The Levi form on a complex manifold #
Let M be a complex manifold modelled on ๐(โ, E), and let f : M โ โ. The Levi form
TauCeti.manifoldLeviForm E f x of f at x is the Levi form TauCeti.leviForm of the
coordinate expression of f in the preferred chart at x. It is a sesquilinear form on the model
space E, which is the tangent space at x.
Unlike the real Hessian, the Levi form of a function which is real Cยฒ at x is intrinsic at
every point: in any chart e at x of the holomorphic Cยฒ maximal atlas, the Levi form of the
coordinate expression f โ e.symm, evaluated on the images of tangent vectors under the derivative
of e, is TauCeti.manifoldLeviForm E f x. This is because the second derivative of a holomorphic
chart transition is complex bilinear and does not contribute to the Levi form.
The underlying real manifold of M is modelled on ๐(โ, E), with the same charts, and its
holomorphic charts are real Cยฒ charts (IsManifold.maximalAtlas_subset_restrictScalars). Its
critical Hessian TauCeti.criticalHessian ๐(โ, E) f x is the real second derivative in the
preferred chart, and the Levi form of f is the Levi form of this real bilinear form. In
particular, for Cยฒ f the diagonal of the Levi form is one quarter of
Hess f(v, v) + Hess f(I โข v, I โข v). At a critical point the critical Hessian is intrinsic as
well, and this diagonal may be computed from the real Hessian in any real Cยฒ chart.
Strict plurisubharmonicity and strong q-pseudoconvexity of f at x are positivity conditions
on TauCeti.manifoldLeviForm E f x, so for f real Cยฒ at x they may be checked in any
holomorphic chart. Consequently the index bounds of strongly q-pseudoconvex functions hold for
the manifold Morse index: at a critical point where f is strongly q-pseudoconvex, the Morse
index is at most dim_โ E + q.
Main definitions #
TauCeti.manifoldLeviForm E f x: the Levi form offatx, computed in the preferred chart.TauCeti.IsManifoldStrictlyPlurisubharmonicAt E f x,TauCeti.IsManifoldStronglyPseudoconvexAt E q f x: positive definiteness of the Levi form atx, on all ofEor on a complex subspace of codimension at mostq.
Main results #
TauCeti.leviForm_comp_symm_apply_of_mem_maximalAtlas: forfrealCยฒatx, the Levi form of the coordinate expression in any holomorphicCยฒchart atxis the manifold Levi form.TauCeti.re_manifoldLeviForm_apply,TauCeti.im_manifoldLeviForm_apply: the real and imaginary parts of the Levi form in terms of the real critical Hessian.TauCeti.manifoldLeviForm_apply_self: forCยฒf, its diagonal is the real number(Hess f(v, v) + Hess f(I โข v, I โข v)) / 4.TauCeti.manifoldLeviForm_add,TauCeti.manifoldLeviForm_neg,TauCeti.manifoldLeviForm_mul_apply,TauCeti.manifoldLeviForm_scalar_comp_apply: the sum, negation, product and composition rules.TauCeti.manifoldLeviForm_comp_apply: the pullback rule under a complexCยฒmap.TauCeti.le_re_manifoldLeviForm_scalar_comp_apply_self: post-composition with an increasing convex function preserves positivity of the Levi form.TauCeti.isManifoldStrictlyPlurisubharmonicAt_iff_criticalHessian: strict plurisubharmonicity in terms of the real critical Hessian.TauCeti.IsManifoldStrictlyPlurisubharmonicAt.scalar_compandTauCeti.IsManifoldStronglyPseudoconvexAt.scalar_comp: increasing convex post-composition preserves the corresponding positivity conditions.TauCeti.IsManifoldStrictlyPlurisubharmonicAt.compandTauCeti.IsManifoldStronglyPseudoconvexAt.comp: the positivity conditions restrict along holomorphic immersions.TauCeti.re_manifoldLeviForm_apply_of_mem_maximalAtlasandTauCeti.im_manifoldLeviForm_apply_of_mem_maximalAtlas: at a critical point, the real and imaginary parts may be computed from the real Hessian in any realCยฒchart.TauCeti.manifoldLeviForm_apply_self_of_mem_maximalAtlas: at a critical point the same holds with the real Hessian computed in any realCยฒchart.TauCeti.isManifoldStrictlyPlurisubharmonicAt_iff_of_mem_maximalAtlasandTauCeti.isManifoldStronglyPseudoconvexAt_iff_of_mem_maximalAtlas: forfrealCยฒatx, these positivity conditions may be checked in any holomorphic chart.TauCeti.IsManifoldStronglyPseudoconvexAt.manifoldMorseIndex_le,TauCeti.IsManifoldStronglyPseudoconvexAt.manifoldMorseIndex_le_of_mem_criticalPoints, andTauCeti.IsManifoldStrictlyPlurisubharmonicAt.manifoldMorseIndex_le: the manifold Morse index boundsฮป โค n + qandฮป โค nat critical points.
