Holomorphic Sectional Curvature

• Holomorphic sectional curvature (HSC) is a measure of curvature in complex geometry, specifically applied to complex lines (J-invariant planes) in Hermitian and almost Hermitian manifolds. It differs between Kähler and non-Kähler geometries, and in the Kähler case, it determines the full curvature tensor.
• Holomorphic sectional curvature has significant applications in global geometry, particularly in distinguishing between Kähler and non-Kähler Hermitian metrics, as well as in understanding the geometry and curvature of specific manifolds like the complex projective space, Euclidean space, and hyperbolic space.
• The curvature associated with the Chern connection often influences signatures, transformations like complexification, and properties such as positive, nonpositive, nonnegative, and identically vanishing curvature.

Holomorphic sectional curvature is the curvature assigned to a complex one-dimensional tangent direction, namely the complex line generated by a nonzero vector. For a Hermitian metric $h$ and $X\in T^{1,0}_pM\setminus\{0\}$, its Chern holomorphic sectional curvature is

$$H(X)=\frac{R(X,\overline X,X,\overline X)}{|X|_h^4} =\frac{R_{i\bar j k\bar l}X^i\overline{X^j}X^k\overline{X^l}} {\left(h_{i\bar j}X^i\overline{X^j}\right)^2}.$$

In real notation on an almost Hermitian manifold it is the sectional curvature of the $J$-invariant plane $\operatorname{span}\{X,JX\}$. The quantity is homogeneous of degree zero and therefore depends only on the complex line $\mathbb C X$. Its behavior differs fundamentally between Kähler and non-Kähler Hermitian geometry: in the Kähler case it determines the full curvature tensor, whereas for a general Hermitian metric it determines only a symmetrized component of the Chern curvature.

1. Definitions, conventions, and curvature models

Let $(M,J,g)$ be a Hermitian manifold. The complex structure satisfies $J^2=-\operatorname{id}$, and the metric is $J$-invariant. The associated $(1,1)$-form is

$X\in T^{1,0}_pM\setminus\{0\}$0

The metric is Kähler precisely when $X\in T^{1,0}_pM\setminus\{0\}$1.

For the Chern connection, the curvature components in holomorphic coordinates are

$X\in T^{1,0}_pM\setminus\{0\}$2

The holomorphic sectional curvature is then

$X\in T^{1,0}_pM\setminus\{0\}$3

The sign conditions are:

• $X\in T^{1,0}_pM\setminus\{0\}$4: positive holomorphic sectional curvature;
• $X\in T^{1,0}_pM\setminus\{0\}$5: nonnegative or semi-positive holomorphic sectional curvature;
• $X\in T^{1,0}_pM\setminus\{0\}$6: negative holomorphic sectional curvature;
• $X\in T^{1,0}_pM\setminus\{0\}$7: nonpositive or semi-negative holomorphic sectional curvature;
• $X\in T^{1,0}_pM\setminus\{0\}$8: identically vanishing holomorphic sectional curvature.

A distinction is required between pointwise constant and globally constant curvature. Pointwise constancy means that there is a function $X\in T^{1,0}_pM\setminus\{0\}$9 such that

$$H(X)=\frac{R(X,\overline X,X,\overline X)}{|X|_h^4} =\frac{R_{i\bar j k\bar l}X^i\overline{X^j}X^k\overline{X^l}} {\left(h_{i\bar j}X^i\overline{X^j}\right)^2}.$$0

for every nonzero $$H(X)=\frac{R(X,\overline X,X,\overline X)}{|X|_h^4} =\frac{R_{i\bar j k\bar l}X^i\overline{X^j}X^k\overline{X^l}} {\left(h_{i\bar j}X^i\overline{X^j}\right)^2}.$$1. Global constancy means that $$H(X)=\frac{R(X,\overline X,X,\overline X)}{|X|_h^4} =\frac{R_{i\bar j k\bar l}X^i\overline{X^j}X^k\overline{X^l}} {\left(h_{i\bar j}X^i\overline{X^j}\right)^2}.$$2 is one number on all of $$H(X)=\frac{R(X,\overline X,X,\overline X)}{|X|_h^4} =\frac{R_{i\bar j k\bar l}X^i\overline{X^j}X^k\overline{X^l}} {\left(h_{i\bar j}X^i\overline{X^j}\right)^2}.$$3. On Kähler manifolds of complex dimension at least two, Schur-type arguments imply that pointwise constant holomorphic sectional curvature is globally constant. This implication can fail for general Hermitian metrics (Chen et al., 2019).

