Appendix
1.1 The first variation of Perelman’s \(\mathcal W \) functional along the Kähler-Ricci flow
Analogues of the following evolution formulas were obtained by Perelman [5] in the Ricci flow case. For notation convenience we denote by \((g_t)_{t \geqslant 0}\) the Kähler-Ricci flow and with \(\omega _t\) the corresponding symplectic forms.
Theorem 2
(Perelman) Let \(X\) be a Fano manifold and let \(f\) be a solution of the conjugate heat equation
$$\begin{aligned} 2 \dot{f} = - \Delta f + | \nabla f|^2 + 2 n - \mathrm{Scal } , \end{aligned}$$
(18)
along the Kähler-Ricci flow \((g_t)_{t \geqslant 0}\), over a time interval \(\left[ 0, T \right] \). Then the function
$$\begin{aligned} 2 H :=2 \Delta f - | \nabla f|^2 + \mathrm{Scal } + 2 f - 2 n , \end{aligned}$$
satisfies the evolution equation
$$\begin{aligned} 2 \dot{H} = - \Delta H + 2 \nabla H \cdot \nabla f + | \mathrm{Ric } + i \partial \overline{\partial } f - \omega _t |^2 + | \nabla ^{1, 0} \partial f|^2 , \end{aligned}$$
(19)
over the time interval \(\left[ 0, T \right] \). Moreover on this interval hold the variation formula
$$\begin{aligned} \frac{d}{dt} \mathcal W (g_t, f_t) = \int \limits _X \Big [ | \mathrm{Ric } + i \partial \overline{\partial } f - \omega _t |^2 + | \nabla ^{1, 0} \partial f|^2 \Big ] e^{- f} d V_{g_t}. \end{aligned}$$
(20)
Proof
We remind first that for any function \(f \in C^{\infty } (X \times \mathbb R _{\geqslant 0} , \mathbb R )\) hold the evolution formulas along the Kähler-Ricci flow
$$\begin{aligned} \frac{\partial }{\partial t} | \nabla f|^2&= - | \nabla f|^2 + \mathrm{Ric } (\nabla f, J \nabla f) + 2 \nabla \dot{f} \cdot \nabla f , \end{aligned}$$
(21)
$$\begin{aligned} \Delta | \nabla f|^2&= 2 \nabla \Delta f \cdot \nabla f + 2 | \nabla ^{1, 0} \partial f|^2 + 2 | \partial \overline{\partial } f|^2 \nonumber \\&\quad + 2 \mathrm{Ric } (\nabla f, J \nabla f) , \end{aligned}$$
(22)
$$\begin{aligned} \frac{\partial }{\partial t} \Delta f&= - \Delta f + \left\langle \mathrm{Ric }, i \partial \overline{\partial } f \right\rangle + \Delta \dot{f}. \end{aligned}$$
(23)
We remind also that the scalar curvature evolves by the formula
$$\begin{aligned} 2 \frac{\partial }{\partial t} \mathrm{Scal } = \Delta \mathrm{Scal } + 2 | \mathrm{Ric } |^2 - 2 \mathrm{Scal }. \end{aligned}$$
(24)
Furthermore we observe the identity
$$\begin{aligned} 2 H = \Box f + 2 f . \end{aligned}$$
(25)
By using the evolution Eq. (23) we get the equality
$$\begin{aligned} \Box \Delta f&= \Delta ^2 f + 2 \Delta f - 2 \left\langle \mathrm{Ric }, i \partial \overline{\partial } f \right\rangle - 2 \Delta \dot{f}\\&= \Delta \Box f + 2 \Delta f - 2 \left\langle \mathrm{Ric }, i \partial \overline{\partial } f \right\rangle \\&= 2 \Delta H - 2 \left\langle \mathrm{Ric }, i \partial \overline{\partial } f \right\rangle , \end{aligned}$$
thanks to the identity (25). Moreover if we combine the evolution Eqs. (21) and (22) we obtain
$$\begin{aligned} \Box | \nabla f|^2&= 2 \nabla \Box f \cdot \nabla f + 2 | \nabla ^{1, 0} \partial f|^2 + 2 | \partial \overline{\partial } f|^2 + 2 | \nabla f|^2\\&= 4 \nabla H \cdot \nabla f + 2 | \nabla ^{1, 0} \partial f|^2 + 2 | \partial \overline{\partial } f|^2 - 2 | \nabla f|^2 , \end{aligned}$$
