The main result is the following.
Theorem 2.
Let be a logCY pair and its dual complex. Assume that . Then the following hold.

has the same dimension at every point.

for .

There is a natural surjection .

The profinite completion is finite.

The cover corresponding to is the dual complex of a quasiétale cover .
Part (1) is a restatements of earlier results; see [Kol11] or [Kol13, 4.40]. The relationship between the rational homology of the dual complex and the coherent cohomology of has been understood for a long time; thus (2) has been known in many cases. Our main contribution is to understand the connection between the fundamental group of the dual complex and the fundamental group of the smooth locus . The main conclusion is (3) which in turn implies (5). The finiteness of follows from [Xu14, GKP13]; it is conjectured that is finite.
If then is either the 1simplex or . In the latter case is infinite but usually has only trivial quasiétale covers.
On a general logCY pair, contains divisors with different coefficients, but it turns out that is rather special if not all coefficients are equal to 1.
Proposition 3.
Let be a logCY pair such that contains at least one divisor with coefficient . Then

either is contractible

or .
It is straightforward to write down examples where is contractible or a sphere. More generally, at least some quotients of spheres are easy to get; see Section 8. This leads naturally to the following.^{1}^{1}1Many people seem to have been aware of this question, among others M. Gross, S. Keel. V. Shokurov, but we could not find any specific mention in the literature.
Question 4.
Let be a logCY pair of dimension . Is for some finite subgroup of and . (We use to denote PLhomeomorphism.)
The group action may have fixed points, thus in general is only an orbifold. LogCY varieties also appear as compactifications of character varieties. In this context, Question 4 is studied in [GT10, Sim15]. We prove the following in Paragraph 33.
Proposition 5.
The answer to Question 4 is positive if or if and is a simple normal crossing pair.
In many questions involving Mori’s program, the 3 and 4 dimensional cases are good indicators of the general situation. However, the proof of Proposition 5 relies on several special low dimensional topological facts, thus it gives only very weak evidence for the general problem. We do not see any good heuristic reason why the answer to Question 4 should be affirmative. From the technical point of view, at least 3 problems remain to be settled.

Finiteness of the fundamental group of . We do not know how to prove it but Theorem 2.3 reduces it to the finiteness of .

Torsion in the integral homology of . Our methods do not say anything about it.

Starting in dimension 5 we have to deal with the possibility that is singular but itself has no obvious quasiétale covers. This seems to us the most likely approach to construct a counter example to Question 4.
It is also very unclear which triangulations of a sphere can be realized as dual complexes of logCY pairs. This is quite hard even in dimension 2; see [Liu15] for partial results.
6Degenerations of Calabi–Yau varieties.
Studying degenerations of Calabi–Yau varieties naturally leads to log Calabi–Yau pairs. Let be a CYdegeneration over the unit disk . That is, is proper, and is dlt where is the central fiber. Thus the fiber is a Calabi–Yau variety for and the central fiber is a union of logCY pairs where is the intersection of with the other irreducible components. We are interested in the dual complex of the central fiber especially when the following two conditions are satisfied

The general fiber is an dimensional Calabi–Yau variety in the strict sense, that is, is simply connected and for .

