Differential geometric global smoothings of simple normal crossing complex surfaces with trivial canonical bundle
-
Mamoru Doi
Abstract
Let
1 Introduction
This is a sequel to the first author’s paper [4], where he obtained a construction of compact complex surface with trivial canonical bundle by gluing together compact complex surfaces with an anticanonical divisor. As an application, he proved that if a simple normal crossing (SNC) complex surface
Throughout this article,
Definition 1.1
Let
In particular, the above
Then
Definition 1.2
Let
where
This is a differential geometric interpretation of Friedman’s original complex analytic (and also algebro-geometric) definition that an SNC complex surface
for the singular locus
Now the main theorem of this article is described as follows.
Theorem 1.3
Let
each
there exists a meromorphic volume form
For each
The complex structure on
Remark 1.4
Conditions (ii) and (iii) of Theorem 1.3 are equivalent to the condition that the canonical bundle
the normal crossing complex surface
we are given
Let us compare Theorem 1.3 with Friedman’s smoothability result. According to [20], Theorem II, the central fiber of a semistable degeneration of
From the modern viewpoint of logarithmic geometry which is a central tool for smoothings, Kawamata and Namikawa generalized Friedman’s smoothability result in higher complex dimensions
A major breakthrough in algebraic extensions of Friedman’s smoothability result has been achieved by Felten et al. in [8]. Conceptually,
Meanwhile, our result of differential geometric smoothings using the gluing technique brings some insight into both differential and algebraic geometry. As mentioned above, the smoothing technique in complex analytic and algebraic geometry is particularly effective for constructing special smooth complex manifolds such as Calabi-Yau manifolds. For this purpose, our construction of differential geometric smoothings is also useful because we need a complex structure not on the total space
The proof of Theorem 1.3 is based on an explicit construction of global smoothings of
By gluing together
This article is organized as follows. In Section 2, we briefly state the results in [4] which will be used in the proof of Theorem 1.3. We introduce the notions of
The first author is mainly responsible for Sections 1, 2.1–2.3, and 3, and the second author mainly for Sections 1, 2.4, 4, and 5.
2 A brief review of complex surfaces with trivial canonical bundle
In dealing with complex surfaces with trivial canonical bundles in differential geometry, it is crucial to note that a complex structure of such a surface is characterized by a
2.1
SL
(
2
,
C
)
- and
SU
(
2
)
-structures
In this subsection, we briefly review the notions and results in [4] without proofs. (See also [12] for reference.)
Definition 2.1
Let
An
and
satisfies
Let
Definition 2.2
Let
We define
Since an
Lemma 2.3
(Goto [12]) Let
The above lemma gives the following characterization of complex surfaces with trivial canonical bundle by
Proposition 2.4
Let M be an oriented 4-manifold. Then M admits a complex structure with trivial canonical bundle if and only if M admits a
Thus, if we say that
Let
Definition 2.5
Let
a bilinear form
Conditions (ii) and (iii) imply that
For
where
We also have the orthogonal decomposition
where
Lemma 2.6
The tangent space of
Proof
See [4], Lemma 2.6.□
Definition 2.7
Let
where
Lemma 2.8
There exists a positive constant
Proof
Let
Then by the
Lemma 2.8 implies that for
Definition 2.9
Let
Define
If
Definition 2.10
Let
2.2 Existence theorem of
d
-closed
SL
(
2
,
C
)
-structures
We are now in a position to state the following existence theorem of a complex structure with trivial canonical bundle.
Theorem 2.11
([4], Theorems 4.1 and 4.2) Let M be a smooth 4-manifold and
Let
Let
Note that the Hermitian form
The proof of Theorem 2.11 is summarized as follows. The equation
when
We note that
with
2.3 Canonical bundle of an SNC complex surface
According to [10], Remark 2.11, the canonical bundle
Definition 2.12
Let
By the adjunction formula, we calculate
Also, the restriction of
The minus sign in the gluing isomorphisms
Then
which locally generate
which explains the minus sign in the gluing isomorphisms
We immediately obtain the following result from Definition 2.12 because a section of
Lemma 2.13
Let
each
there exists a meromorphic volume form
This lemma implies that if we have
Proposition 2.14
Let
where
Hence,
We will use Proposition 2.14 in Example 3.9.