For a Kähler metric, the curvature tensor satisfies

$$H(X)=\frac{R(X,\overline X,X,\overline X)}{|X|_h^4} =\frac{R_{i\bar j k\bar l}X^i\overline{X^j}X^k\overline{X^l}} {\left(h_{i\bar j}X^i\overline{X^j}\right)^2}.$$4

With the normalization in which the constant holomorphic sectional curvature is $$H(X)=\frac{R(X,\overline X,X,\overline X)}{|X|_h^4} =\frac{R_{i\bar j k\bar l}X^i\overline{X^j}X^k\overline{X^l}} {\left(h_{i\bar j}X^i\overline{X^j}\right)^2}.$$5,

$$H(X)=\frac{R(X,\overline X,X,\overline X)}{|X|_h^4} =\frac{R_{i\bar j k\bar l}X^i\overline{X^j}X^k\overline{X^l}} {\left(h_{i\bar j}X^i\overline{X^j}\right)^2}.$$6

The complete simply connected Kähler models are $$H(X)=\frac{R(X,\overline X,X,\overline X)}{|X|_h^4} =\frac{R_{i\bar j k\bar l}X^i\overline{X^j}X^k\overline{X^l}} {\left(h_{i\bar j}X^i\overline{X^j}\right)^2}.$$7, $$H(X)=\frac{R(X,\overline X,X,\overline X)}{|X|_h^4} =\frac{R_{i\bar j k\bar l}X^i\overline{X^j}X^k\overline{X^l}} {\left(h_{i\bar j}X^i\overline{X^j}\right)^2}.$$8, and complex hyperbolic space $$H(X)=\frac{R(X,\overline X,X,\overline X)}{|X|_h^4} =\frac{R_{i\bar j k\bar l}X^i\overline{X^j}X^k\overline{X^l}} {\left(h_{i\bar j}X^i\overline{X^j}\right)^2}.$$9, according to whether $J$0 is positive, zero, or negative. Under the normalization $J$1, the Fubini–Study metric satisfies

$J$2

Holomorphic sectional curvature is weaker than holomorphic bisectional curvature,

$J$3

because $J$4 only tests repeated directions. It is also distinct from Ricci and scalar curvature, which are contractions of the curvature tensor. Nonnegative holomorphic sectional curvature does not generally imply nonnegative Ricci curvature, as shown by explicit complete Kähler examples with indefinite Ricci curvature (Chen et al., 2023).

2. Algebraic determination and spherical averages

At a point, let $J$5 be an $J$6-dimensional Hermitian vector space. The curvature polynomial

$J$7

is real-valued and bihomogeneous of bidegree $J$8. In the Kähler case, the curvature tensor can be viewed as a Hermitian form on $J$9. The diagonal values

$\operatorname{span}\{X,JX\}$0

therefore determine that Hermitian form by polarization.

A representation-theoretic projection identity makes this precise. If $\operatorname{span}\{X,JX\}$1 denotes orthogonal projection onto $\operatorname{span}\{X,JX\}$2, then

$\operatorname{span}\{X,JX\}$3

For Kähler curvature, the first spherical moment is

$\operatorname{span}\{X,JX\}$4

where $\operatorname{span}\{X,JX\}$5 is the scalar curvature. The second moment is

$\operatorname{span}\{X,JX\}$6

where $\operatorname{span}\{X,JX\}$7 is the Ricci tensor. Consequently, if $\operatorname{span}\{X,JX\}$8, then

$\operatorname{span}\{X,JX\}$9

and hence $\mathbb C X$0. Thus holomorphic sectional curvature determines the entire Kähler curvature tensor (Magnússon, 2023).

For a general Hermitian metric, the Chern curvature is a Hermitian form on

$\mathbb C X$1

The expression $\mathbb C X$2 only probes the $\mathbb C X$3-component. It does not determine the $\mathbb C X$4-block or the off-diagonal blocks between $\mathbb C X$5 and $\mathbb C X$6. If $\mathbb C X$7 denotes the symmetric block, then the Hermitian first and second moments take the form

$\mathbb C X$8

and

$\mathbb C X$9

Thus $(M,J,g)$0 forces the symmetric component to vanish but does not force the entire Chern curvature to vanish.

