The Minimum Principle for Convex Subequations

Abstract

A subequation, in the sense of Harvey–Lawson, on an open subset $X\subset \mathbb R^n$ is a subset F of the space of 2-jets on X with certain properties. A smooth function is said to be F-subharmonic if all of its 2-jets lie in F, and using the viscosity technique one can extend the notion of F-subharmonicity to any upper-semicontinuous function. Let $\mathcal P$ denote the subequation consisting of those 2-jets whose Hessian part is semipositive. We introduce a notion of product subequation $F\#\mathcal P$ on $X\times \mathbb R^{m}$ and prove, under suitable hypotheses, that if F is convex and f(x, y) is $F\#\mathcal P$-subharmonic then the marginal function

$$\begin{aligned} g(x):= \inf _y f(x,y) \end{aligned}$$

is F-subharmonic. This generalises the classical statement that the marginal function of a convex function is again convex. We also prove a complex version of this result that generalises the Kiselman minimum principle for the marginal function of a plurisubharmonic function.

Appendices

Appendix A: F-Subharmonic Functions

1.1 A.1 Types of Subequations

Definition A.1

Let $F\subset J^2(X)$.

• (1) We say F is constant coefficient if $F_x$ is independent of x, i.e.

  $$\begin{aligned} (x,r,p,A) \in F_x \Leftrightarrow (x',r,p,A)\in F_{x'} \text { for all }x,x',r,p,A. \end{aligned}$$

• (2) We say F is independent of the gradient part (or gradient-independent) if each $F_x$ is independent of p, i.e.

  $$\begin{aligned} (r,p,A)\in F_x \Leftrightarrow (r,p',A) \in F_x \text { for all } x,r,p,p',A. \end{aligned}$$

• (3) We say F depends only on the Hessian part if each $F_x$ is independent of (r, p), i.e.

  $$\begin{aligned} (r,p,A) \in F_x \Leftrightarrow (r',p',A)\in F_{x} \text { for all }x,r,r',p,p',A. \end{aligned}$$

Definition A.2

(G-Invariance)The group $GL_n(\mathbb R)$ acts on $J^2(X)$ by

$$\begin{aligned} g^*(x,r,p,A):= (x,r,g^t p, g^t Ag) \text { for } g\in GL_n(\mathbb R). \end{aligned}$$

If G is a subgroup of $GL_n(\mathbb R)$ we say $F\subset J^2(X)$ is G-invariant if $g^*\alpha \in F$ for all $\alpha \in F$ and all $g\in G$.

Remark A.3

Our action of $GL_n$ comes from thinking of the jet space using the cotangent space to X, and is different in convention to that of [13].

1.2 A.2 Complex Subequations

Set

$$\begin{aligned} \mathbb J = \left( \begin{array}{cc} 0&{} -{\text {Id}}_{n} \\ {\text {Id}}_n &{} 0\end{array} \right) \in M_{2n\times 2n}(\mathbb R). \end{aligned}$$

If $A\in M_{2n\times 2n}(\mathbb R)$ commutes with $\mathbb J$ then making the standard identification $\mathbb C\simeq \mathbb R^2$ we think of A as a complex matrix $\hat{A} \in M_{n\times n}(\mathbb C)$. Explicitly if in block form

$$\begin{aligned} A = \left( \begin{array}{cc} a&{} c \\ b &{} d \end{array} \right) \end{aligned}$$

where $a,b,c,d\in M_{n\times n}(\mathbb R)$ then A commutes with $\mathbb J$ if and only if $a=d$ and $b=-c$, in which case

$$\begin{aligned} \hat{A}: = a+ib \in M_{n\times n}(\mathbb C). \end{aligned}$$

Observe $\widehat{AB} = \hat{A} \hat{B}$ and $\widehat{A^t} = \hat{A}^*$.

Let ${\text {Herm}}_n$ be the set of hermitian $n\times n$ complex matrices, and

$$\begin{aligned} {\text {Pos}}_n^{\mathbb C} := \{ \hat{A} \in {\text {Herm}}_n : v^* \hat{A} v \ge 0 \text { for all } v\in \mathbb C^n\} \end{aligned}$$

the subset of semipositive hermitian matrices. From the above it easy to check that if $A\mathbb J = \mathbb JA$ then

$$\begin{aligned} A\in {\text {Sym}}^2_{2n}&\Longleftrightarrow \hat{A}\in {\text {Herm}}(\mathbb C^n)\text { and }\\ A \in {\text {Pos}}_{2n}&\Longleftrightarrow \hat{A}\in {\text {Pos}}^{\mathbb C}_{n}. \end{aligned}$$

Now for any $A\in M_{2n\times 2n}(\mathbb R)$ the matrix

$$\begin{aligned} A_{\mathbb C} : = \frac{1}{2} ( A - \mathbb J A \mathbb J) \end{aligned}$$

commutes with $\mathbb J$ and thus we may think of $A_{\mathbb C}$ as an element of $M_{n\times n}(\mathbb C)$. Observe if A is symmetric then $A_{\mathbb C}$ is hermitian.

