Shiyu CAO

Abstract In this paper, the author discusses the deformations of compact complex manifolds with ample canonical bundles.It is known that a complex manifold has unobstructed deformations when it has a trivial canonical bundle or an ample anti-canonical bundle.When the complex manifold has an ample canonical bundle, the author can prove that this manifold also has unobstructed deformations under an extra condition.

Keywords Complex structures, Deformations, Iteration Methods

1 Introduction

Let (X,ω) be a compact Khler manifold with dimension dimCX=n, and we denote its canonical bundle byKX.In the last several decades, there have been a large amount of results about the deformations of complex structures on compact complex manifolds, for example[11, 16].The most fundamental theorem established by Kodaira and Spencer states that on a compact complex manifoldX,an element?∈A0,1(X,T1,0X),which we usually call a Beltrami differential, determines a new complex structure once it solves the Maurer-Cartan equation

They also showed that the obstruction of the deformations lies in the cohomology group H2(X,T1,0X).Consequently, whenXis a Fano manifold, i.e.,is ample, by the Kodaira vanishing theorem, we see that

because of the negativity of the line bundleKX, which yields that all Fano manifolds have unobstructed deformations.

When the manifoldXis Calabi-Yau, i.e., the canonical bundleKXis trivial, the deformations are also unobstructed according to Bogomolov, Tian and Todorov, which is now widely known as the Bogomolov-Tian-Todorov theorem (see [2, 24–25]).Besides, there are also many noteworthy results concerning the deformations of logarithmic Calabi-Yau pairs, for example,[9,13].It is worth pointing out that the research concerning the deformations of other interesting structures in complex geometry also have a lot of breakthrough in recent years,for example[19–21].Note that in [15] there is a more global method to deal with the deformation theory.

WhenKXis ample,it is much more complicated.There are examples that the deformations may be obstructed.For example, Horikawa [8, Section 10] constructed an example as follows.First,by studying the deformations of holomorphic maps,he got that the monoidal transformationYof the complex projective space CP3has obstructed deformations, where the centerCis a curve of degree 14 and of genus 24 in CP3which was constructed by Mumford [17].Horikawa then showed that ifXis a general element of a sufficiently ample linear system onY, thenXis non-singular, irreducible, and has an ample canonical bundle, and then he showed thatXhas obstructed deformations by showing that its Kodaira-Spencer map is not surjective.

However,there are also examples that some certain compact complex manifolds with ample canonical bundles have unobstructed deformations, such as ample hypersurfaces in an Abelian variety (see [4]) and surfaces of type IIb, which are birational to the quintic hypersurface in CP3(see [7]).

Thus, it is natural to ask what the obstruction of the deformations is and whether it has a Hodge theoretic characterization when the canonical bundle is ample.

In this paper,we use the Hodge theory and the iteration method to explore the obstruction.We will solve (1.1) and express the solution as a formal power series

whenKXis ample.

Explicitly speaking, we begin with an arbitrary harmonic initial value?1∈H0,1(X,T1,0X)and solve the reduced equations (2.8) by induction with an extra condition that the essential obstruction vanishes:

The solution at step 2 (which means the coefficient oft2in?(t)) is expressed as

Here H is the orthogonal projection of differential forms to their harmonic parts,G is the Green operator of, ?′is the (1,0)-component of the Chern connection on the anticanonical bundle and ?0is a globally defined and nowhere vanishing element in An,0(X,), which can be written as

under a local coordinate (z1,···,zn).

The notion?┙?0denotes the contraction between elements in A0,q(X,T1,0X)and ?0,which induces an isomorphism

And we denote the inverse by.

By running induction, the solution we obtain at theN-th step can be expressed as

for any positive integerN.

The solutions we have obtained till theN-th step can be put together and written as

where?iis the solution at stepi(which means the coefficient oftiin?(t)),1 ≤i≤N?1.Here?N=?1t+···+?NtN.

By doing so, the solution?(t) can eventually be expressed as

which is uniquely determined by the harmonic initial value?1.

Note that at each step the condition(1.2)means H(?′?i?i?i?j?0)=0 for the correspondingi,j.

In conclusion, we obtain the following theorem.

Theorem 1.1Let X be a compact complex manifold with an ample canonical bundle KX.IfH(?′?i??i??0) = 0, where ? is defined by(1.4), then X has unobstructed deformations.Here?0is a nowhere vanishing element inAn,0(X,)defined in(1.3).

Remark 1.1There are examples satisfying our condition H(?′?i??i??0)=0, e.g.

(1) Compact Riemann surfaces with genus at least 2.

