[過去ﾛｸﾞ] 純粋・応用数学・数学隣接分野（含むガロア理論）13 (1002ﾚｽ)
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979: 2023/07/31(月)10:30 ID:jznoxopE(29/50) AA×


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979: [] 2023/07/31(月) 10:30:48.69 ID:jznoxopE Let us denote by ¯∂ (resp. ∂) the complex exterior derivative of type (0, 1) (resp. (1, 0)). Then the correspondence uα 7→ ¯∂uα defines a linear differential operator ¯∂ : C p,q(M, E) → C p,q+1(M, E). The Chern connection Dh is defined to be ¯∂ + ∂h, where ∂h is defined by uα 7→ h −1 α ∂(hαuα). Since ¯∂ 2 = ∂ 2 h = ∂ ¯∂ + ¯∂∂ = 0, there exists a E ∗ ⊗ E-valued (1, 1)-form Θh such that D2 hu = Θh ∧u holds for all u ∈ C p,q(M, E). Θh is called the curvature form of h. Note that Θhe−φ = Θh+IdE ⊗∂ ¯∂φ. Θh is said to be positive (resp. semipositive) at x ∈ M if Θh = ?n j,k=1 Θjk¯dzj ∧ dzk in terms of a local coordinate (z1, . . . , zn) LEVI PROBLEM UNDER THE NEGATIVITY 5 around x and (Θjk¯(x))j,k = (Θµ νjk¯ (x))j,k,µ,ν is positive (semipositive) in the sense (of Nakano) that the quadratic form ?( ? µ hµκ¯Θ µ νjk¯ )(x)ξ νj ξ κk is positive definite (resp. positive semidefinite). http://rio2016.5ch.net/test/read.cgi/math/1674527723/979

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