Definition A.4

Let $X\subset \mathbb R^{2n}\simeq \mathbb C^n$ be open. We say $F\subset J^2(X)$ is a complex subequation if $(x,r,v,A)\in F$ if and only if $(x,r,v,A_{\mathbb C})\in F$.

So by abuse of notation if F is a complex subequation we may equivalently consider it as a subset

$$\begin{aligned} F\subset J^{2,\mathbb C}(X) := X\times \mathbb R \times \mathbb C^n \times {\text {Herm}}(\mathbb C^n) =: X\times J^{2,\mathbb C}_n \end{aligned}$$

without any loss of information. The group $GL_n(\mathbb C)$ acts on $J^{2,\mathbb C}(X)$ by

$$\begin{aligned} g^*(x,r,p,A) = (x,r,g^* p, g^* Ag). \end{aligned}$$

Observe also if F is complex, then having the Positivity property (1) is equivalent to

$$\begin{aligned} (x,r,p,A)\in F \Longrightarrow (x,r,p,A+P)\in F \text { for all }P\in {\text {Pos}}_n^{\mathbb C}. \end{aligned}$$

Example A.5

Let

$$\begin{aligned} \mathcal P^{\mathbb C}_X : = X\times \mathbb R\times \mathbb C^n \times {\text {Pos}}^{\mathbb C}_n \end{aligned}$$

which is a convex complex subequation. We will write $\mathcal P^{\mathbb C}$ for $\mathcal P^{\mathbb C}_X$ when X is clear from context.

Example A.6

(Convex and Plurisubharmonic) Recall $\mathcal P_X = X\times \mathbb R\times \mathbb R^n\times {\text {Pos}}_n$. Then $\mathcal P_X(X)$ consists of locally convex functions on X [13, Example 14.2]. Similarly if $X\subset \mathbb C^n$ is open then $\mathcal P^{\mathbb C}_X(X)$ consists of the plurisubharmonic functions on X [13, p. 63].

1.3 A.3 Basic Properties of F-Subharmonic Functions

The following lists some of the basic limit properties satisfied by F-subharmonic functions (under very mild assumptions on F).

Proposition A.7

Let $F\subset J^2(X)$ be closed. Then

• (1) (Maximum Property) If $f,g\in F(X)$ then $\max \{f,g\}\in F(X)$.

• (2) (Decreasing Sequences) If $f_j$ is decreasing sequence of functions in F(X) (so $f_{j+1}\le f_j$ over X) then $f:=\lim _j f_j$ is in F(X).

• (3) (Uniform limits) If $f_j$ is a sequence of functions on F(X) that converge locally uniformly to f then $f\in F(X)$.

• (4) (Families locally bounded above) Suppose $\mathcal F\subset F(X)$ is a family of F-subharmonic functions locally uniformally bounded from above. Then the upper-semicontinuous regularisation of the supremum

  $$\begin{aligned} f:= {\sup }^*_{f\in \mathcal F} f \end{aligned}$$

  is in F(X).

• (5) If F is constant coefficient and f is F-subharmonic on X and $x_0\in \mathbb R^{n}$ is fixed, then the function $x\mapsto f(x-x_0)$ is F-subharmonic on $X-x_0$.

Proof

See [13, Theorem 2.6] for (1-4). Item (5) is immediate from the definition. $\square$

Lemma A.8

(Limits under perturbations of subequations) Let X be open and $F\subset J^2(X)$ be a primitive subequation. For $\delta >0$ let $F^\delta \subset J^2(X)$ be defined by

$$\begin{aligned} F^{\delta } = \{ (x,r,p,A) : \exists r',p' \text { with } (x,r',p',A)\in F \text { and } |r-r'|\le \delta \text { and } \Vert p-p'\Vert \le \delta \}. \end{aligned}$$

Then

• (1) $F^{\delta }$ is a primitive subequation.

• (2) If F satisfies the Negativity property then so does $F^\delta$.

