On the Futaki Invariants of Complete Intersections
Key words and phrases:
Futaki invariant, complete intersection, KählerEinstein metric1991 Mathematics Subject Classification:
Primary: 32F07; Secondary: 58G031. Introductions
In 1983, Futaki [2] introduced his invariants which generalize the obstruction of KazdanWarner to prescribe Gauss curvature on . The Futaki invariants are defined for any compact Kähler manifold with positive first Chern class that has nontrivial holomorphic vector fields. Their vanishing are necessary conditions to the existence of KählerEinstein metric on the underlying manifold.
Let be a compact Kähler manifold with positive first Chern class . Choosing an arbitrary positive form in as a Kähler metric on , we can find a smooth function on , determined up to a constant, such that the following
(1.1) 
holds. Let be the Lie algebra of holomorphic vector fields on . The Futaki invariants are defined as
Ding and Tian [1] took a further step in introducing the Futaki invariants to Fano normal varieties. This is not only a generalization of Futaki invariants to singular varieties, but also has important application in KahlerEinstein geometry. In [8], the generalized Kenergy on normal varieties was introduced and a stability criteria for the hypersurface or complete intersection was established by using the notion of the generalized Kenergy. The Futaki invariants on singular varieties are related to the stability of Fano manifolds due to the work of Tian [9]. To be more precise, checking the stability of a Fano manifold is the same as checking the sign of the real part of the Futaki invariants on the degenerations of the Fano manifold. Because of this, we need an effective way to compute the Futaki invariants on singular varieties.
In this paper, we give a simple formula for the Futaki invariants of Fano complete intersections. The main theorem of this paper is:
Theorem 1.1.
Let be the dimensional normal Fano variety in defined by the homogeneous polynomials of degree respectively. Let be a holomorphic vector field on such that
(1.2) 
for constants . Then the Futaki invariant is
(1.3) 
where .
Corollary 1.1.
If is a hypersurface in defined by the homogeneous polynomial of degree and if , then
In particular, and have the same sign.
The formula is new even in the case when is a hypersurface or an orbifold. If the zero locus of the holomorphic vector field on is a smooth manifold, then using the residue formula of the AtiyahBottLefschetz type, Futaki was able to develop a method to compute his invariants by the information of the vector field and the manifold near the zero locus of the vector field [3]. In [1], the authors developed the method to compute the Futaki invariants on 2dimensional Kähler orbifolds. In [6], the Futaki invariants for toric varieties were calculated.
Acknowledgment. The author thanks G. Tian for his mathematical insights and encouragements during the preparation of this paper. He also thanks D. Phong, M. Kuranishi, H. Pinkham, L. Borisov and Z. Wu for the discussions of this topic.
2. Preliminaries
Let be a traceless matrix. defines a holomorphic vector field
(2.1) 
on and a smooth function
(2.2) 
on , where is the coordinates of .
Both and descends to a vector field and a smooth function on the projective space , respectively. Let be the FubiniStudy metric of . Then we have the following relation:
(2.3) 
Suppose is an dimensional Fano normal variety in and suppose that for a constant . If is a vector field on such that the one parameter group generated by the real part of leaves invariant, we say that is tangent to . Suppose is the local holomorphic coordinates at some smooth point of . Equation (2.3) can be written as
(2.4) 
where is the inverse matrix of and is the metric matrix of .
We define the divergence of on by
(2.5) 
The following lemma is the observation on which the whole paper is based.
Lemma 2.1 ([9]).
If is a normal projective variety, then
(2.6) 
where the function is defined as
(2.7) 
Proof: A straightforward computation yields
By Equation (2.7), we see that
Thus is a holomorphic function on the normal variety which must be a constant.
∎
Corollary 2.1.
The Futaki invariant can be written as
∎
3. An Explicit Expression of the Function
Suppose is a complete intersection of . That is, is the zero locus of homogeneous polynomials in with degree , respectively and the dimension of is . By the adjunction formula, the anticanonical bundle of is
where is the hyperplane bundle of . We assume that is a normal variety. There is a unique function (up to a constant), defined on the regular part of , such that if , then
In this section, we write out the above function explicitly. The idea is to trace the proof of the well known adjunction formula. But here we work on the metric level rather than the cohomological level. This makes the notations a little bit complicated.
We begin by the following general setting: Let be a Kähler manifold of dimension and be a submanifold of dimensional defined by holomorphic functions . Suppose is an open set of such that

is a local holomorphic coordinate system of ;

On , we have

There are holomorphic functions on such that
In particular, is the local holomorphic coordinate system of .
Suppose is the restriction of the Kähler metric of on . Define . Of course, is not a global function on . In order to study the change of the with respect to the change of the local holomorphic coordinates, we assume that there is another neighborhood of such that . As before, we assume that
and on , we have
for holomorphic functions on . is the local holomorphic coordinate system of . Let
be the restriction of the Kähler metric of on . Define . Then we have
Proposition 3.1 (Adjunction Formula).
With the above notations, on , we have
Proof: Let
and
Then is locally defined by or . In particular, on we have
(3.1) 
Before going further, we make the following conventions:



Since and are local coordinates of and respectively, there is the transform by
for and
If , then
Using Equation (3.1), we have
(3.2) 
∎
We are going to use the above proposition in the case of complete intersections of . Since is defined by the zero locus of homogeneous functions, we must make some necessary adjustment because homogeneous polynomials are not functions on .
Let be the standard covering of defined by where is the homogeneous coordinates of . Suppose be the standard coordinates on . Let
For each , define
Then it is clear that .
At each point , can be used as local coordinate system at . Let be the corresponding metric matrix and let be its determinant. Define
(3.4) 
Then we have
Lemma 3.1.
defines a global positive function of .
Proof: A straightforward computation shows (cf. [4, pp. 146])
The lemma follows from Proposition 3.1, Equation (3.4) and the above equation.
∎
Theorem 3.1.
Let be the function on , defined by
(3.5) 
Then
where .
∎
4. The Trace of the Action on
Let be the variety defined in the previous section. The vector field naturally acts on by
(4.1) 
Suppose is the vector space spanned by . Since is tangent to , is an automorphism on .
The main result of this section is,
Theorem 4.1.
Let be the trace of the automorphism of on . Then
Proof: We adopt all notations from last section. Consider a smooth point of in . From Equation (2.2), the function in the local coordinates is
(4.2) 
By (2.1), let
(4.3) 
Let the holomorphic vector field on be written as
(4.4) 
If is the embedding, then .
Before going on, we need a general elementary lemma. To begin, we use the general setting on page 3. In addition, we let be a holomorphic vector field of such that is tangent to . In what follows, we temporary distinguish the on and the on . So let’s denote the on to be . In the local coordinates, is
Then on can be written as
where from the chain rule,
(4.6) 
If is the embedding, then .
We have the following elementary lemma:
Lemma 4.1.
Let
and let for and be the inverse matrix of . Then on , we have
Proof: By definition,
We can write the above equation as
(4.7) 
By the implicit differentiation, we see that on ,
(4.8) 
Using Equation (4.7) and (4.8),
The lemma follows from the above identity and the fact that
∎
5. The Computation of the Invariants
Let be the complete intersection defined in §3. Let . . Then .
We assume that .
Let be the homogeneous coordinates of . Define
Then ’s are global smooth functions on .
In this section, we compute the invariant , where for simplifying the notations, we assume that is the FubiniStudy metric of the . The key result is the following:
Lemma 5.1.
For , we have
(5.1) 
and in addition, we have
(5.2) 
Proof: We have the following identities for
(5.3) 
Integration against gives
Since for ,
where is the divisor of the zero locus of , we have