2.4 Semistable degenerations of
K
3
surfaces
In this subsection, we give a summary of the classification of degenerations of
for each
the total space
the central fiber
The new family
Theorem 2.15
([20], Theorem II. See also [16], Chapter 6, Theorem 5.2) Let
If one omits the assumption that all irreducible components of
Definition 2.16
Let
For a
Theorem 2.17
([10], Theorem 5.10) Any d-semistable K3 surface is smoothable to K3 surfaces. More precisely, there exists a semistable degeneration
When we posted a preprint of this article on arXiv, we did not know whether there exists an SNC complex surface
Starting from a
which yields that
3 Explicit construction of differential geometric global smoothings
In this section, we explicitly construct differential geometric global smoothings of a given SNC complex surface
3.1 Local coordinates on an SNC complex surface
Let
if
if
Then we choose
if
the meromorphic volume form
where
In terms of the above coordinate system, we define a new one
if
if
Condition (E) is rephrased as
Letting
The meromorphic volume form
(3.2)where
(3.3)Also, the Poincaré residue
(3.4)
from which (3.3) follows. Let
which is locally represented on
Example 3.1
Here we suppose indices
We will see that
(1) X is an SNC complex surface.
Letting
by the coordinate transformation
(2) X satisfies condition (ii) of Theorem 1.3.
Letting
where
(3) Meromorphic volume forms
Using (3.6), we obtain well-defined meromorphic volume forms
where
by the Poincaré residue map as
so that we have
(4) X does not satisfy condition (i) of Theorem 1.3.
Note that
where
Although the above
Let
the map
(3.7)is a well-defined isomorphism for
Now consider a Hermitian metric around
on
Thus, it follows from (3.1) and (3.9) that
In the rest of this subsection, we assume
which makes sense because
In particular, using (3.11) we have
Note that
For later convenience, we further extend
Now setting
we redefine the neighborhood
Accordingly, redefining the neighborhood
Thus,
Then we see from (3.7) that (3.13) and (3.14) still hold if we interchange
Then by (3.8), (3.14), and (3.18), we have for all
In the same way as above, we can define a neighborhood
For later use, we define
Then we have
3.2 Local smoothings of
X
i
∪
X
j
around
D
i
j
without a triple point
Here we suppose
We define a smooth complex 2-form
Then
Lemma 3.2
The 2-form
Proof
Since the bundle norm
for some smooth function
which shows that
Note in particular that
where
Now we construct local smoothings of
where
and also define
from which it follows that
Also, we see from (3.20) that
Next suppose
Hence, we see from (3.27) that
Then we see from (3.24) and (3.25) that
We have injective diffeomorphisms
which are compatible with the projections to
we see from (3.27) that the image of
Now let
we see from (3.28) that
(3.30)
where the last line yields the fiber
3.3 Local smoothings of
X
i
∪
X
j
∪
X
k
around
D
i
j
∪
D
j
k
∪
D
k
i
Here we suppose
Recall that we have a holomorphic volume form
Then as in Lemma 3.2, we see that
where we used in (3.31)
which follows from (3.4) and (3.13), respectively. As in Section 3.2, we can also regard
We see from (3.1), (3.9), (3.10), and (3.32) that
where
We also see from (3.20) and (3.31) that
Now we construct a family of local smoothings of
Fix
Let
For
We define a diffeomorphism
We define an injective diffeomorphism
Step 1
Fix
Note that the projection
Also, in the same way as we obtained (3.25) and (3.26) in Section 3.2, we have for
Thus, it follows from (3.39) that
Now if we restrict the domain of
which satisfies
due to (3.35) and (3.38). In particular, the induced
Step 2
Let us study
Thus, we can extend
where
Now define
so that we have
Step 3
Let
Then
where
To define an
where
is a diffeomorphism for
Meanwhile, if
Thus to make
We have an alternative expression of
where
and (3.41) follows from
Step 4
Let
which is compatible with the projections to
these satisfy a cocycle condition
Let
so that
(3.42)
where all arrows are diffeomorphisms. By this diagram, we can regard each
Step 5
As in Section 3.2, we have an injective diffeomorphism
Meanwhile
and under the identification of
For
under which
Hence, regarding
(3.44)
where
Note that at this point we have only constructed differential geometric smoothings, and thus each fiber over
3.4 Existence of holomorphic volume forms on global smoothings
Here we shall prove Theorem 1.3.