thanks to the identity (25). We infer the expressions
$$\begin{aligned} 2 \Box H&= 2 \Box \Delta f - \Box | \nabla f|^2 + \Box \mathrm{Scal } + 2 \Box f\\&= 4 \Delta H - 4 \left\langle \mathrm{Ric }, i \partial \overline{\partial } f \right\rangle \\&\quad - 4 \nabla H \cdot \nabla f - 2 | \nabla ^{1, 0} \partial f|^2 - 2 | \partial \overline{\partial } f|^2 + 2 | \nabla f|^2\\&\quad - 2 | \mathrm{Ric } |^2 + 2 \mathrm{Scal } + 4 H - 4 f\\&= 4 \Delta H + 4 \Delta f + 4 \mathrm{Scal } - 4 \left\langle \mathrm{Ric }, i \partial \overline{\partial } f \right\rangle - 4 \nabla H \cdot \nabla f\\&\quad - 2 | \nabla ^{1, 0} \partial f|^2 - 2 | \partial \overline{\partial } f|^2 - 2 | \mathrm{Ric } |^2 - 4 n\\&= 4 \Delta H - 4 \nabla H \cdot \nabla f\\&\quad - 2 (B - 2 \Delta f + 2 n - 2 \mathrm{Scal }). \end{aligned}$$
where
$$\begin{aligned} B : = | \nabla ^{1, 0} \partial f|^2 + | \partial \overline{\partial } f|^2 + 2 \left\langle \mathrm{Ric }, i \partial \overline{\partial } f \right\rangle + | \mathrm{Ric } |^2. \end{aligned}$$
Arranging the terms by means of the trivial identity \(\mathrm{Tr }_{\omega _t} \alpha = \left\langle \omega _t, \alpha \right\rangle \), with \(\alpha \) a real \((1, 1)\)-form, we obtain the evolution equation
$$\begin{aligned} 2 \Box H&= 4 \Delta H - 4 \nabla H \cdot \nabla f\\&\quad - 2 | \mathrm{Ric } + i \partial \overline{\partial } f - \omega _t |^2 - 2 | \nabla ^{1, 0} \partial f|^2 , \end{aligned}$$
which implies the evolution formula (19). We remind now that the evolution Eq. (18) rewrites as \(\Box ^{*} e^{- f} = 0\). Thus time deriving the identity
$$\begin{aligned} \mathcal W (g_t, f_t) = \int \limits _X 2 H e^{- f} d V_g , \end{aligned}$$
we infer
$$\begin{aligned} \frac{d}{dt} \mathcal W (g_t, f_t)&= - \int \limits _X \Box H e^{- f} d V_{g} , \end{aligned}$$
which implies Perelman’s variation formula (20). \(\square \)
1.2 Local expression of the complex anti-linear part of the Hessian
Let \((X, J, \omega )\) be a Kähler manifold and \(u \in C^2 (X, \mathbb R )\). Let \((z_1, \ldots , z_n)\) be \(J\)-holomorphic coordinates and consider the local expression
$$\begin{aligned} \overline{\partial }_{{T_{X, J}}} \nabla _g u = A_{k, \bar{l}} \bar{\zeta }_l^{*} \otimes \zeta _k + \overline{A_{k, \bar{l}}} \zeta _l^{*} \otimes \bar{\zeta }_k , \end{aligned}$$
where \(\zeta _k : = \frac{\partial }{\partial z_k}\). We want to find the expression of the coefficients \(A_{k, \bar{l}}\) with respect to \(u\). For this purpose we consider the identities
$$\begin{aligned} \nabla _g u = \nabla ^{1, 0}_{g, J} u + \nabla ^{0, 1}_{g, J} u \end{aligned}$$
and
$$\begin{aligned} \nabla ^{1, 0}_{g, J} u \lnot \omega = i \overline{\partial }_{J} u. \end{aligned}$$
If we write locally \(\nabla ^{1, 0}_{g, J} u = \xi _k \zeta _k\) then the last identity writes locally as
$$\begin{aligned} \frac{i}{2} \omega _{l, \bar{k}} \xi _l \bar{\zeta }_k^{*} = i ( \bar{\zeta }_k. u) \bar{\zeta }_k^{*}. \end{aligned}$$
We infer the expression \(\xi _l = 2 \omega ^{k, \bar{l}} \bar{\zeta }_k . u\). Moreover by the definition of the operator \(\overline{\partial }_{{T_{X, J}}}\) hold the identities