The corresponding failure is visible in complete non-Kähler examples. A conformal deformation of the Euclidean metric on $(M,J,g)$1,

$(M,J,g)$2

has pointwise negative holomorphic sectional curvature

$(M,J,g)$3

when $(M,J,g)$4, but the value is not globally constant. Other complete non-Kähler conformal metrics satisfy

$(M,J,g)$5

Therefore $(M,J,g)$6 does not imply Chern flatness in the general Hermitian category (Chen et al., 2019).

3. Constancy, boundedness, and rigidity

On an almost Hermitian manifold, holomorphic planes are those invariant under $(M,J,g)$7, while antiholomorphic planes satisfy $(M,J,g)$8. For a unit vector $(M,J,g)$9,

$J^2=-\operatorname{id}$0

In indefinite signature, holomorphic sectional curvature is considered on nonisotropic vectors and nondegenerate holomorphic planes.

A fundamental rigidity phenomenon is that pointwise boundedness can force pointwise constancy. If an indefinite almost Hermitian manifold has signature

$J^2=-\operatorname{id}$1

and, at each point,

$J^2=-\operatorname{id}$2

then it has pointwise constant holomorphic sectional curvature. The estimate is tested on mixed spacelike-timelike vectors

$J^2=-\operatorname{id}$3

The resulting curvature expression is a quartic polynomial in $J^2=-\operatorname{id}$4, while its bound contains the factor $J^2=-\operatorname{id}$5. Passing to $J^2=-\operatorname{id}$6 produces algebraic curvature identities, including

$J^2=-\operatorname{id}$7

Polarization then yields equality of holomorphic sectional curvatures in spacelike and timelike directions and propagates it to all directions. The argument is pointwise and algebraic rather than differential (Borisov et al., 2010).

For definite metrics, the corresponding construction takes place in the complexified tangent bundle. Boundedness of either the real or imaginary part of the holomorphic sectional curvature on complex holomorphic planes of signature $J^2=-\operatorname{id}$8 likewise implies pointwise constancy. Complex directions such as

$J^2=-\operatorname{id}$9

replace real spacelike-timelike pairs.

Related results hold for antiholomorphic sectional curvature. In the indefinite case, an upper or lower pointwise bound on antiholomorphic sectional curvatures implies pointwise constant antiholomorphic sectional curvature when $J$0. Weakly isotropic antiholomorphic planes are used to obtain identities such as

$J$1

Analogous statements apply in the definite case to the real or imaginary parts of complex sectional curvature. The paper also treats totally real biholomorphic sectional curvature

$J$2

on antiholomorphic orthonormal pairs.

A different algebraic characterization occurs on generalized globally framed $J$3-manifolds. Such a manifold carries

$J$4

with

$J$5

On the horizontal distribution

$J$6

the tensor $J$7 acts as an almost-complex structure. A $J$8-holomorphic plane is

$J$9

and its curvature is

$(1,1)$0

For generalized $(1,1)$1-manifolds of dimension $(1,1)$2, $(1,1)$3, constant $(1,1)$4-holomorphic sectional curvature is equivalent to

$(1,1)$5

for every horizontal $(1,1)$6. This generalizes Tanno’s characterizations for almost Hermitian and Sasakian manifolds (Lee et al., 2011).

4. Hermitian connections and compact classification

The holomorphic sectional curvature depends on the connection used. For a Hermitian manifold, Gauduchon’s canonical family is

$(1,1)$7

where $(1,1)$8 is the Chern connection and $(1,1)$9 is the Lichnerowicz connection. Important members are

$X\in T^{1,0}_pM\setminus\{0\}$00

The curvature of $X\in T^{1,0}_pM\setminus\{0\}$01 contains Chern curvature, covariant derivatives of Chern torsion, and quadratic torsion terms with coefficients depending polynomially on $X\in T^{1,0}_pM\setminus\{0\}$02.

For compact Hermitian surfaces, pointwise constant Gauduchon holomorphic sectional curvature is highly restrictive. Either the metric is Kähler, or the surface is an isosceles Hopf surface with an admissible metric and

$X\in T^{1,0}_pM\setminus\{0\}$03

An isosceles Hopf manifold is a quotient

$X\in T^{1,0}_pM\setminus\{0\}$04

with

$X\in T^{1,0}_pM\setminus\{0\}$05

An admissible metric is conformal to the standard Hopf metric and has conformal factor

$X\in T^{1,0}_pM\setminus\{0\}$06

The exceptional values arise from conformal curvature equations on the Hopf cover. Their integrability forces the conformal factor to be quadratic, while the Hopf dilation leaves only $X\in T^{1,0}_pM\setminus\{0\}$07 and $X\in T^{1,0}_pM\setminus\{0\}$08. For the standard Hopf metric, the Strominger/Bismut curvature vanishes,

$X\in T^{1,0}_pM\setminus\{0\}$09

although the $X\in T^{1,0}_pM\setminus\{0\}$10-curvature tensor is nonzero. Thus vanishing holomorphic sectional curvature for a particular Hermitian connection need not imply flatness of that connection.