(2)The manifolds likeX=X1×···×Xmfor any integerm≥2 where eachXiis a compact Riemann surface with genus at least 2,i=1,···,m.

Both of them have ample canonical bundles and thus by Theorem 1.1 they have unobstructed deformations.

In addition, we need to point out that our method also works whenc1(X) = 0, i.e., whenKXis a torsion line bundle.

Corollary 1.1(see [24–25])If c1(X) = 0, i.e., KXis a torsion line bundle, then X has unobstructed deformations.

This paper is organized as follows.In Section 2, we present some basic notions and reduce the Maurer-Cartan equation (1.1) into two equations (2.8).In Section 3, we solve the reduced equations when the canonical bundle is ample and discuss some examples about the obstruction.Besides, we also show that our method still works whenKXis a torsion line bundle.

2 Reduction of the Equation

Inspired by the work of Liu, Rao and Wan[13], we first reduce the Maurer-Cartan equation(1.1) into two equations.

Let (X,ω) be a compact K?hler manifold.In terms of a local coordinate,

Selecting a nowhere vanishing section ? of An,0(X,), we have an isomorphism obtained by contraction:

And we denote the inverse by

Here the notion?┙(?) denotes the contraction between tangent vectors and differential forms that dual to each other.Sometimes we also use the notioni?(?) to denote the same operation.

Throughout this paper, we need the following technical lemma.

Lemma 2.1For any ?,ψ∈A0,1(X,T1,0X)and? ∈An,q(X), we have

For the proof, the generalizations and further applications of this lemma, one can refer to[12, 14].

There is a unique Chern connection ?=?′+on the Hermitian line bundle(det()).Therefore, similar to Lemma 2.1, we have the following Tian-Todorov lemma (e.g.in [12,Theorem 3.4])

Before reducing the Maurer-Cartan equation, we need some preparations.

Definition 2.1For an element ?∈A0,1(X,T1,0X), the divergence operator is defined by

In terms of a local coordinate(z1,···,zn), we write.Thus

Since div(?) is a (0,1)-form, it is obvious that

Proposition 2.1Let ? be an element inA0,1(X,T1,0X)and?be a nowhere vanishingelement inAn,0(X,).

ProofWe assume that the equations in (2.3) hold.Note that

By the assumption, the left-hand side of (2.4) is

and the right-hand side of (2.4) is

Comparing the two sides of (2.4) we have

and we get

since the operation ?┙? is an isomorphism.

In order to simplify the subsequent calculations, we need the following lemma.

Lemma 2.2Denote

whereΦ(z)∈A0,1(X)and

where?(z)is a smooth function on X.Then we have

Here the notionsdz andcan be locally written as

ProofOn one hand, we know that

On the other hand,

Hence

which implies the conclusion.

From now on,our aim is to solve equations(2.3)by using the Hodge theory and the iteration method.To do this, following the approach of Kodaira and Spencer [11, 16], we expand the terms?and ? into power series

Thus the terms Φ(z) and ?(z) defined in Lemma 2.2 can also be expanded into power series int.

Throughout this paper, we usually choose a harmonic?1as the initial value, i.e.,= 0 and=0.

The following proposition reveals the legality of the iteration method in the study of deformation theory.

Proposition 2.2If for any k≤N?1we have

we then derive that

Here the subscript[?]kdenotes the coefficient of tkonce we expand both equations in(2.3)into power series of the variant t.

ProofAccording to Proposition 2.1, the condition implies that

for anyk≤N?1.

Note that the first equation in (2.6) to be proved is equivalent to(z)N= 0 while the second one in (2.5) that we assumed is equivalent to ((z)?Φ(z))N?1=0.Then by explicit calculations we have

where in the third equality we used the Tian-Todorov lemma.

Meanwhile, we have

and then

where in the first equality we used the Tian-Todorov lemma and in the third equality we used the assumption that the equations in (2.5) hold in lower degrees and the fact that the initial value?1is harmonic so that=0.

Although Proposition 2.2 enables us to solve the equations(2.3)by induction and then solve the Maurer-Cartan equation

there is a straightforward way to deal with the problem.Indeed, as we pointed out in the proof of Proposition 2.2, the second equation in (2.3) is equivalent to

which has a trivial solution.Then the original equation also has a trivial solution

where dzandare defined in Lemma 2.2.

Then it suffices to solve the equation

By direct calculations, we have

Then by (2.2) we have

In conclusion,the equations that we need to solve can be reduced to the following equations

3 Solving the Equations

In this section, we solve the equations (2.8) on a compact K?hler manifold (X,ω) when the canonical bundleKXis ample or a torsion line bundle separately.