$$\begin{aligned} A_{k, \bar{l}} \zeta _k = \left[ \left( \overline{\partial }_{{T_{X, J}}} \nabla _g u \right) \bar{\zeta }_l \right] ^{1, 0}_{{J_t}} = \left[ \bar{\zeta }_l , \nabla ^{1, 0}_{g, J} u \right] ^{1, 0}_{{J_t}} = ( \bar{\zeta }_l . \xi _k) \zeta _k . \end{aligned}$$
We infer the expressions
$$\begin{aligned} A_{k, \bar{l}}&= \bar{\zeta }_l. \xi _k = 2 \bar{\zeta }_l . \left( \omega ^{r, \bar{k}} \bar{\zeta }_r. u \right) \\&= 2 \omega ^{p, \bar{k}} \left[ \bar{\zeta }_l. \bar{\zeta }_p. u - \left( \bar{\zeta }_l . \omega _{j, \bar{p}} \right) \omega ^{r, \bar{j}} \bar{\zeta }_r . u \right] . \end{aligned}$$
1.3 Comparison of norms on \(T_{{X, J}}\)-valued forms
Let \((X, J, g)\) be a hermitian manifold. Let \(\omega : = g J\) and let \(h^{*}\) be the corresponding hermitian metric over the complex vector bundle \(T^{*}_{{X, J}}\). With respect to a local complex frame \((\zeta _k)_k \subset T_{{X, J}}^{1, 0}\) we have the expression
$$\begin{aligned} h^{*} = 4 \sum _{k, l} \omega ^{l \bar{k}} \zeta _k \otimes \bar{\zeta }_l. \end{aligned}$$
We remind that if \((V, J)\) is a complex vector space equipped with a hermitian metric \(h\) then the corresponding hermitian metric \(h_{\mathbb{C }}\) over the complexified vector space \((V \otimes _{\mathbb{R }} \mathbb C , i)\) is defined by the formula
$$\begin{aligned} 2 h_{\mathbb{C }} (v, w) : = h (v, \overline{w}) + \overline{h ( \overline{v}, w)}, v, w \in V \otimes _{\mathbb{R }} \mathbb C , \end{aligned}$$
where we still note by \(h\) the \(\mathbb C \)-linear extension of \(h\). Thus \(h_{\mathbb{C }}\) coincides with the sesquilinear extension over \(V \otimes _{\mathbb{R }} \mathbb C \) of the Riemannian metric associated to \(h\). We infer by the expression (31) in [4] of the Riemannian metric on the exterior products that the induced hermitian product on the vector bundle \(\Lambda ^{p, q}_J T^{*}_X\) is given by the formula
$$\begin{aligned}&\Big \langle \Lambda _{j = 1}^p \alpha _{1, j} \wedge \Lambda _{j = 1}^q \beta _{1, j} , \Lambda _{j = 1}^p \alpha _{2, j} \wedge \Lambda _{j = 1}^q \beta _{2, j} \Big \rangle \\&\quad = (p + q) ! \det \left( 2^{- 1} h^{*} (\alpha _{1, j}, \bar{\alpha }_{2, l}) \right) \overline{\det \left( 2^{- 1} h^{*} ( \bar{\beta }_{1, j}, \beta _{2, l}) \right) }. \end{aligned}$$
Consider now an element
$$\begin{aligned} A \in T^{*}_{{X, - J}} \otimes _{\mathbb{C }} T_{{X, J}} \cong \Lambda ^{0, 1}_J T^{*}_X \otimes _{\mathbb{C }} T_{{X, J}} , \end{aligned}$$
and let \((e_k)_k \subset T_{{X, J}}, e_k :=\zeta _k + \bar{\zeta }_k\) be the \(J\)-complex basis associated to \((\zeta _k)_k\). Then hold the local expression
$$\begin{aligned} A = A_{k, \bar{l}} \bar{\zeta }^{*}_l \otimes _{J} e_k = A_{k, \bar{l}} \bar{\zeta }^{*}_l \otimes \zeta _k + \mathrm{Conjugate }. \end{aligned}$$
Assume from now on that the frame \((e_k)_k\) is \(h\)-orthonormal. On one side if one think of \(A\) as an element in \(\mathrm{End }_{\mathbb{R }} (T_X)\) then
$$\begin{aligned} |A|^2_g = \mathrm{Tr }_{\mathbb{R }} \left( A A_g^T \right) = 2 |A_{k, \bar{l}} |^2 , \end{aligned}$$
since
$$\begin{aligned} A_g^T = A_{l, \bar{k}} \bar{\zeta }^{*}_l \otimes \zeta _k + \mathrm{Conjugate }. \end{aligned}$$
On the other side