• (3) $\bigcap _{\delta >0} (F^\delta (X))=F(X)$.

Proof

That $F^{\delta }$ has the Positivity property is immediate from the definition, and $F^{\delta }$ is closed as F is closed giving (1). Statement (2) is also immediate from the definition. Finally using F is closed, $\bigcap _{\delta >0} F^{\delta }_x = F_x$, and thus $\bigcap _{\delta >0} (F^{\delta }(X)) = F(X).$ $\square$

1.4 F-Subharmonicity in Terms of Second Order Jets

It is useful to understand the property of being F-subharmonic in terms of second order jets. To do so we first discuss what it means to be twice differentiable at a point. Again let $X\subset \mathbb R^n$ be open.

Definition A.9

(Twice differentiability at a point) We say that $f:X\rightarrow \mathbb R$ is twice differentiable at $x_0\in X$ if there exists a $p\in \mathbb R^n$ and an $L\in {\text {Sym}}_n^2$ such that for all $\epsilon >0$ there is a $\delta >0$ such that for $\Vert x-x_0\Vert <\delta$ we have

$$\begin{aligned} |f(x) - f(x_0) - p.(x-x_0) - \frac{1}{2} (x-x_0)^tL (x-x_0) | \le \epsilon \Vert x-x_0\Vert ^2. \end{aligned}$$

(52)

When f is twice differentiable at $x_0$ then the p, L in (52) are unique, and moreover in this case f is differentiable at $x_0$ and

$$\begin{aligned} p = \nabla f|_{x_0}= \left( \begin{array}{c} \frac{\partial f}{\partial x_1} \\ \frac{\partial f}{\partial x_2} \\ \vdots \\ \frac{\partial f}{\partial x_n} \end{array}\right) |_{x_0}\in \mathbb R^n. \end{aligned}$$

When f is twice differentiable at $x_0$ we shall refer to L as the Hessian of f at $x_0$ and denote it by ${\text {Hess}}(f)|_{x_0}$. Of course, by Taylor’s Theorem, when f is $\mathcal C^{2}$ in a neighbourhood of $x_0$ then ${\text {Hess}}_{x}(f)$ is the matrix with entries

$$\begin{aligned} ({\text {Hess}}(f)_{x_0} )_{ij}: = \frac{\partial ^2 f}{\partial x_i\partial x_j}|_{x_0}. \end{aligned}$$

Definition A.10

(Second order jet) Suppose that $f: X\rightarrow \mathbb R$ is twice differentiable at $x_0$. We denote the second order jet of f at $x_0$ by

$$\begin{aligned} J^2_{x_0}(f):= (f(x_0), \nabla f|_{x_0}, {\text {Hess}}(f)|_{x_0}) \in J^2_{n} = \mathbb R\times \mathbb R^n\times {\text {Sym}}_n^2. \end{aligned}$$

(53)

The importance of the Positivity property is made apparent by the following that shows that F-subharmonicity behaves as expected for sufficiently smooth functions.

Lemma A.11

Let $F\subset J^2(X)$ satisfy the Positivity assumption (1) and suppose $f:X\rightarrow \mathbb R$ is $\mathcal C^2$. Then $f\in F(X)$ if and only if $J^2_{x}(f)\in F_x$ for all $x\in X$.

Proof

The reader may easily verify this, or consult [13, Equation 2.4 and Proposition 2.3]. $\square$

The definition F-subharmonicity given above says that at any upper-contact point x, with upper-second order jet (p, A), the quadratic function

$$\begin{aligned} y\mapsto f(y) + p.(x-y) + \frac{1}{2} (y-x)^tA (y-x) \end{aligned}$$

has second-order jet lying in $F_x$. The next statement says that this is equivalent to the more classical “viscosity definition". Given an upper-semicontinuous f we say that $\phi$ is a $\mathcal {C}^2$-test function touching f from above at $x_0$ if $\phi \in \mathcal C^2$ in a neighbourhood of $x_0$ with $\phi \ge f$ on this neighbourhood and $\phi (x_0) = f(x_0)$.

Lemma A.12

(Viscosity definition of F-subharmonicity) An upper-semicontinuous $f:X\rightarrow \mathbb R\cup \{-\infty \}$ is in F(X) if and only if for all $x_0\in X$ and test-functions $\phi$ touching f from above at $x_0$ it holds that $J^2_{x_0}(\phi )\in F_{x_0}.$

Proof

See [13, Lemma 2.4]. $\square$

It takes some work to understand how F-subharmonicity interacts with linearity in the space of functions. However when F is constant-coefficient and convex the following is true:

Proposition A.13

(Convex combinations of F-subharmonic functions) Let F be a constant coefficient convex primitive subequation. Then any convex combination of F-subharmonic functions is again F-subharmonic.