. This is a combinatorial version of the large complex structure limit or maximal unipotent degeneration conditions.
Question 7.
Let be a CYdegeneration of relative dimension satisfying the conditions (6.1–2). Is ?
For a positive answer is given by [Kul77]; the general case is proposed in [KS01]. It is easy to see that is a simply connected rational homology sphere but it is not clear that is a manifold. Using Theorem 2, we prove the following in Paragraph 34.
Proposition 8.
The answer to Question 7 is positive if or if and the central fiber is a simple normal crossing divisor.
Acknowledgments.
We thank M. Gross, A. Levine, J. Nicaise, S. Payne, N. Sibilla and C. Simpson for comments, discussions, references and J. M^{c}Kernan for sharing with us early versions of [BMSZ15]. We also want to thank the anonymous referee for many helpful remarks.
Partial financial support to JK was provided by the NSF under grant number DMS1362960. Partial financial support to CX was provided by The National Science Fund for Distinguished Young Scholars. A large part of this work was done while CX enjoyed the inspiring environment at the Institute for Advanced Studies.
1. Volume preserving maps
For logCY pairs, volume preserving maps, also called crepant birational maps, form the most important subclass of birational equivalences.
Definition 9.
Let be normal pairs. A proper, birational morphism is called crepant if and . An arbitrary birational map between proper pairs is called crepant if it can be factored as
where the are proper, birational, crepant morphisms. In characteristic 0 this is equivalent to having a common crepant log resolution. (If the are not proper, the above definition still works and defines proper and crepant birational maps. Note that itself is not proper, so the terminology is somewhat confusing.)
The main requirement is the natural linear equivalence
A proper, crepant birational map between log canonical pairs is called thrifty if there are closed subsets of codimension that do not contain any of the log canonical centers of the such that restricts to an isomorphism .
If is logCY then a global section of determines a natural volume form on , up to scalar. Then a map is crepant birational iff it is volume preserving, up to a scalar.
It is important to note that in (9.1) usually one can not choose such that is effective, not even if we allow to be singular. However, as we see next, such a choice of is possible if we allow the to be rational contractions.
Definition 10.
Let be normal, proper varieties. A birational map is called a contraction if does not have any exceptional divisors. Observe that is an isomorphism in codimension 1 iff both and are contractions. (Note that a birational morphism is always a contraction. For a birational map there does not seem to be a generally accepted notion of contraction; the above definition is natural and quite useful.)
Let be a birational contraction that is crepant. Then and . The converse is usually not true. (For example, a flop is crepant birational but a flip is not.) However, if the are logCY then the linear equivalence (9.2) is automatic and the converse holds.
The following lemma gives several useful ways of factoring a crepant birational map. In the logCY case, a detailed understanding of volume preserving maps is given in [CK15], which also suggested to us (11.4).
Lemma 11.
Given two proper, log canonical pairs , the following are equivalent.

There is a crepant birational map .

There is a log canonical pair and crepant, birational contraction maps

There is a pair and crepant, birational contraction maps
where is factorial, dlt and is a morphism.

There are factorial, dlt pairs and crepant, birational maps
where the are morphisms, is crepant birational, thrifty and an isomorphism in codimension 1.
Proof. It is clear that each assertion implies the previous one.
In order to see (1) (3), let be a crepant, birational map and the exceptional divisors of . The discrepancy equals minus the coefficient of in , thus it is .
By [Kol13, 1.38] there is a factorial, dlt, crepant modification that extracts precisely the and possibly some other divisors with discrepancy . Then is a crepant, birational contraction map.
The proof of (4) is similar. We take a common log resolution and write where consists of the set of divisors on that are in one of the following three sets: birational transforms of , divisors that are exceptional for exactly one of the or divisors with coefficient 1 in . Let be the sum of all other exceptional divisors.
As in [Kol13, 1.35], for run the MMP over for to get . If is an extremal ray that we contract then . Since is numerically trivial, this is equivalent to
By our choice of , is effective and its support is . Thus the extremal rays contracted are always contained in and the MMP contracts all the divisors contained in . Note also that and have the same lc centers and they are not contained in . Thus is a local isomorphism at all the lc centers of hence the induced birational map is thrifty and an isomorphism in codimension 1. ∎
Definition 12 (Dual complex).
Let be a simple normal crossing variety over a field with irreducible components . (Our main interest is in the case but sometimes it is convenient to allow to be arbitrary.)
A stratum of is any irreducible component for some .
The dual complex of , denoted by , is a CWcomplex whose vertices are labeled by the irreducible components of and for every stratum we attach a dimensional cell. Note that for any there is a unique irreducible component of that contains ; this specifies the attaching map. (The dual complex is a regular complex in the terminology of [Hat02].)
It is very important for us that crepant birational equivalence does not change the dual complex.
Theorem 13.
[dFKX12] The dual complexes of proper, log canonical, crepant birational pairs are PLhomeomorphic to each other.∎
Using Theorem 13 our aim is to study the dual complex of logCY pairs in 2 steps. First we show that is crepant birational to a “Fano–type” logCY pair. There are several natural ways to define “Fano–type.” For our current purposes the best seems to be to assume that supports a big and semiample divisor.
The second step is the study of in the Fano–type cases.
Definition 14.
Let be a variety and an effective divisor. We say that fully supports a divisor that is ample (or big, or semiample or …) if there is an effective divisor that is ample (or big, or semiample or …) such that .
Example 63 shows the important difference between supporting a big and semiample divisor and fully supporting a big and semiample divisor.
15Dlt singularities.
Divisorial log terminal singularities form a very convenient class to work with; see [KM98, 2.37] or [Kol13, 2.8]. The precise definition is important for the proof of Theorem 49, but for most everything else all one needs to know is that dlt pairs behave very much like simple normal crossing pairs.
If is dlt then, by [dFKX12, Thm.3], can be computed directly from as in Definition 12. In particular, and has maximal intersection iff there are divisors such that is nonempty (hence necessarily 0dimensional).
If is dlt then for every stratum there is a natural divisor such that is dlt and ; see [Kol13, Sec.4.1] for the definition and basic properties. Thus is also logCY.
In general is somewhat complicated to determine, but if all divisors in have coefficient 1 and is Cartier then the same holds for and consists of the intersections of with those irreducible components of that do not contain .
This implies that for every irreducible divisor , the link of is PLhomeomorphic to . Thus the local structure of is determined by the lower dimensional logCY pairs .
2. Reduction steps
We discuss various steps that simplify without changing the dual complex or changing it in simple ways. The final conclusion is that one should focus on logCY pairs such that