Proof of Theorem 1.3
Let
For each double curve
Lemma 3.3
There exists a Hermitian form
Proof
We shall follow the argument in [4], Section 3.3. Let
Letting
Lemma 3.4
There exists a smooth complex 1-form
In particular, we have
Proof
The first assertion follows from (3.23), (3.33), and [4], Proposition 3.4. Then the second assertion follows from (3.34).□
Hence for
where
Then
Thus,
where the norm is measured by the associated metric
Now recall that
under
Lemma 3.5
We have estimates
for some positive constants C which are independent of
Proof
See [4], Proposition 3.6.□
Under a diffeomorphism
where
Since one can see that as
for some positive constants
which implies that the complex structure
3.5 Examples of
d
-semistable SNC complex surfaces with trivial canonical bundle without triple points
Here we shall give some examples of
Example 3.6
For
(see, e.g., [13], p. 608), we calculate the canonical bundle
so that
Meanwhile, using the adjunction formula and (3.48), we have
Since
where
Thus, putting (3.51) and (3.52) into (3.50) gives that
Now for the above cubic curve
we see that the Euler characteristic of the general fiber
where we used the invariance of homology under continuous deformations. Hence, we see from the Enriques-Kodaira classification of compact complex surfaces with trivial canonical bundle [1] that
Example 3.7
For
under which isomorphisms we have
Thus by the adjunction formula, we can calculate
by considering the transition functions.
Also, for a cubic curve
Now consider a local coordinate system
Also, let
Letting
Thus, we see that
so that
For
together with local embeddings into
Consequently, the general fiber
Hence by the classification of compact complex surfaces with trivial canonical bundle [1], we see that the general fiber
Example 3.8
Let
Here we shall give some examples of SNC complex surfaces for which condition (ii) holds but the canonical bundle is not trivial. The following example for the case of
Example 3.9
Let
Now let
Note that if
As we saw in Example 3.7,
Claim 3.10
We have
Proof
Applying Proposition 2.14 to our example, we see that
where
Noting
Hence, we find that
as desired. This completes the proof.□
We can modify the above example as follows. For a modification, we can take any number
4 Examples of
d
-semistable
K
3
surfaces of Type III
In this section, we provide several examples of
Let us recall Example 3.1, where we considered an SNC complex surface
We will rename
4.1 Blow-up of an SNC complex surface at finite points in double curves excluding triple points
Here we shall prove Proposition 4.1, which will be used in constructing examples of
Consider an SNC complex surface
We may assume
Then the blow-up
where
whereas
Note that the natural projections
Now if we define
then we have an open covering
we have
Meanwhile, the meromorphic volume form
Thus, we see that
which leads to condition (ii) of Theorem 1.3. Also, we have
which leads to condition (iii) of Theorem 1.3.
Defining
we see that
Extending the above local argument to the whole of the blow-up of
Proposition 4.1
Let
4.2 A
d
-semistable
K
3
surface of Type III
Here and hereafter, we will denote tensor products of line bundles by their sums. Also, we will denote the divisor class
Example 4.2
As we mentioned above, we rewrite the SNC complex surface with triple points in Example 3.1 as
which are shown in the Figure 1.
Double lines and triple points in
To make
for the SNC variety
The following description of
Lemma 4.3
An SNC complex surface X is d-semistable if and only if the collective normal class of X is trivial.