Proof

This is implied by [11, Theorem 5.1 ] (apply the cited theorem to $F_x:=\lambda H_x$ and $G_x:= (1-\lambda )H_x$ for a given $\lambda \in [0,1]$) $\square$

Appendix B: Associativity of Products

We prove Proposition 3.4 which states that if $X_i\subset \mathbb R^{n_i}$ are open and $F_i\subset J^2(X_i)$ for $i=1,2,3$ then

$$\begin{aligned} (F_1\#F_2)\#F_3 = F_1\#(F_2\#F_3). \end{aligned}$$

Let x, y, z be coordinates on $\mathbb R^{n_1},\mathbb R^{n_2},\mathbb R^{n_3}$ respectively. We will consider certain linear mappings

$$\begin{aligned} \Gamma&:\mathbb R^{n_1}\rightarrow \mathbb R^{n_2+n_3}\\ \Phi&:\mathbb R^{n_1+n_2}\rightarrow \mathbb R^{n_3}\\ \Psi&:\mathbb R^{n_1} \rightarrow \mathbb R^{n_2}\\ \Upsilon&:\mathbb R^{n_2}\rightarrow \mathbb R^{n_3} \end{aligned}$$

and write

$$\begin{aligned} \Phi (x,y) = \Phi _1(x) + \Phi _2(y) \end{aligned}$$

where $\Phi _i:\mathbb R^{n_i} \rightarrow \mathbb R^{n_3}$ is linear. Recall that $\iota _{\Gamma }:\mathbb R^{n_1} \rightarrow \mathbb R^{n_1+n_2+n_3}$ is $\iota _{\Gamma }(x) = (x,\Gamma (x))$ and similarly for $\iota _\Phi :\mathbb R^{n_1+n_2}\rightarrow \mathbb R^{n_1+n_2+n_3}$ and $\iota _{\Psi }:\mathbb R^{n_1}\rightarrow \mathbb R^{n_1+n_2}$.

Lemma B.1

Suppose

$$\begin{aligned} \Gamma = (\Psi , \Phi _1 + \Phi _2\circ \Psi ). \end{aligned}$$

(54)

Then

$$\begin{aligned} \iota _{\Gamma } = \iota _{\Phi }\circ \iota _{\Psi } \end{aligned}$$

Proof

$$\begin{aligned} \iota _{\Phi }(\iota _{\Psi }(x))&= \iota _{\Phi }(x,\Psi (x)) = (x,\Psi (x),\Phi (x,\Psi (x))) \\ {}&= (x,\Psi (x),\Phi _1(x) + \Phi _2\circ \Psi (x)) = \iota _{\Gamma }(x).\end{aligned}$$

$\square$

Now set

$$\begin{aligned} \begin{array}{ll} j_2: \mathbb R^{n_2} \rightarrow \mathbb R^{n_1+n_2} &{}\text { } j_2(y) = (0,y)\\ j_3: \mathbb R^{n_3} \rightarrow \mathbb R^{n_1+n_2+n_3} &{}\text { } j_3(z) = (0,0,z)\\ j_{23}:\mathbb R^{n_2+ n_3} \rightarrow \mathbb R^{n_1+n_2+n_3} &{}\text { } j_{23}(y,z) = (0,y,z)\\ k:\mathbb R^{n_3} \rightarrow \mathbb R^{n_2+n_3} &{}\text { } k(z) = (0,z). \end{array} \end{aligned}$$

Fix $(x,y,z)\in X_1\times X_2\times X_3$. By definition of the product subequation we know $\alpha \in (F_1\#(F_2\#F_3))_{(x,y,z)}$ if and only if

$$\begin{aligned} \forall \Gamma \text { we have }\iota _\Gamma ^*\alpha \in (F_1)_{x} \text { and } j_{23}^*\alpha \in (F_2\#F_3)_{(y,z)}. \end{aligned}$$

(55)

Observe for every $\Gamma$ there is a pair $(\Phi ,\Psi )$ such that (54) holds. Thus by Lemma B.1, condition (55) is equivalent to

$$\begin{aligned} \forall \Psi ,\Phi \text { we have }\iota ^*_\Psi \iota ^*_\Phi \alpha \in (F_1)_{x} \text { and } j_{23}^*\alpha \in (F_2\#F_3)_{(y,z)}. \end{aligned}$$