For the two-parameter Zhao–Zheng family

$X\in T^{1,0}_pM\setminus\{0\}$11

the non-Kähler Hopf alternative occurs precisely when

$X\in T^{1,0}_pM\setminus\{0\}$12

At $X\in T^{1,0}_pM\setminus\{0\}$13, this reduces to $X\in T^{1,0}_pM\setminus\{0\}$14 or $X\in T^{1,0}_pM\setminus\{0\}$15. At $X\in T^{1,0}_pM\setminus\{0\}$16, the connection is Levi–Civita, and no non-Kähler Hopf exception occurs. In particular, a compact Hermitian surface with pointwise constant Lichnerowicz holomorphic sectional curvature is Kähler (Chen et al., 2022).

The classification problem remains open in higher-dimensional compact Hermitian geometry. The conjectural picture is that nonzero constant holomorphic sectional curvature should force the metric to be Kähler, while zero constant holomorphic sectional curvature should force Chern flatness. In complex dimension three, vanishing real bisectional curvature provides a partial result: if

$X\in T^{1,0}_pM\setminus\{0\}$17

vanishes for every unitary frame and every nonzero nonnegative $X\in T^{1,0}_pM\setminus\{0\}$18, then the metric is Chern flat. The hypothesis is stronger than $X\in T^{1,0}_pM\setminus\{0\}$19, and the implication

$X\in T^{1,0}_pM\setminus\{0\}$20

remains open for arbitrary compact Hermitian manifolds in complex dimension at least three (Zhou et al., 2021).

5. Curvature signs and global complex geometry

Holomorphic sectional curvature has strong consequences for canonical positivity in the Kähler category. If a compact projective manifold carries a Kähler metric with semi-negative holomorphic sectional curvature,

$X\in T^{1,0}_pM\setminus\{0\}$21

then it contains no rational curves and its canonical bundle is nef. The argument uses the Schwarz lemma for holomorphic maps from compact curves, followed in the projective case by Mori’s bend-and-break principle. The vanishing case $X\in T^{1,0}_pM\setminus\{0\}$22 forces the curvature tensor to vanish, and the manifold is an abelian variety up to a finite unramified covering (Heier et al., 2014).

To quantify degeneracy, define

$X\in T^{1,0}_pM\setminus\{0\}$23

$X\in T^{1,0}_pM\setminus\{0\}$24

The invariant $X\in T^{1,0}_pM\setminus\{0\}$25 measures the number of directions not contained in a maximal $X\in T^{1,0}_pM\setminus\{0\}$26-flat tangent subspace. For semi-negative curvature,

$X\in T^{1,0}_pM\setminus\{0\}$27

when $X\in T^{1,0}_pM\setminus\{0\}$28, where $X\in T^{1,0}_pM\setminus\{0\}$29 is the nef dimension of $X\in T^{1,0}_pM\setminus\{0\}$30. Under the Abundance Conjecture, this yields

$X\in T^{1,0}_pM\setminus\{0\}$31

In complex dimension two, semi-negative curvature leads to a trichotomy: flat or abelian geometry when $X\in T^{1,0}_pM\setminus\{0\}$32, elliptic fibrations when $X\in T^{1,0}_pM\setminus\{0\}$33, and ample canonical bundle when $X\in T^{1,0}_pM\setminus\{0\}$34.

The weaker cohomological condition of almost nonpositive holomorphic sectional curvature allows the Kähler classes and metrics to vary. For a Kähler class $X\in T^{1,0}_pM\setminus\{0\}$35, set

$X\in T^{1,0}_pM\setminus\{0\}$36

Almost nonpositivity requires classes $X\in T^{1,0}_pM\setminus\{0\}$37 satisfying

$X\in T^{1,0}_pM\setminus\{0\}$38

This condition implies nefness of $X\in T^{1,0}_pM\setminus\{0\}$39, absence of rational curves, and Miyaoka–Yau type inequalities. A quantitative comparison with the nef threshold $X\in T^{1,0}_pM\setminus\{0\}$40 is

$X\in T^{1,0}_pM\setminus\{0\}$41

The resulting Chern-class inequality is

$X\in T^{1,0}_pM\setminus\{0\}$42

for compact Kähler manifolds with almost nonpositive holomorphic sectional curvature (Zhang, 2018).