First, we state a technical lemma about the divergence of the Beltrami differential div(?)which is known to experts in this area(see[22–23,28]).For the readers’convenience,we present the proof here.

Lemma 3.1(see[22,28])Let(X,ω)be a compact K?hler manifold.Let ?∈A0,1(X,T1,0X)and?′′be the Laplacian operator of.Then we have

ProofLocally we writeThe lemma can be proved by direct calculations.

(1) For the first term, we have

where in the last equality we used the condition

(2) For the second term, we have

3.1 When KX is ample

LetXbe a compact K?hler manifold with an ample canonical bundleKX.SinceKXis ample, there is a Hermitian metrichonKXsuch that its curvature form gives rise to a K?hler metric

onX.For any harmonic initial value?1∈H0,1(X,T1,0X), we try to construct a power series

satisfying the Maurer-Cartan equation

As we did in the last section, we denote

which gives rise to an isomorphism between A0,q(X,T1,0X) and An?1,q(X,) through contraction?┙?0.The inverse is denoted by ??0┙?.Clearly,for any elementsα, β∈A0,q(X,T1,0X),we have the following equalities

where 〈·,·〉 denotes the inner product on the space of (bundle-valued) differential forms.Then the operation ?┙?0preserves the inner product and the Hodge decomposition

where H is the orthogonal projection of a(bundle valued)differential form to its harmonic part,?′′is the Laplacian operator ofand G is the Green operator of ?′′.

In other words, we have an isomorphism between two spaces of harmonic forms

The following lemma wonderfully reflects the spirit of the iteration method and is of significant importance in the proof of the main theorem.

Lemma 3.2Assume that for ?ν∈A0,1(X,T1,0X), ν=2,···,K,

Then one has

The readers who are interested in the proof can refer to [14, Lemma 4.2].

Now we are ready to solve the reduced equations (2.8) whenKXis ample with an extra condition which is an essential obstruction in this case.

Theorem 3.1Let X be a compact complex manifold with an ample canonical bundle.IfH(?′?i??i??0) = 0for any ?1∈H0,1(X,T1,0X), where ? is defined by(1.4), then there exists a power series solving(2.8).Therefore, X has unobstructed deformations.

ProofAs we are going to solve the equations (2.8) upward from?1with respect to the degree of the formal variantt, the condition H(?′?i??i??0)=0 means

for any positive integersiandj,where?iis what we get at thei-th step of the iteration process as the coefficient ofti.

For anyα∈Ap,q(X,), the Bochner-Kodaira identity states that

where ?′is the Laplacian operator of ?′.

Since?1∈H0,1(X,T1,0X), so?1┙?0∈Hn?1,1(X,).By (3.1), we have

i.e., ?′(?1┙?0)=0 and ?′?(?1┙?0)=0.Then by the Tian-Todorov lemma, we have

Thus

According to the Hodge theorem [6, p.84], the condition H(?′?i?1?i?1?0) = 0 implies that we can take the solution?2as

This is the solution of the first equation in (2.8) at the second step.

As a consequence, we have

where in the fourth equality we used the fact thatThis is the second equation of(2.8) at the second step.

By running induction, we assume that we have obtained the solutions up to theN-th step,i.e., we have already constructed?k, 1 ≤k≤N.The proof will be accomplished as soon as we construct the solution?N+1.

As the?′ksare assumed to be constructed (k≤N), by the Tian-Todorov lemma again we have

for any positive integersi,jsuch thati+j=N+1.

Then combining Lemma 3.2 with the calculations above, one has

Since H(?′?i?i?i?j?0)=0, we can take?N+1as

Then, similar to (3.4), it holds that

Remark that in the view point of iteration one has

where?N=?1t1+···+?NtNcan be treated as the truncation of?(t) at theN-th step.

Therefore, we eventually obtain a solution given by

which is uniquely determined by the chosen harmonic initial value?1.

Remark 3.1Due to (3.3) and (3.7), we have?k∈fork≥2.

Example 3.1It is clear that on a compact Riemann surface, the condition H(?′?i??i??0)=0 holds due to the dimension.

LetX=X1×X2, where eachXiis a compact Riemann surface of genusgi≥2,i= 1,2.ThenKXis clearly ample.

We take a local coordinate {z1,z2} onXsuch that eachziis the local coordinate ofXi,i=1,2.Then we have

From the last example,we know that there is a Beltrami differential∈A0,1(Xi,T1,0Xi)on eachXidetermining the unobstructed deformations ofXi,i=1,2.Under the local coordinate we can write them asis a smooth function only inzi(i= 1,2).Then we have

Repeating the calculations in Section 2 we have

on eachXi,i=1,2.