is dlt, factorial,

is rationally connected,

,

fully supports a big, semiample divisor and

.
16Disconnected case.
17Index 1 cover.
Assume that and let be the smallest natural number such that . Correspondingly there is a degree quasiétale (that is, étale in codimension 1) cover and .
18Rational connectedness.
Proposition 19.
Let be a dlt logCY pair and a dominant map to a nonuniruled variety . Then every irreducible component dominates .
Proof. We may assume that is smooth and projective. Let be a general complete intersection curve, in particular is defined along . Then and .
If does not dominate then we reach a contradiction by proving that . By blowing up we may assume that dominates a divisor ; see [Kol13, 2.22].
Let be the normalization of the closure of the graph of . Choose such that the first projection is crepant. Since is defined along we may identify with its preimage in and so .
We apply the canonical class formula (20.4) to the second projection and write .
Note that since is not uniruled [MM86] and by (20.5). Finally, (20.8) shows that a divisor appears in with positive (resp. nonnegative) coefficient if there is divisor dominating that appears in with positive (resp. nonnegative) coefficient. Thus is disjoint from the noneffective part of and intersects nontrivially. Furthermore, appears in with positive coefficient. Thus . Adding these together we get that , a contradiction.∎
20Kodaira–type canonical class formula.
The original formula for elliptic surfaces was further developed by [Fuj86, Kaw98]. The following is a somewhat simplified version of the form given in [Kol07, 8.5.1].
Let be a log canonical pair where is not assumed effective. Let be a proper morphism to a normal variety with geometrically connected generic fiber . Assume that

is effective,

is logCY and

is linearly equivalent to the pullback of some Cartier divisor from . (This seems like a strong restriction but it is easy to achieve by changing .)
Let be the largest open set such that is flat over with logCY fibers and set . Then one can write
where and have the following properties.

is a linear equivalence class, called the modular part. It depends only on the generic fiber and it is the pushforward of a nef class by some birational morphism . In particular, is pseudoeffective.

is a divisor, called the boundary part. It is supported on .

Let be an irreducible divisor. Then
where the supremum is taken over all divisors over that dominate .
This implies the following.

If is dominated by a divisor such that (resp. ) then (resp. ).

If is effective then so is .

If is lc then for every and iff is dominated by a divisor such that .
The strongest reduction assertion is the following, which is a weak form of our main technical result; see Corollary 58 for the general case.
Theorem 21.
Let be a logCY pair. Then there is a volume preserving birational map such that

in the maximal intersection case fully supports a big and semiample divisor and

in general there is a morphism with generic fiber such that

and

fully supports a big and semiample divisor.