(56)

Using the definition of $F_2\#F_3$, condition (56) is in turn equivalent to

$$\begin{aligned} \forall \Psi ,\Phi ,\Upsilon \text { we have }\iota ^*_\Psi \iota ^*_\Phi \alpha \in (F_1)_{x} \text { and } \iota _{\Upsilon }^* j_{23}^*\alpha \in (F_2)_{y} \text { and }k^*j_{23}^*\alpha \in (F_3)_{z}. \end{aligned}$$

(57)

Now $j_{23}\circ k=j_3$, and a simple check yields $j_{23}\circ \iota _{\Phi _2} = \iota _{\Phi }\circ j_2$. Thus (57) is equivalent to

$$\begin{aligned} \forall \Psi ,\Phi \text { we have }\iota ^*_\Psi \iota ^*_\Phi \alpha \in (F_1)_{x} \text { and } j_{2}^*\iota _{\Phi }^*\alpha \in (F_2)_{y} \text { and }j_3^*\alpha \in (F_3)_{z}. \end{aligned}$$

(58)

So from the definition of $(F_1\#F_2)$, condition (58) is equivalent to

$$\begin{aligned} \forall \Phi \text { we have }\iota ^*_\Phi \alpha \in (F_1\#F_2)_{x} \text { and }j_3^*\alpha \in (F_3)_{z} \end{aligned}$$

(59)

which, by definition, is equivalent to $\alpha \in ((F_1\#F_2)\#F_3)_{(x,y,z)}$.

Appendix C: Products of Gradient-Independent Subequations

Recall we say that a subequation $F\subset J^2(X)$ has Property (P⁺⁺) if the following holds. For all $x\in X$ and all $\epsilon >0$ there exists a $\delta >0$ such that

$$\begin{aligned} (x,r,p,A)\in F_x \Rightarrow (x',r-\epsilon ,p, A+\epsilon {\text {Id}})\in F_{x'} \text { for all } \Vert x'-x\Vert <\delta \ \end{aligned}$$

($\hbox {P}^{++}$)

or said another way,

$$\begin{aligned} F_x + (x'-x,0,-\epsilon , \epsilon {\text {Id}}) \subset F_{x'} \text { for all } \Vert x'-x\Vert <\delta . \end{aligned}$$

($\hbox {P}^{++}$)

Lemma C.1

Assume that F and G have property ($\text{ P}^{++}$) and are independent of the gradient part. Then $H: = F\#G$ is a subequation

Proof

We have already seen in Lemma 3.2 that H is closed, and satisfies the Positivity and Negativity properties (1) and (2). It remains to prove the Topological property (3) which we break up into a number of pieces. Since F, G are independent of the gradient part, so is H, and thus the only non-trivial part of the topological property is to show [13, Section 4.8]

$$\begin{aligned} {\text {Int}}(H_{(x,y)}) =({\text {Int}}H)_{(x,y)} \end{aligned}$$

The fact that ${\text {Int}}(H_{(x,y)}) \subset ({\text {Int}}H)_{(x,y)}$ is obvious, so the task is to prove the other inclusion.

Let

$$\begin{aligned} \alpha \in {\text {Int}}( H_{(x,y)}), \end{aligned}$$

so there exists a $\delta _1>0$ such that

$$\begin{aligned} \Vert \hat{\alpha } - \alpha \Vert <\delta _1 \text { and } \hat{\alpha } \in J^2(X\times Y)|_{(x,y)} \Rightarrow \hat{\alpha }\in H_{(x,y)}.\end{aligned}$$

(60)

By hypothesis there is a $\delta _2>0$ such that

$$\begin{aligned}&F_x +\left( x'-x, -\frac{\delta _1}{4}, 0,\frac{\delta _1}{4} {\text {Id}}\right) \subset F_{x'} \text { for } \Vert x-x'\Vert <\delta _2 \end{aligned}$$

(61)

$$\begin{aligned}&G_y +\left( x'-x, -\frac{\delta _1}{4}, 0,\frac{\delta _1}{4} {\text {Id}}\right) \subset G_{y'} \text { for } \Vert y-y'\Vert <\delta _2. \end{aligned}$$

(62)

Set $\delta = \min \{ \delta _1/2,\delta _2/2\}$ and pick any $\alpha '\in J^2(X\times Y)$ with

$$\begin{aligned} \Vert \alpha '-\alpha \Vert <\delta . \end{aligned}$$

We will show that $\alpha ' \in H$.