Then

which implies that H(?′?i??i??0)=0.ThenX=X1×X2has unobstructed deformations.Throughout the calculations above,the notion ?′idenotes the covariant derivative inzi,i=1,2.

By the same arguments,one easily knows that the manifolds of the formX=X1×···×Xmalso have ample canonical bundles and satisfy the condition H(?′?i??i??0)=0, where eachXiis a compact Riemann surface with genusgi≥2 (i= 1,···,m).Therefore they have unobstructed deformations.

Remark 3.2The condition H(?′?i??i??0) = 0 is essential in the proof of our main theorem.It may look a little complicated at first,but it can be improved into a somewhat more geometric form.

First, we claim that

Indeed, for anyα∈An?1,0(), one has

which implies the claim.HereVis some vector field of (1,0)-type.

Proposition 3.1If KXis ample, and X satisfiesHn?1,2(X,K?1X )?Ker(?′?), then X hasunobstructed deformations.

ProofIf Hn?1,2() ?Ker(?′?), for any harmonic elementγ∈Hn?1,2(), we have

which implies H(?′?i??i??0)=0.By Theorem 3.1, the deformations are unobstructed.

Recall that the contraction ?┙?0and its inverse ??0┙? give rise to an isomorphism between harmonic spaces

Locally the operator Λ can be written as

Then we have

Thus the condition Hn?1,2(X,K?1X)?Ker(?′?) is equivalent to

The characterization(3.18)seems make more sense in geometry than the original one since the harmonic space H0,2(X,T1,0X)is isomorphic to the cohomology group H2(X,T1,0X),which contains the obstructions of the deformations (see [11, 16]).

Remark 3.3Note that a projective varietyXis said to satisfy the Bott vanishing theorem,if Hi(X,?j(L))=0 for all the ample line bundles overX,wherei>0,j≥0.Bott showed that it holds for projective spaces.A good reference about it is[10,Chapter 3.4].Later this theorem was generalized to the toric case (the proof can be found in [1, 3, 5, 18]) and some certain Del Pezzo surfaces andK3 surfaces (see [26]).But they are all beyond our consideration.We remark that any smooth variety with ample canonical bundle has unobstructed deformations,once it satisfies the Bott vanishing theorem.

Remark 3.4IfXis a nonsingular irreducible hypersurface of CP3of degreed.According to [11, (6.49)], we have the fact that dimHn?1,2() =(d?2)(d?3)(d?5).Whend= 5, by the adjunction formula, we see thatKXOX(1), which is ample.In this case, the cohomology group containing the obstruction H(?′?i??i??0) vanishes.So we see that the quintic surface in CP3has unobstructed deformations.

3.2 When KX is a torsion line bundle

In this subsection, we show that our method also works when the compact K?hler manifoldXhas a torsion canonical bundleKX, i.e., there is an integermsuch that, the trivial line bundle overX.

Corollary 3.1If c1(X) = 0, i.e., KXis a torsion line bundle, then X has unobstructed deformations.

ProofAccording to Yau’s celebrated work [27], there exists a K?hler metricωonXsuch that Ric(ω) = 0.Similar to the ample case, we start with an arbitrary harmonic initial value?1∈H0,1(X,T1,0X) and try to construct a power series

which satisfies the Maurer-Cartan equation

By the arguments in Section 2, it suffices for us to solve the following equations

Under the Ricci-flat setting, the Bochner-Kodaira identity states that

for any- valued differential forms.Since the two Laplacian operators coincide, it follows that G?′=?′G, which, together with the fact thatimplies that

Thus ?′(i?2?0) = 0.By running induction, we assume that the solutions?ksatisfying?′(i?k?0) = 0 have already been constructed fork≤N?1.By the same operation in the last subsection, we obtain the solution?Ngiven by

such thati?N?0∈Im(?′).Hence the proof is completed.

Remark 3.5For the convergence and the regularity of the solution?(t) in both theKXample case and theKXtorsion case,there are many works concerning this,for example,[11,16]and more recently, [14, Theorem 4.3, Theorem 4.4] or [13, Proposition 4.10], etc.By repeating the calculations therein, one can obtain the convergence and the regularity of?(t) by standard analytic theory.

AcknowledgementsThe author would like to express his gratitude to Professors Huitao Feng and Kefeng Liu for their support, encouragement and guidance over years.And he would like to thank Professor Xueyuan Wan for his unselfish help and many stimulating discussions.