22Fractional part of .
Assume that . Then, by [BCHM10, HX13], the MMP terminates with a Fanocontraction . Note that by Theorem 13.
If Proposition 24 applies then is collapsible to a point and it remains to compare and . In general this seems rather difficult and it can happen that is empty yet is not. There is, however, one case when the two are closely related.
Assume that fully supports a big and semiample divisor over . Then fully supports a big and mobile divisor over for any dlt modification . This implies that dominates and
Thus [dFKX12, Thm.3] proves that collapses to . (We do not known whether they are PLhomeomorphic or not.)
Thus, in this case, is collapsible to a point.
Corollary 23.
Let be a logCY pair such that is not collapsible to a point.
Then the model obtained in Theorem 21 also satisfies in the maximal intersection case and in general. ∎
Let be a dlt pair and an irreducible divisor such that is irreducible and nonempty for every log canonical center . Then is the cone over . If is a Fano contraction and dominates then it has irreducible and nonempty intersection with every log canonical center; this is a special case of [Kol13, Thm.4.40]. Applying these repeatedly, we obtain the following.
Proposition 24.
Let be a dlt pair and a Fano contraction. Assume that dominates . Then there is a unique smallest lc center dominating and
the join of a simplex of dimension and of .
In particular, is contractible, even collapsible. ∎
Note also that the simplex is PLhomeomorphic to where is a reflection on a hyperplane. Thus, as in Example 64, if is the quotient of a sphere then so is .
3. Basic results on the dual complex
We need two results that connect the topology of and the algebraic geometry of . The following homological lemma is essentially proved in [GS75, pp.68–72]; see also [FM83, pp.26–27] and [Kol13, 3.63]. The fundamental group result is rather straightforward; [KK14, Lem.25].
Lemma 25.
Let be a proper, simple normal crossing variety over . Then there are natural injections
Furthermore, if for every and every then (25.1) is an isomorphism.∎
Lemma 26.
Let be a connected, simple normal crossing variety over . Then there is a natural surjection
Furthermore, if the are simply connected then (26.1) is an isomorphism.∎
In some cases one can describe a dual complex using a fibration and finite group actions as in Theorem 21.2.
27.
Let be a simplicial complex and a finite group acting on . Let denote the barycentric subdivision of . If is a simplex and such that then also fixes every vertex of . Thus the action on naturally extends to a simplicial action on the topological realization .
The quotient is a regular complex denoted by . There is a natural map whose fibers are exactly the orbits on .
Such branched covering spaces are discussed in [Fox57]; we need the following properties.

There are natural isomorphisms for every .

There is an exact sequence
In particular, if is finite then so is .
The quotient construction naturally arises for families of simple normal crossing varieties.
Lemma 28.
be a simple normal crossing variety and a morphism such that every stratum of dominates . Let be a general point and the fiber over . Then is a simple normal crossing variety and there is a finite group acting on such that .
Proof. By shrinking we may assume that is smooth on every stratum of . Then defines a locally trivial fiber bundle. Since is a finite simplicial complex, the monodromy of the fiber bundle is a finite group and .∎
Algebraically minded readers may prefer to think of Lemma 28 as a combination of the next two claims.
Lemma 29.
be a simple normal crossing variety over a field and a morphism. Then the generic fiber is a simple normal crossing variety over the function field and is a subcomplex of . Furthermore, if every stratum dominates then . ∎
Lemma 30.
Let be a Galois extension with Galois group . Let be a simple normal crossing variety over . Then acts on and . ∎
4. Homology of the dual complex
The following proves (2.2).
Proposition 31.
Let be a logCY pair. Then
Proof. Assume first that is rationally connected, and . Set . Then for by [Cam91, KMM92] and
by Serre duality. The long cohomology sequence of the exact sequence
now implies that for . Thus for by Lemma 25.
We try to use a similar argument in general; the problem is that, in (31.1), instead of we have and is linearly equivalent to . The presence of the fractional part and the linear (as opposed to linear) equivalence both cause problems.
Assume next that is dlt and fully supports a big and semiample divisor . Note that
thus by vanishing we see that
Using (31.1) these imply that for .
Finally, by Theorem 13, we are free to replace with any other logCY pair that is crepant birational to it. We first apply Theorem 21, then Lemmas 28 and 27 to reduce to the already established case when fully supports a big and semiample divisor.∎
Next we study the top cohomology of the structure sheaf and of the dual complex.
32The top cohomology of logCY pairs.
Let be a dlt, logCY pair of dimension . Then is Serre dual to hence save when and .
Let be an irreducible component such that . As we noted above, then , thus is a connected component of . As we noted in Paragraph 16, there are 2 possibilities. Either or has 2 irreducible components which are crepant birational to each other.
Let be a dlt pair. Set and assume that for every curve . We can then view as a reducible logCY pair. A nonembedded definition of such pairs, called semidlt pairs, is given in [Kol13, Sec.5.4]. The precise definition is not important for now, we will only use the case when as above.
Using these observations inductively, we get the following.
Claim 32.1. Let be a connected, semidlt, logCY pair of dimension . Assume that it has a stratum of dimension such that . Then