Denote the space coordinate of $\alpha '$ by $(x',y')$, so $\alpha '\in J^2(X\times Y)|_{(x',y')}$. Thus we certainly have $\Vert x'-x\Vert<\delta <\delta _2$ and $\Vert y-y'\Vert <\delta _2$. Define

$$\begin{aligned} \hat{\alpha } : = \alpha ' +\left( (x-x',y-y'),-\frac{\delta _1}{2},0, -\frac{\delta _1}{2} {\text {Id}}_{n+m}\right) . \end{aligned}$$

Then $\hat{\alpha } \in J^2(X\times Y)_{(x,y)}$ and

$$\begin{aligned} \Vert \hat{\alpha } - \alpha \Vert \le \Vert \alpha ' - \alpha \Vert + \Vert \hat{\alpha } - \alpha '\Vert < \delta _1. \end{aligned}$$

Thus (60) applies, so $\hat{\alpha }\in H_{(x,y)}$ which means

$$\begin{aligned} j^* \hat{\alpha } \in G_{y} \text { and } i_U^* \hat{\alpha }\in G_{x} \text { for all } U. \end{aligned}$$

Now using (62).

$$\begin{aligned} j^*\alpha ' = j^*\hat{\alpha } +\left( y'-y,\frac{\delta _1}{2},0, \frac{\delta _1}{2} {\text {Id}}_{m}\right) \in G_y +\left( y'-y,\frac{\delta _1}{2},0, \frac{\delta _1}{2} {\text {Id}}_{m}\right) \subset G_{y'}. \end{aligned}$$

Similarly using the Positivitiy property of F and (61)

$$\begin{aligned} i_U^*\alpha '&= i_U^*\hat{\alpha } + \left( x-x',\frac{\delta _1}{2},0,\frac{\delta _1}{2} {\text {Id}}_{n} + \frac{\delta _1}{2} U^tU\right) \\&\subset i_U^*\hat{\alpha } + \left( x-x',\frac{\delta _1}{2},0,\frac{\delta _1}{2} {\text {Id}}_{n}\right) \subset F_{x'}. \end{aligned}$$

Thus $\alpha '\in H_{(x',y')}$. As this holds for all such $\alpha '$ we conclude $\alpha \in {\text {Int}}(H)$ completing the proof. $\square$

1.1 C.1 The Complex Case

Let $X\subset \mathbb R^{2n}\simeq \mathbb C^n$ be open. If $f:X\rightarrow \mathbb R$ is twice differentiable at a point $z\in X$ its complex Hessian is

$$\begin{aligned} {\text {Hess}}^{\mathbb C}_z(f) = \frac{1}{2} ( {\text {Hess}}(f) - \mathbb J {\text {Hess}}_x(f) \mathbb J) \in {\text {Herm}}(\mathbb C^n). \end{aligned}$$

When f is sufficiently smooth we have

$$\begin{aligned} ( {\text {Hess}}^{\mathbb C}_z(f))_{jk} = 2 \frac{\partial ^2 f}{\partial z_j\partial \overline{z}_k}|_z \end{aligned}$$

where, as usual,

$$\begin{aligned} \frac{\partial }{\partial z_j} = \frac{1}{2} \left( \frac{\partial }{\partial x_j} - i \frac{\partial }{\partial y_j}\right) \text { for } z_j = x_j + i y_j. \end{aligned}$$

In terms of the gradient, under the identification $\mathbb R^{2n}\simeq \mathbb C^n$ we have

$$\begin{aligned} \nabla f|_z = \left( \begin{array}{c} \frac{\partial f}{\partial x}|_z \\ \frac{\partial f}{\partial {y}}|_z \end{array}\right) = 2\frac{\partial f}{\partial \overline{z}}|_z. \end{aligned}$$

Definition C.2

(Complex 2-jet) The complex 2-jet of f at $z\in X$ is

$$\begin{aligned} J^{2,\mathbb C}_{z} (f) := \left( f(z), 2\frac{\partial f}{\partial \overline{z}}|_z,{\text {Hess}}^{\mathbb C}_z(f)\right) \in J^{2,\mathbb C}_{z} = \mathbb R\times \mathbb C^n\times {\text {Herm}}_n. \end{aligned}$$

So if $F\subset J^2(X)$ is complex then

$$\begin{aligned} J^2_{z} (f)\in F_z \Longleftrightarrow J^{2,\mathbb C}_{z} (f)\in F_z. \end{aligned}$$