In fact, Lee showed that
Now we return to our example. In Example 3.1,
We choose two points
Let
Claim 4.4
The SNC complex surface
Proof
We use the same notation as above. Then we see that
where
for all
By Proposition 4.1 and Claim 4.4, we can apply Theorem 1.3 to obtain a family of smoothings
where
5 Two hyperplanes and a quadric surface in
C
P
3
We apply the argument in the previous section to more general SNC complex surfaces and will provide a more technical example. In this case, we encounter the issue that one cannot glue all components together along their intersections because the intersection parts are not isomorphic in general (see Section 5.2). This kind of problem does not happen in the case of the doubling construction [4,5]. However, we will see that there is a good way to handle this sort of mismatch problem by choosing carefully where and in what order we take the blow-ups.
5.1 Notation
Let
For
that is,
5.2 The mismatch problem
We denote the blow-up of
and take the proper transforms
5.3 A
d
-semistable SNC complex surface
As we saw in the previous section, if we choose the blow-up locus in a symmetric way, the mismatch problem may occur. Hence, we shall take the blow-up of each component
For
We take the blow-up of
Finally, we construct the desired
Step 1
For
Moreover, we claim the following isomorphism between
Claim 5.1
Proof
For
as desired.□
Let
The Blow-up
Step 2
Next we take the blow-up of
and consider the proper transform
Meanwhile,
The SNC complex surface
Step 3
Now we construct an SNC complex surface by gluing
Eventually, we obtain the following induced isomorphisms
We use (5.5) and (5.6) for gluing all components
Setting
Proposition 5.2
X is d-semistable.
The rest of this subsection is devoted to prove Proposition 5.2. In the light of Lemma 4.3, Proposition 5.2 is an immediate consequence of the following.
Claim 5.3
Let
Proof of Claim 5.3
For the first part of the statement, we saw already in Section 5.1. For the proof of the second part, it suffices to show that
for our purpose. Recalling
(i) Let
Thus, we have
(iii) Finally, we show that
as we mentioned in Step 2. Therefore, we have
This completes the proof of the claim.□
5.4 Computation of the Euler characteristic
Applying Theorem 1.3 to
Proposition 5.4
X is a d-semistable
Proof
We remark that the Euler characteristic of
(See Proposition 3.2 in [22].) By taking a sequence of blow-ups in Steps 1 and 2, we see that
Plugging these results into (5.8), we find that
Hence, one can reduce the problem to the computation of the Euler characteristic for each piece. For
Similarly, we have
Consequently, (5.9) yields that
Hence, the assertion is verified.□
Acknowledgements
The authors would like to thank Professor Kento Fujita for allowing us to use his example which is the source of Example 3.9. Naoto Yotsutani also thank Professors Nam-Hoon Lee, Taro Sano, and Yuji Odaka for fruitful discussions through e-mails. Finally, we are grateful to the referee for valuable comments which improved the presentation of our manuscript. This work was partially supported by JSPS KAKENHI Grant Number 18K13406 and Young Scientists Fund of Kagawa University Research Promotion Program 2021 (KURPP).
-
Conflict of interest: The authors have no competing interests to declare that are relevant to the content of this article.