Almost quasi-negative curvature permits a small positive part while retaining some negative curvature. To prevent the negative region from becoming negligible, a capacity is defined from

$X\in T^{1,0}_pM\setminus\{0\}$43

If the positive maxima tend to zero and the negative curvature has uniformly positive capacity, then $X\in T^{1,0}_pM\setminus\{0\}$44 is ample. The capacity condition supplies the strict positivity of the canonical volume needed to upgrade nefness to ampleness (Zhang et al., 2020).

For nonnegative holomorphic sectional curvature, a pseudo-effective holomorphic tensor forces a parallel space of truly flat tangent vectors. If no nonzero truly flat vector exists at some point, then a compact Kähler manifold is projective and rationally connected. This includes the quasi-positive case. A compact simply connected Kähler manifold with $X\in T^{1,0}_pM\setminus\{0\}$45 is likewise projective and rationally connected. In complex dimension three, non-projective compact Kähler manifolds with $X\in T^{1,0}_pM\setminus\{0\}$46 are either flat or $X\in T^{1,0}_pM\setminus\{0\}$47-bundles over a two-dimensional torus (Zhang et al., 2023).

For compact Kähler manifolds equipped only with a Hermitian metric, nonpositive Chern holomorphic sectional curvature still implies nefness of the canonical bundle. In complex dimension two, strict negativity implies ampleness. If

$X\in T^{1,0}_pM\setminus\{0\}$48

then

$X\in T^{1,0}_pM\setminus\{0\}$49

so $X\in T^{1,0}_pM\setminus\{0\}$50 admits a Ricci-flat Kähler metric, although the original Hermitian metric need not be Kähler or flat. Conversely, if the first Bott–Chern class vanishes,

$X\in T^{1,0}_pM\setminus\{0\}$51

then every Hermitian metric with $X\in T^{1,0}_pM\setminus\{0\}$52 must satisfy $X\in T^{1,0}_pM\setminus\{0\}$53 (Tang, 25 Jul 2026).

6. Positivity, constructions, and geometric applications

Positive holomorphic sectional curvature is stable under several geometric constructions, although the mechanism differs from that for negative curvature. Let

$X\in T^{1,0}_pM\setminus\{0\}$54

be a compact holomorphic fibration with positive holomorphic sectional curvature on the base and on every fiber. A warped metric is constructed from a relative fiber form $X\in T^{1,0}_pM\setminus\{0\}$55 and the base metric: $X\in T^{1,0}_pM\setminus\{0\}$56 The vertical curvature approaches the positive fiber curvature, while the horizontal curvature is amplified by $X\in T^{1,0}_pM\setminus\{0\}$57. Mixed curvature terms remain controlled and are absorbed for sufficiently large $X\in T^{1,0}_pM\setminus\{0\}$58. This produces a Hermitian metric with positive holomorphic sectional curvature on $X\in T^{1,0}_pM\setminus\{0\}$59. If the relative form and base form are Kähler, the resulting metric can be chosen Kähler (Chaturvedi et al., 2017).

A degenerate-form formulation uses

$X\in T^{1,0}_pM\setminus\{0\}$60

Generalized Codazzi–Griffiths identities give

$X\in T^{1,0}_pM\setminus\{0\}$61

where the final term is nonpositive and measures the failure of the vertical and horizontal structures to be parallel. The same domination principle yields positivity. Applying it to Grassmannian fibers gives positive holomorphic sectional curvature on Grassmannian bundles over bases with positive holomorphic sectional curvature. The canonical Grassmannian metric satisfies

$X\in T^{1,0}_pM\setminus\{0\}$62

Flag-manifold fibrations follow by iterating Grassmannian fibrations (Magnússon, 2022).

In the toric setting, Delzant’s construction realizes a projective toric manifold as a Kähler quotient of $X\in T^{1,0}_pM\setminus\{0\}$63. The quotient curvature is expressed through a quadratic term $X\in T^{1,0}_pM\setminus\{0\}$64: $X\in T^{1,0}_pM\setminus\{0\}$65 Compactness of the moment level set implies $X\in T^{1,0}_pM\setminus\{0\}$66 for every nonzero horizontal lift, giving strict positivity. Every smooth projective toric manifold therefore admits a canonical toric Kähler metric with positive holomorphic sectional curvature.

