Chern number inequalities of deformed Hermitian-Yang-Mills metrics on four dimensional Kaehler manifolds
aa r X i v : . [ m a t h . DG ] A ug CHERN NUMBER INEQUALITIES OF DEFORMEDHERMITIAN-YANG-MILLS METRICS ON FOUR DIMENSIONALK ¨AHLER MANIFOLDS
XIAOLI HAN AND XISHEN JIN
Abstract.
In this paper, we investigate the Chern number inequalities on 4-dimensional K¨ahler manifolds admitting the deformed Hermitian-Yang-Millsmetrics under the assumption ˆ θ ∈ ( π, π ). Introduction
Let (
X, ω ) be a compact K¨ahler manifold of complex dimension n and L be aholomorphic line bundle over X . Given a Hermitian metric h on L , we define acomplex function ζ on X by ζ := ( ω − F h ) n ω n where F h = − ∂ ¯ ∂ log h is the curvature of the Chern connection with respect tothe metric h . It is easy to see that the average of this function is a fixed complexnumber Z L, [ ω ] := Z X ζ ω n n !depending only on the cohomology classes c ( L ) and [ ω ] ∈ H , ( X, R ). Let ˆ θ bethe argument of Z L, [ ω ] . Definition 1.1.
A Hermitian metric h on L is said to be a deformed Hermitian-Yang-Mills (dHYM) metric if it satisfies(1.1) Im( ω − F ) n = tan(ˆ θ ) Re( ω − F ) n . The equation (1.1) is called dHYM equation. The dHYM equation was firstdiscovered by physical scientists Marino et all [16] as the requirement for a D-braneon the B-model of mirror symmetry to be supersymmetric. This phenomenon wasexplained by Leung-Yau-Zaslow [14] in mathematical language. From a viewpointof differential geometry, this might be thought of a relationship between the ex-istence of ‘nice’ metrics on the line bundle over Calabi-Yau manifolds and specialLagrangian submanifolds in another Calabi-Yau manifolds. Recently, the dHYMmetrics have been studied actively(e.g. [5], [7], [8], [10], [9], [12], [17], [18], [20] etc).See also [6] for a survey.We define the Lagrangian Phase operator θ : ∧ , X → R by θ ( F ) = n X j =1 arctan λ j , here λ j ( j = 1 , · · · , n ) are the eigenvalues of ω − F ∈ End( T , X ). Then accordingto arguments in [12], the equation (1.1) is equivalent to(1.2) θ ( F ) = ˆ θ ( mod 2 π ) . In particular, we always change the interval of argument such that ˆ θ = θ ( F ) if F isa dHYM metric in the whole paper. This angle ˆ θ is also called the analytic liftedangle.Given two (1 , Z [ ω ] ,c ( L ) ( t ) = − Z X e − t √− ω ch ( L )as t runs from + ∞ to 1 if Z [ ω ] ,c ( L ) ( t ) does not cross 0 ∈ C . Here ch ( L ) = e c ( L ) is the Chern character of L . More precisely, Z [ ω ] ,c ( L ) ( t ) = − Z X ( c ( L ) − t √− ω ) n n ! . This “winding angle” is also called algebraic lifted angle. The motivation to definethe algebraic lifted angle is to compute the analytic lifted angle whenever the laterexists. As discussed in [7], in dimension less than 4, if F is a dHYM metric andˆ θ ∈ ( ( n − π , nπ ), then the analytic lifted angle ˆ θ equals the algebraic lifted angle.As pointed in [6], the only obstacle to define the algebraically lifted angle is thepossibility that Z [ ω ] ,c ( L ) ( t ) = 0 for some T ∈ [1 , + ∞ ), since if this case occurs, the“winding angle” is no-longer well-defined. When dim C X = 1, this can not occursince Im( Z [ ω ] ,c ( L ) ( t )) >
0. When dim C X = 2, Z [ ω ] ,c ( L ) ( t ) ∈ C ∗ = C \ { } by theHodge index theorem as in [7]. In the case of dim C X >