all strata have dimension ,

the dimensional strata are crepant birational to each other and

. ∎
Let be a simple normal crossing variety. The cohomology of is computed by a spectral sequence whose terms are
In the bottom row we find the complex that computes the cohomology of . Note also that if holds for every positive dimensional stratum then the only term that contributes to is . We have thus proved the following.
Claim 32.3. Let be a connected, semidlt, logCY pair of dimension such that . Then

either ,

or . ∎
Next we prove Proposition 5.
33Dimension induction.
Let be a dlt logCY pair of dimension such that . As we noted in Paragraph 15, the local structure of is determined by the lower dimensional logCY pairs.
Assume first that has maximal intersection. Using Proposition 31 and Paragraph 32 we see that hence is a rational homology sphere. In low dimensions we obtain complete answers to Question 4.
If then and .
If then is a rational surface and .
If then is a 2–manifold that is a rational homology sphere. Thus .
If then is a 3–manifold that is a rational homology sphere. The fundamental group of a 3–manifold is residually finite [Hem87], thus (2.4) implies that the fundamental group itself is finite. By (2.5) the universal cover is a simply connected homology sphere, thus . (This uses the Poincaré conjecture.) Note however that we do not claim that is the sphere .
Thus, starting with we do not claim that is a manifold.
We can, however, do better if is a simple normal crossing pair. In this case Theorem 36 implies that and its links are simply connected. Thus we get that is homeomorphic to a sphere if . (Conjecturally, PLhomeomorphic to a sphere.)
If then is a 5–manifold that is a simply connected rational homology sphere. Our results say nothing about the torsion group .
Finally consider the case when does not have maximal intersection. A similar induction shows that, for , the universal cover is a simply connected manifold, possibly with boundary. Thus is either a sphere or a ball.
34.
Let be a CYdegeneration of relative dimension satisfying the conditions (6.1–2).
If is an irreducible component and is the intersection of with the other components then is a logCY pair and is PLhomeomorphic to the link of . Thus if or if and the central fiber is a simple normal crossing divisor then is a manifold using the results of Paragraph 33.
Since has Du Bois singularities (cf. [Kol13, Chap.6]) we see that for , thus is a rational homology sphere.
Finally, there are natural surjections
We assumed that is simply connected for , hence is a quotient of . Since is reduced, has a section, thus gets killed in .
These imply that is trivial and so is a simply connected rational homology sphere. For this implies that it is homeomorphic to a sphere (conjecturally PLhomeomorphic to a sphere). This completes the proof of Proposition 8.
5. Fundamental groups of logCY pairs
In this section we study various fundamental groups associated to a logCY pair. It is easy to see that usually itself is simply connected. It is much more interesting to understand the fundamental group of the smooth locus and the fundamental group of the dual complex. Note that while the latter is a crepantbirational invariant, the fundamental group of the smooth locus is not; see Example 62.
35General setup.
Let be a normal, projective variety and a divisor. Fix a smooth metric on and for let denote the neighborhood of . Then is a deformation retract of , hence . If is snc or dlt then we can form the dual complex and we have a surjection
If is any closed subset then there is a natural map . Very little is known about this map in general but if is ample and then it is an isomorphism by the Lefschetz hyperplane theorem for large enough finite