A degeneration argument transfers this positivity to successive point blow-ups of projective toric manifolds. A smooth projective family is constructed whose noncentral fibers are the given blow-up and whose central fiber is a toric blow-up. Positivity is open under smooth Kähler deformation. Consequently, every projective manifold obtained from a projective toric manifold by finitely many point blow-ups admits a Kähler metric with positive holomorphic sectional curvature. In particular, every rational surface admits such a metric. Combined with Hitchin’s theorem, this gives

$X\in T^{1,0}_pM\setminus\{0\}$67

The higher-dimensional analogue for blow-ups along curves or higher-dimensional centers remains unresolved (Zhang, 22 Jun 2026).

Positive holomorphic sectional curvature also has boundary and CR-geometric consequences. For a strictly pseudoconvex real hypersurface semi-isometrically immersed in a Kähler manifold, the Tanaka–Webster holomorphic sectional curvature satisfies

$X\in T^{1,0}_pM\setminus\{0\}$68

where $X\in T^{1,0}_pM\setminus\{0\}$69 is ambient holomorphic sectional curvature, $X\in T^{1,0}_pM\setminus\{0\}$70 is the $X\in T^{1,0}_pM\setminus\{0\}$71-mean curvature vector, and $X\in T^{1,0}_pM\setminus\{0\}$72 is holomorphic sectional torsion. Pseudohermitian $X\in T^{1,0}_pM\setminus\{0\}$73-convexity is equivalent to

$X\in T^{1,0}_pM\setminus\{0\}$74

Thus, under $X\in T^{1,0}_pM\setminus\{0\}$75,

$X\in T^{1,0}_pM\setminus\{0\}$76

in the strictly convex case. In CR dimension one, $X\in T^{1,0}_pM\setminus\{0\}$77, so the Tanaka–Webster scalar curvature is positive. The estimate implies $X\in T^{1,0}_pM\setminus\{0\}$78-positivity in the sense of Cao, Chang, and Chen and yields positive scalar curvature for an adapted Riemannian metric on compact three-dimensional hypersurfaces (Son, 2020).

Positive holomorphic sectional curvature also admits sharp global volume estimates. If a compact connected $X\in T^{1,0}_pM\setminus\{0\}$79-dimensional Kähler manifold satisfies

$X\in T^{1,0}_pM\setminus\{0\}$80

then

$X\in T^{1,0}_pM\setminus\{0\}$81

equivalently,

$X\in T^{1,0}_pM\setminus\{0\}$82

Equality holds precisely for the normalized Fubini–Study metric on $X\in T^{1,0}_pM\setminus\{0\}$83. More generally, if $X\in T^{1,0}_pM\setminus\{0\}$84, then

$X\in T^{1,0}_pM\setminus\{0\}$85

The proof is based on vanishing of tensor powers, jet counting, Riemann–Roch, and a mean RC-curvature condition rather than Bishop–Gromov comparison. The same sharp volume bound follows from

$X\in T^{1,0}_pM\setminus\{0\}$86

a condition implied by both

$X\in T^{1,0}_pM\setminus\{0\}$87

Under the mean RC condition alone, the sharp volume estimate is established, but the corresponding projective-space rigidity statement remains open (Datar et al., 16 Aug 2026).

Finally, positive and negative holomorphic sectional curvature can coexist with substantial degeneracy in the opposite sign directions. For complete Kähler metrics constructed on powers of tautological line bundles over projective space, one has

$X\in T^{1,0}_pM\setminus\{0\}$88

with many zero directions. The curvature profile is governed by a discriminant equality

$X\in T^{1,0}_pM\setminus\{0\}$89

and the resulting metrics may have indefinite Ricci curvature despite semi-positive holomorphic sectional curvature. Conversely, in semi-negative geometry, ample canonical bundles can coexist with large linear subspaces of zero holomorphic sectional curvature. Algebraic square-decomposition methods bound the dimension of such zero subspaces in terms of the holomorphic sectional curvature square decomposition length (Chen et al., 2023). These phenomena demonstrate that holomorphic sectional curvature is a directional invariant whose global implications depend decisively on Kähler symmetries, sign definiteness, torsion, and the geometry of its zero locus.