2, Collins-Yau [8] gavean example model on Bl p P such that Z [ ω ] ,c ( L ) ( t ) can pass through the origin.Collins-Xie-Yau [7] proved the following Chern number inequality for dim C X = 3. Theorem 1.1 ([7]) . Suppose ( X, ω ) is -dimensional K¨ahler manifold and L admitsa dHYM metric with analytic lifted angle ˆ θ ∈ ( π , π ) . Then (cid:18)Z X ω (cid:19) (cid:18)Z X c ( L ) (cid:19) < (cid:18)Z X c ( L ) ∧ ω (cid:19) (cid:18)Z X c ( L ) ∧ ω (cid:19) . The Chern number inequalities play very important roles in the study of canoni-cal metrics. For example, in the study of Hermian-Einstein metrics, Bogomolov [2]obtained the Bogomolov inequality for semi-stable holomorphic vector bundle andSimpson [19] proved the Bogomolov inequality for stable Higgs bundles on compactK¨ahler manifolds by constructing Higgs Hermitian-Einstein metrics. Furthermore,in the study of K¨ahler-Einstein metrics, Miyaoka [15] and Yau [26] proved the fa-mous Miyaoka-Yau inequality. The Miyaoka-Yau inequality can also be obtained byconstructing Higgs structure on T , ( X ) ⊗O X . A celebrated conjecture by Thomas-Yau [23, 24] is that the existence of special Lagrangian submanifolds in Calabi-Yaumanifolds is equivalent to some stability conditions. In mirror symmetry, some sta-bility conditions are needed in order to obtain the existence of dHYM metrics. Thiswas studied in a series of works [5, 7, 8]. The Chern number inequality in Theorem1.1 is precisely the line bundle case of the inequality conjectured by Bayer-Macri-Toda [1] in their construction of Bridgeland stability conditions. Furthermore, asdescribed in [8], the Chern number inequalities ensure that Z [ ω ] ,c ( L ) ( t ) ∈ C ∗ for all ∈ [1 , + ∞ ). So the “winding angle” can be well-defined and also is the algebraiclifted angle.In [8], Collins-Yau proposed the following conjecture of Chern number inequali-ties on the 4-dimension case. Conjecture 1.2 ([8]) . Suppose ( X, ω ) is a four dimensional compact K¨ahler man-ifold and L admits a dHYM metric F with constant angle ˆ θ ∈ ( π , π ) . Then thefollowing Chern number inequalities hold c ( L ) · ωc ( L ) · ω > and ( c ( L ) · ω )( ω ) c ( L ) · ω − c ( L ) · ω ) + ( c ( L ) · ω )( c ( L ) ) c ( L ) · ω < . Let us recall the motivation of Conjecture 1.2 in [8]. If dim C X = 4, the path Z [ ω ] ,c ( L ) ( t ) can be expressed by − Z X ( t ω − t ω ∧ c ( L ) + c ( L ) ) − t √− Z X ( t c ( L ) ∧ ω − c ( L ) ∧ ω ) . For t ≈ + ∞ , Z [ ω ] ,c ( L ) ( t ) lies near the negative real axis. Furthermore, at t = 1, Z [ ω ] ,c ( L ) (1) = − Z X ( c ( L ) − √− ω ) = − e √− θ R ([ ω ] , c ( L )) , where R ([ ω ] , c ( L )) is a positive constant. Hence, if ˆ θ ∈ ( π, π ), then Z [ ω ] ,c ( L ) (1)lies on the upper half-plane. Then Z [ ω ] ,c ( L ) ( t ) must cross the positive real axis atsome t = T ∗ >
1, i.e. we must have Im( Z [ ω ] ,c ( L ) ( T ∗ )) = 0. So we should have c ( L ) · ωc ( L ) · ω > . Furthermore, at T ∗ we must have that Re( Z [ ω ] ,c ( L ) ( T ∗ )) > L admits a dHYM metric F with constant angle ˆ θ ∈ ( π, π ).In this paper, we prove the Chern number inequalities under the assumptionˆ θ ∈ ( π, π ) using the Khovanskii-Teissier inequalities in [4, 11]. Actually, we canprove the following theorem. Theorem 1.3.