References #
- J. Milnor, Morse Theory, Princeton University Press, 1963, Section 7.
- R. Hain, Notes on the Morse theory of complex manifolds, Section 2.
The Levi form of a real function f on a complex manifold at x: the Levi form of the
coordinate expression of f in the preferred chart at x. It is a sesquilinear form on the model
space E, which is the tangent space TangentSpace ๐(โ, E) x. By
TauCeti.leviForm_comp_symm_apply_of_mem_maximalAtlas, when f is real Cยฒ at x it may be
computed in any chart at x of the holomorphic Cยฒ maximal atlas.
Instances For
The manifold Levi form is the Levi form of the coordinate expression in the preferred chart.
On the model space itself, whose preferred charts are the identity, the manifold Levi form is the Levi form.
The Levi form of a function depends only on its germ.
Comparison with the real critical Hessian #
The real part of the Levi form is one quarter of the sum
Hess f(v, w) + Hess f(I โข v, I โข w) of critical-Hessian values, that is, one half of the
I-invariant part of the critical Hessian.
The imaginary part of the Levi form is one quarter of the mixed difference
Hess f(I โข v, w) - Hess f(v, I โข w) of critical-Hessian values. Only the I-invariant part of
the critical Hessian contributes; its I-anti-invariant part cancels in this difference.
The diagonal of the Levi form. For a function which is Cยฒ at x, the diagonal value of
the Levi form on v is the real number (Hess f(v, v) + Hess f(I โข v, I โข v)) / 4, where Hess
is the critical Hessian of f on the real manifold underlying M.
The Levi form of a function which is Cยฒ at x is Hermitian.
Algebraic rules #
The sum rule for the Levi form on a complex manifold.
Negating a function negates its Levi form on a complex manifold.
The product rule for the Levi form on a complex manifold. Writing โf for the
complex-linear extension StrongDual.extendRCLike of the real differential mvfderiv ๐(โ, E) f x,
the Levi form of f g at x is f(x) L g + g(x) L f + (conj (โf v) โg w + conj (โg v) โf w) / 4.
The composition rule for the Levi form on a complex manifold, for post-composition with a
function of one real variable. Writing โf for the complex-linear extension
StrongDual.extendRCLike of mvfderiv ๐(โ, E) f x, the Levi form of ฯ โ f at x is
ฯ'(f x) L f + ฯ''(f x) conj (โf v) โf w / 4.
Post-composition with a convex function on a complex manifold. If ฯ'' โฅ 0 at f x, then
the diagonal of the Levi form of ฯ โ f dominates ฯ'(f x) times the diagonal of the Levi form
of f. In particular, if moreover ฯ'(f x) > 0, post-composition with ฯ preserves strict
positivity of the Levi form at x.
The holomorphic pullback rule for the Levi form on complex manifolds. If g is complex
Cยฒ at x and u is real Cยฒ at g x, then the Levi form of u โ g is the pullback of the
Levi form of u by the complex manifold derivative of g.
Independence of the holomorphic chart #
The Levi form in an arbitrary holomorphic chart. Let e be a chart of the Cยฒ maximal
atlas of the complex manifold M, with x in its source, and let f be Cยฒ at x. The Levi
form of the coordinate expression f โ e.symm at e x, evaluated on the images of v and w
under the derivative of e, is the Levi form of f at x. Unlike the real Hessian, the Levi form
thus transforms as a Hermitian form under holomorphic changes of coordinates at every point.
Strict plurisubharmonicity and strong pseudoconvexity #
A real function f on a complex manifold is strictly plurisubharmonic at x if its Levi
form at x is positive definite: 0 < Re (manifoldLeviForm E f x v v) for every nonzero v
(TauCeti.isManifoldStrictlyPlurisubharmonicAt_iff). As for TauCeti.manifoldLeviForm, it is
defined in the preferred chart at x; by
TauCeti.isManifoldStrictlyPlurisubharmonicAt_iff_of_mem_maximalAtlas, for f real Cยฒ at x it
may be checked in any chart at x of the holomorphic Cยฒ maximal atlas.