References
[1] W. Barth, K. Hulek, C. Peters, and A. Van de Ven, Compact complex surfaces, Second edition, A Series of Modern Surveys in Mathematics, vol. 4, Springer-Verlag, Berlin, 2004. 10.1007/978-3-642-57739-0Search in Google Scholar
[2] R. Clancy, New examples of compact manifolds with holonomy Spin, Ann. Glob. Anal. Geom. 40 (2011), 203–222. 10.1007/s10455-011-9254-4Search in Google Scholar
[3] K. Chan, N.C. Leung, Z.N. Ma, Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties, J. Differential Geom. arXiv: 1902.11174v5. Search in Google Scholar
[4] M. Doi, Gluing construction of compact complex surfaces with trivial canonical bundle, J. Math. Soc. Japan 61 (2009), 853–884. 10.2969/jmsj/06130853Search in Google Scholar
[5] M. Doi and N. Yotsutani, Doubling construction of Calabi-Yau threefolds, New York J. Math. 20, (2014), 1203–1235. Search in Google Scholar
[6] M. Doi and N. Yotsutani, Doubling construction of Calabi-Yau fourfolds from toric Fano fourfolds, Commun. Math. Stat. 3, (2015), 423–447. 10.1007/s40304-015-0066-xSearch in Google Scholar
[7] M. Doi and N. Yotsutani, Gluing construction of compact Spin(7)-manifolds, J. Math. Soc. Japan 71 (2019), 349–382. 10.2969/jmsj/77007700Search in Google Scholar
[8] S. Felten, M. Filip, and H. Ruddat, Smoothing Toroidal Crossing Spaces, arXiv: 1908.11235, to appear in Forum of Mathematics Pi. 10.1017/fmp.2021.8Search in Google Scholar
[9] R. Friedmann and D. Morrison, (Eds.), The Birational Geometry of Degenerations, Progress in Mathematics (PM, volume 29), Birkhäuser, Boston, 1983. Search in Google Scholar
[10] R. Friedman, Global smoothings of varieties with normal crossings, Ann. Math. 118 (1983), 75–114. 10.2307/2006955Search in Google Scholar
[11] K. Fujita, Private Communication, 2021. Search in Google Scholar
[12] R. Goto, Moduli spaces of topological calibrations, Calabi-Yau, hyperKähler, G2 and Spin (7) structures, Int. J. Math. 15 (2004), 211–257. 10.1142/S0129167X04002296Search in Google Scholar
[13] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library. John Wiley and Sons, Inc., New York, 1994, xiv+813 pp. 10.1002/9781118032527Search in Google Scholar
[14] V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, New Jersey, 1974, xvi+222 pp. Search in Google Scholar
[15] K. Hashimoto and T. Sano, Examples of non-Kähler Calabi-Yau 3-folds with arbitrarily large b2, arXiv:1902.01027v2, to appear in Geom. Topol. Search in Google Scholar
[16] D. Huybrechts, Lectures on K3 Surfaces, Cambridge Studies in Advanced Mathematics, vol. 158, Cambridge University Press, Cambridge, 2016, xi+485 pp. 10.1017/CBO9781316594193Search in Google Scholar
[17] D. D. Joyce, Compact manifolds with special holonomy, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000, xii+436 pp. 10.1093/oso/9780198506010.001.0001Search in Google Scholar
[18] Y. Kawamata and Y. Namikawa, Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties, Invent. Math. 118 (1994), 395–409. 10.1007/BF01231538Search in Google Scholar
[19] A. Kovalev, Twisted connected sums and special Riemannian holonomy, J. Reine Angew. Math. 565 (2003), 125–160. 10.1515/crll.2003.097Search in Google Scholar
[20] V. Kulikov, Degenerations of K3 surfaces and Enriques surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 5, 1008–1042, 1199. 10.1070/IM1977v011n05ABEH001753Search in Google Scholar
[21] N.-H. Lee, Constructive Calabi-Yau Manifolds, Ph.D. Thesis, University of Michigan, 2006. Search in Google Scholar
[22] N.-H. Lee, d-semistable Calabi-Yau threefolds of Type 3, Manuscripta Math. 161 (2020), 257–281. 10.1007/s00229-018-1097-xSearch in Google Scholar
[23] K. Nishiguchi, Degeneration of K3 surfaces, J. Math. Kyoto Univ. 28 (1988), 267–300. 10.1215/kjm/1250520482Search in Google Scholar
[24] U. Persson, On degenerations of algebraic surfaces, Memoirs A.M.S. 189 (1977), xv+144 pp. 10.1090/memo/0189Search in Google Scholar
[25] H. Ruddat and B. Siebert, Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerations, Publ. Math. IHES 132 (2020), 1–82.10.1007/s10240-020-00116-ySearch in Google Scholar
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Articles in the same Issue
- Research Articles
- Differential geometric global smoothings of simple normal crossing complex surfaces with trivial canonical bundle
- Chow transformation of coherent sheaves
-
On the algebra generated by
- Second Chern-Einstein metrics on four-dimensional almost-Hermitian manifolds
- Towards the cosymplectic topology
- Quot schemes and Fourier-Mukai transformation
- Deformations of astheno-Kähler metrics
- Partial slice regularity and Fueter's theorem in several quaternionic variables
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