Suppose ( X, ω ) is a four dimensional K¨ahler manifold and L admitsa dHYM metric F with constant angle ˆ θ ∈ ( π, π ) . Then the following Chernnumber inequalities hold c ( L ) · ωc ( L ) · ω > and ( c ( L ) · ω )( ω ) c ( L ) · ω − c ( L ) · ω ) + ( c ( L ) · ω )( c ( L ) ) c ( L ) · ω < . Acknowledgements:
The authors would like to thank T. Collins and J. Xiaofor some helpful discussions. The research is partially supported by NSFC 11721101.The second author is also supported by the Fundamental Research Funds for theCentral Universities and the Research Funds of Renmin University of China. . Preliminaries on Khovanskii-Teissier inequalities
In this section, we review the Khovanskii-Teissier inequality. We always assumethat (
X, ω ) is an n -dimension compact K¨ahler manifold. The original Khovanskii-Teissier inequality was discovered by Khovanskii [13] and Teissier [21, 22]. TheKhovanskii-Teissier inequality tells us that for any β ∈ H , ( X, R ), there holds (cid:18)Z X β ∧ ω n − (cid:19) (cid:18)Z X ω n (cid:19) ≤ (cid:18)Z X β ∧ ω n − (cid:19) . This inequality can be viewed as a generalization of the Hodge Index Theorem. Infact, the Khovanskii-Teissier inequality has been extended beyond the K¨ahler cone,see [3, 4, 11]. In this paper, we need the generalized Khovanskii-Teissier inequalitiesrelated to the complex Hessian equations. These generalized Khovanskii-Teissierinequalities can be found in [4, 11]. We first recall the definition of k -positive cone K Γ k ,ω in H , ( X, R ) as in [4]. Definition 2.1 ([4]) . We denote K Γ k ,ω ⊂ H , ( X, R ) the cone of classes admittinga k -positive representative with respect to ω , i.e., for any [ α ] ∈ K Γ k ,ω , there existsa closed (1 , α ∈ [ α ] such that the eigenvalues Λ = ( λ , · · · , λ n ) satisfies σ i (Λ) > i = 1 , · · · , k , where σ i is the i -th elementary symmetric polynomial. Remark . K Γ k ,ω is an open convex cone in H , ( X, R ). It is easy to see K Γ n ,ω ⊂ K Γ n − ,ω ⊂ · · · ⊂ K Γ ,ω and K Γ n ,ω is the K¨ahler cone.Collins [4] and Xiao [11] proved the following generalized Khovanskii-Teissierinequalities on K Γ k ,ω by using the complex Hessian equations and the concavityinequalities for elementary symmetric polynomials. Theorem 2.1 ([4, 11]) . Let ( X, ω ) be a compact n -dimensional K¨ahler manifold.For any [ α ] ∈ K Γ k ,ω and [ β ] ∈ H , ( X, R ) , there holds the following generalizedKhovanskii-Teissier type inequalities (cid:0) ω n − m · α m − · β (cid:1) (cid:0) ω n − m · α m (cid:1) ≤ (cid:0) ω n − m +1 · β · α m − (cid:1) , for all m = 2 , · · · , k . Equalities hold if and only if [ β ] = λ [ α ] for some λ ∈ R . Chern number inequalities in four dimension
In this section, we first prove an inequality in the K¨ahler cone.
Theorem 3.1.