Instances For
A real function f on a complex manifold modelled on a finite-dimensional space E is
strongly q-pseudoconvex at x if its Levi form at x is positive definite on a complex
subspace P of E with dim_โ E โค dim_โ P + q (TauCeti.isManifoldStronglyPseudoconvexAt_iff).
As for TauCeti.manifoldLeviForm, it is defined in the preferred chart at x; by
TauCeti.isManifoldStronglyPseudoconvexAt_iff_of_mem_maximalAtlas, for f real Cยฒ at x it
may be checked in any chart at x of the holomorphic Cยฒ maximal atlas.
Instances For
Strict plurisubharmonicity on a complex manifold is positive definiteness of the Levi form.
Strong q-pseudoconvexity on a complex manifold is positive definiteness of the Levi form on
a complex subspace of codimension at most q.
Strict plurisubharmonicity on a complex manifold in terms of the real critical Hessian: f is
strictly plurisubharmonic at x exactly when Hess f(v, v) + Hess f(I โข v, I โข v) > 0 for every
v โ 0.
Strong q-pseudoconvexity on a complex manifold is weaker for larger q.
Strong 0-pseudoconvexity on a complex manifold is strict plurisubharmonicity.
A strictly plurisubharmonic function on a complex manifold is strongly q-pseudoconvex for
every q.
Post-composition with a function ฯ of one real variable with ฯ'(f x) > 0 and
ฯ''(f x) โฅ 0 preserves strict plurisubharmonicity on a complex manifold at x.
Post-composition with a function ฯ of one real variable with ฯ'(f x) > 0 and
ฯ''(f x) โฅ 0 preserves strong q-pseudoconvexity on a complex manifold at x.
Strict plurisubharmonicity on a complex manifold at x depends only on the germ of the
function at x.
Strong q-pseudoconvexity on a complex manifold at x depends only on the germ of the
function at x.
The sum of two Cยฒ functions strictly plurisubharmonic at x on a complex manifold is
strictly plurisubharmonic at x.
Restriction along a holomorphic immersion of complex manifolds. If u is strictly
plurisubharmonic at g x and the complex manifold derivative of the complex Cยฒ map g at x is
injective, then u โ g is strictly plurisubharmonic at x.
Restriction along a holomorphic immersion of complex manifolds. If u is strongly
q-pseudoconvex at g x and the complex manifold derivative of the complex Cยฒ map g at x is
injective, then u โ g is strongly q-pseudoconvex at x: the preimage of a positive subspace of
codimension at most q is a positive subspace of codimension at most q.
Strict plurisubharmonicity in an arbitrary holomorphic chart. For f real Cยฒ at x and
a chart e at x of the holomorphic Cยฒ maximal atlas, f is strictly plurisubharmonic at x
if and only if its coordinate expression f โ e.symm is strictly plurisubharmonic at e x.
Strong q-pseudoconvexity in an arbitrary holomorphic chart. For f real Cยฒ at x and
a chart e at x of the holomorphic Cยฒ maximal atlas, f is strongly q-pseudoconvex at x
if and only if its coordinate expression f โ e.symm is strongly q-pseudoconvex at e x.
Critical points #
The real part of the Levi form at a critical point, in an arbitrary real chart. For a
real Cยฒ chart e, it is the quarter-sum of the real Hessian on the chart derivatives of
(v, w) and (I โข v, I โข w).
The imaginary part of the Levi form at a critical point, in an arbitrary real chart. For
a real Cยฒ chart e, it is one quarter of the difference of the two mixed real-Hessian terms on
the chart derivatives of (I โข v, w) and (v, I โข w).
The Levi form at a critical point, from the real Hessian in any chart. At a critical point
x of a function f which is Cยฒ at x, the diagonal value of the Levi form on v is
(Hess f(v, v) + Hess f(I โข v, I โข v)) / 4, where the real Hessian may be computed in any chart
e of the real Cยฒ maximal atlas, holomorphic or not, after transporting the tangent vectors by
the derivative of e.
The index bound for strongly q-pseudoconvex functions on a complex manifold. At a
critical point x of a function f which is Cยฒ at x and strongly q-pseudoconvex at x, the
Morse index of f is at most dim_โ E + q.
A version of TauCeti.IsManifoldStronglyPseudoconvexAt.manifoldMorseIndex_le taking membership
in TauCeti.criticalPoints and smoothness of order r โฅ 2.
The index bound for strictly plurisubharmonic functions on a complex manifold. At a
critical point x of a function f which is Cยฒ at x and strictly plurisubharmonic at x, the
Morse index of f is at most dim_โ E.