$$\begin{aligned} |A|^2_{\omega } \equiv |A|_{\Lambda _J^{0, 1} T^{*}_X \otimes _{\mathbb{C }} T_{X, J}, \omega }^2&= \left\langle A_{k, \bar{l}} \bar{\zeta }^{*}_l , A_{k, \bar{p}} \bar{\zeta }^{*}_p \right\rangle \\&= \frac{1}{2} \overline{h^{*} \left( \overline{A}_{k, \bar{l}} \zeta ^{*}_l , A_{k, \bar{p}} \bar{\zeta }^{*}_p \right) }\\&= \frac{1}{2} |A_{k, \bar{l}} |^2 \cdot 4 , \end{aligned}$$
Thus \(|A|^2_g = |A|^2_{\omega }.\) The same identity hold true for any
$$\begin{aligned} A \in T^{*}_{{X, J}} \otimes _{\mathbb{C }} T_{{X, J}} \cong \Lambda ^{1, 0}_J T^{*}_X \otimes _\mathbb{C } T_{{X, J}}. \end{aligned}$$
In higher degrees there is a multiplicative factor involved. We consider for example \(A \in \Lambda ^{1, 1}_J T^{*}_X \otimes _\mathbb{C } T_{{X, J}}\) and its local expression
$$\begin{aligned} A = i A_{p, k, \bar{l}} \left( \zeta _p^{*} \wedge \bar{\zeta }_l^{*} \right) \otimes \zeta _k + \mathrm{Conjugate }. \end{aligned}$$
Then
$$\begin{aligned} |A|_{\Lambda _J^{1, 1} T^{*}_X \otimes _\mathbb{C } T_{X, J}, \omega }^2&= \left\langle i A_{p, k, \bar{l}} \zeta _p^{*} \wedge \bar{\zeta }_l^{*} , i A_{r, k, \bar{h}} \zeta _r^{*} \wedge \bar{\zeta }_h^{*} \right\rangle \\&= A_{p, k, \bar{l}} \overline{A}_{r, k, \bar{h}} \left\langle \zeta _p^{*} \wedge \bar{\zeta }_l^{*} , \zeta _r^{*} \wedge \bar{\zeta }_h^{*} \right\rangle \\&= \frac{1}{2} |A_{p, k, \bar{l}} |^2 h^{*} \left( \zeta _p^{*} , \bar{\zeta }_p^{*} \right) \overline{h^{*} \left( \zeta _l^{*} , \bar{\zeta }_l^{*} \right) }\\&= 8 |A_{p, k, \bar{l}} |^2. \end{aligned}$$
On the other side if we think of \(A\) as an element of \(\Lambda ^2 T_X \otimes _\mathbb{R } T_X\) then
$$\begin{aligned} |A|_{\Lambda ^2 T_X \otimes _\mathbb{R } T_X, g}^2&= \mathrm{Tr }_\mathbb{R } \left[ \left( e_r \lnot A \right) \left( e_r \lnot A \right) _g^T \right] \\&\quad + \mathrm{Tr }_{\mathbb{R }} \left[ \left( J e_r \lnot A \right) \left( J e_r \lnot A \right) _g^T \right] , \end{aligned}$$
and
$$\begin{aligned} e_r \lnot A&= i A_{r, k, \bar{l}} \bar{\zeta }_l^{*} \otimes \zeta _k - i A_{l, k, \bar{r}} \zeta _l^{*} \otimes \zeta _k + \mathrm{Conjugate } ,\\ J e_r \lnot A&= - A_{r, k, \bar{l}} \bar{\zeta }_l^{*} \otimes \zeta _k - A_{l, k, \bar{r}} \zeta _l^{*} \otimes \zeta _k + \mathrm{Conjugate } ,\\ \left( e_r \lnot A \right) _g^T&= i A_{r, l, \bar{k}} \bar{\zeta }_l^{*} \otimes \zeta _k + i \overline{A}_{k, l, \bar{r}} \zeta _l^{*} \otimes \zeta _k + \mathrm{Conjugate } ,\\ \left( J e_r \lnot A \right) _g^T&= - A_{r, l, \bar{k}} \bar{\zeta }_l^{*} \otimes \zeta _k - \overline{A}_{k, l, \bar{r}} \zeta _l^{*} \otimes \zeta _k + \mathrm{Conjugate }. \end{aligned}$$
(By conjugate we mean the complex conjugate of all therms preceding this word). Thus hold the equality
$$\begin{aligned} \mathrm{Tr }_{\mathbb{R }} \left[ \left( e_r \lnot A \right) \left( e_r \lnot A \right) _g^T \right] = 2 |A_{l, k, \bar{r}} |^2 = \mathrm{Tr }_\mathbb{R } \left[ \left( J e_r \lnot A \right) \left( J e_r \lnot A \right) _g^T \right] . \end{aligned}$$
We infer the identity
$$\begin{aligned} 2 |A|_{\Lambda ^2 T_X \otimes _\mathbb{R } T_X, g}^2 = |A|_{\Lambda _J^{1, 1} T^{*}_X \otimes _\mathbb{C } T_{X, J}, \omega }^2. \end{aligned}$$