Suppose α and ω are any two K¨ahler classes on M , then we havethe following inequality (cid:0) α · ω (cid:1) (cid:0) ω (cid:1) α · ω − (cid:0) α · ω (cid:1) + (cid:0) α · ω (cid:1) (cid:0) α (cid:1) α · ω ≤ . Moreover, the equality holds if and only if α and ω are proportional.Proof. According to Theorem 2.1, we have( α k +1 · ω − k )( α k − · ω − k ) ≤ ( α k · ω − k ) , or all k = 1 , ,
3. More precisely, we have(3.1) ( α · ω )( ω ) ≤ ( α · ω ) , (3.2) ( α · ω )( α · ω ) ≤ ( α · ω ) , and(3.3) ( α )( α · ω ) ≤ ( α · ω ) . Multiplying Equation (3.1) and (3.2), we have(3.4) ( ω )( α · ω ) ≤ ( α · ω )( α · ω ) . Also multiplying Equation (3.2) and (3.3), we have(3.5) ( α · ω )( α ) ≤ ( α · ω )( α · ω ) . Since α · ω > α · ω >
0, multiplying them to the both sides of Equation(3.4) and (3.5) respectively, we obtain( α · ω )( ω )( α · ω ) ≤ ( α · ω )( α · ω )( α · ω )and ( α · ω )( α · ω )( α ) ≤ ( α · ω )( α · ω )( α · ω ) . Adding these two inequalities together, we get( ω )( α · ω ) + ( α · ω ) ( α ) ≤ α · ω )( α · ω )( α · ω ) . Hence, we have (cid:0) α · ω (cid:1) (cid:0) ω (cid:1) α · ω − (cid:0) α · ω (cid:1) + (cid:0) α · ω (cid:1) (cid:0) α (cid:1) α · ω < . (cid:3) The range ˆ θ ∈ ( π , π ) implies that c ( L ) is actually a K¨ahler class. ThenConjecture 1.2 can be easily obtained as an application of Theorem 3.1. Proof.
We assume the Hermitian metric h on L is a dHYM metric and the inducedChern curvature of h is denoted by F . We also denote λ i ( i = 1 , · · · ,
4) are theeigenvalues of the endormorphism ω − √− F ∈ End( T , X ). Then the dHYMmetric condition is equivalent to the following equation on λ i :ˆ θ = X i =1 arctan λ i . Since ˆ θ ∈ ( π , π ), we know that λ i > λ i λ j > i = j , i.e., c ( L ) lies inthe K¨ahler cone. Since λ i λ j >
1, we get the first inequality immediately.Choosing α = c ( L ) in Theorem 3.1, we have (cid:0) c ( L ) · ω (cid:1) (cid:0) ω (cid:1) c ( L ) · ω − (cid:0) c ( L ) · ω (cid:1) + (cid:0) c ( L ) · ω (cid:1) (cid:0) c ( L ) (cid:1) c ( L ) · ω ≤ . Then we obtain the second inequality since c ( L ) · ω is strictly positive. (cid:3) At last, we prove the Chern number inequalities while ˆ θ ∈ ( π, π ). Theorem 3.2.
The Chern number inequalities also hold while ˆ θ ∈ ( π, π ) . roof. For convenience, we always normalize the volume of (
X, ω ) is 1, i.e., Z X ω = 1 . We assume that α = √− F ∈ c ( L ) is a dHYM metric and λ ≤ λ ≤ λ ≤ λ are the eigenvalues of ω − α . It is well-know that all λ i are continuous functions.Let σ k be the k -th elementary symmetric polynomial with respect to ( λ , · · · , λ ).Furthermore, according to [25], σ i > i = 1 , ,
3, i.e. α ∈ K Γ ,ω .Case 1: inf X λ ≥
0, i.e., α is semi-positive, then we get the needed inequalitiesby Theorem 3.1. Indeed, the Chern number inequalities hold for any α + εω , ε > ε →
0, we get the Chern number inequalities.Case 2: inf X λ <
0. For convenience, we assume that λ ( x ) < x ∈ X .According to the dHYM equation X i =1 arctan λ i = ˆ θ and ˆ θ ∈ ( π, π ), we know that λ ≥ λ ≥ λ > | λ | ≥ λ λ ≥ λ λ > x , σ − − σ = X i = j λ i λ j − − λ λ λ λ >λ ( λ + λ + λ ) + λ λ + λ λ − λ λ λ λ ≥ λ ( λ + λ ) + λ λ + λ λ ≥ ( λ + λ )( λ + λ ) > . (3.6)Since tan ˆ θ = σ − σ σ − − σ >
0, then σ > σ at x . Furthermore, σ − σ > x ∈ X . Furthermore, at all x ∈ X , there holds σ − σ σ + σ ≤ σ − σ ( σ − λ λ )= − λ λ − λ λ − λ λ − λ λ − λ λ λ − λ λ − λ λ λ − λ λ λ − λ λ − λ λ λ − λ λ − λ λ − λ λ λ − λ λ λ − λ λ − λ λ = − ( λ + λ + λ )( λ + λ )( λ + λ ) − λ λ ( λ + λ ) − λ ( λ + λ ) < . Hence we have σ σ > σ + σ > σ and σ > x ∈ X .In conclusion, we have the following inequalities pointwise on X (3.7) σ − σ > , (3.8) σ − σ − > , and(3.9) σ > . Then we claim that Z X σ − Z X σ Z X σ + Z X σ < , here(3.10) Z X σ = 4 Z X ω ∧ α, (3.11) Z X σ = 6 Z X ω ∧ α , and(3.12) Z X σ = 4 Z X ω ∧ α , according to the definition of elementary symmetric polynomials. Indeed, since α ∈ K Γ ,ω , by Theorem 2.1 we have,0 < Z X ω ∧ α Z X ω ∧ α ≤ ( Z X ω ∧ α ) , and 0 < Z X ω ∧ α Z X ω ≤ ( Z X ω ∧ α ) . Multiplying these two inequalities together and applying the Equations (3.10)-(3.12), we get the following inequality Z X σ ≤ Z X σ Z X σ . Then, Z X σ − Z X σ Z X σ + Z X σ ≤ − Z X σ Z X σ + Z X σ < − Z X σ + Z X σ = − Z X σ < . Since tan ˆ θ = R X σ − R X σ R X σ − − R X σ >
0, we know that(3.13) R X σ − R X σ R X σ − − R X σ ( Z X σ − Z X σ Z X σ + Z X σ ) < < ( Z X σ ) . Multiplying R X σ − − R X σ > Z X σ ) − Z X σ Z X σ Z X σ + ( Z X σ ) Z X σ < (cid:0) c ( L ) · ω (cid:1) (cid:0) ω (cid:1) c ( L ) · ω − (cid:0) c ( L ) · ω (cid:1) + (cid:0) c ( L ) · ω (cid:1) (cid:0) c ( L ) (cid:1) c ( L ) · ω ≤ (cid:3) emark . The key estimates in the proof above are the inequalities (3.7)-(3.9).While proving these inequalities, we do not distinguish the sign of λ . We can alsoprove σ > x ∈ X where λ ( x ) ≤
0, we havearctan λ + arctan λ + arctan λ ∈ ( π , π )according to ˆ θ ∈ ( π, π ). Since0 > tan(arctan λ + arctan λ + arctan λ ) = λ + λ + λ − λ λ λ − λ λ − λ λ − λ λ and λ + λ + λ − λ λ λ >
0, we have λ λ + λ λ + λ λ > . Hence, σ − σ − λ λ + λ λ + λ λ + λ λ + λ λ + λ λ − λ λ λ λ − > ( λ + λ + λ ) λ − λ λ λ λ > . Then we get σ − σ >
0, since ˆ θ ∈ ( π, π ) and tan ˆ θ = σ − σ σ − − σ >
0. Furthermore,since arctan λ + arctan λ ∈ ( π , π ) , and arctan λ + arctan λ ∈ ( π , π ) , we know that λ λ > λ λ >
1. Thus σ = λ λ + λ λ + λ λ + λ λ + λ λ + λ λ > λ + λ ) λ + λ λ > . At the points x ∈ X where λ ( x ) >
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E-mail address : [email protected] Xishen Jin, Department of Mathematics, Renmin University of China, Beijing, 100872,China,
E-mail address : [email protected]@ruc.edu.cn