Theorem [46, Theorem 2.4] (which gives that $f$ is a locally trivial smooth fiber bundle over $Y\S'$) and the fact $Y\S'$ is connected, we immediately conclude that all fibers $X_y, y \in Y\S'$ are diffeomorphic to a torus. But a compact Kähler manifold which is diffeomorphic to a torus must be in fact biholomorphic to a torus, as follows easily using the Albanese map, and we conclude that all fibers $X_y, y \in Y\S'$ are biholomorphic to tori, say $X_y \cong \mathbb{C}^{n-m}/\Lambda_y$, where $\Lambda_y$ is a lattice in $\mathbb{C}^{n-m}$. Since $f$ is a holomorphic submersion over $Y\S'$, we may choose a basis $v_1(y), \dots, v_{2n-2m}(y)$ of the lattice $\Lambda_y$ which varies holomorphically in $y \in B$, for any sufficiently small $B \subset Y\S'$. We can then construct another family $f'$ of tori over $B$, by taking the quotient of $B \times \mathbb{C}^{n-m}$ by the holomorphic free $\mathbb{Z}^{2n-2m}$ action given by

$$ (\ell_1, \dots, \ell_{2n-2m}) \cdot (y, z) = \left( y, z + \sum_{i=1}^{2n-2m} \ell_i v_i(y) \right), $$

where $y \in B, z \in \mathbb{C}^{n-m}$ and $\ell_i \in \mathbb{Z}$. Note that while the choice of the generating vectors $v_i(y)$ is not unique, the quotient does not depend on this choice. This gives us a holomorphic submersion $f' : U' \to B$ with fiber $f'^{-1}(y)$ biholomorphic to $X_y$, for all $y \in B$. A theorem of Wehler [86, Satz 3.6] then shows that the families $f$ and $f'$ are locally isomorphic, so up to shrinking $B$ there is a biholomorphism $U' \to U$, which is compatible with the projections to $B$. Composing the quotient map $B \times \mathbb{C}^{n-m}$ with this biholomorphism, we obtain a local biholomorphism $p : B \times \mathbb{C}^{n-m} \to U$ such that $f \circ p(y, z) = y$ for all $(y, z)$. The map $p$ is thus the universal covering of $U$.

The following is the key tool we need:

**PROPOSITION 5.25 ([29, 31, 38]).** — Up to shrinking $B$, on $U = f^{-1}(B)$ there is a closed semipositive definite real (1, 1) form $\omega_{SF}$ which is semi-flat in the sense that $\omega_{SF}|_{X_y}$ a flat Kähler metric on $X_y$ for all $y \in B$, and such that $p^*\omega_{SF} = \sqrt{-1}\partial\bar{\partial}\eta$ where $\eta \in C^\infty(B \times \mathbb{C}^{n-m}, \mathbb{R})$ satisfies

$$ \eta(y, \lambda z) = \lambda^2 \eta(y, z), \quad (5.72) $$

for all $(y, z) \in B \times \mathbb{C}^{n-m}$ and $\lambda \in \mathbb{R}$.

This was proved in [31, Section 3] when $X$ is projective and $[\omega_0]$ is rational, following the recipe in [29], and was then proved in [38] in general. We will not prove this here, but just say that the function $\eta$ is given explicitly by

$$ \eta(y, z) = -\frac{1}{4} \sum_{i,j=1}^{n-m} (\mathrm{Im}Z(y))_{ij}^{-1} (z_i - \bar{z}_i)(z_j - \bar{z}_j), $$

where $Z$ is a holomorphic period map from $B$ to the Siegel upper half space $\mathfrak{H}_{n-m}$ of symmetric $(n-m) \times (n-m)$ complex matrices with positive definite
imaginary part (so $(\operatorname{Im}Z(y))^{-1}$ in this formula is well-defined). The key reason why this can be done is that $\mathfrak{H}_{n-m}$ classifies complex tori which are polarized by a Kähler class. We refer the reader to [31, 38] for the details of the construction of $Z$ (which is easier under the rationality assumption) and of why this $\eta$ satisfies our requirements.

Now, recall that thanks to Proposition 5.15 (or rather its generalization to the case when $S \neq \emptyset$) we have that

$$C^{-1}(e^{-t}\omega_{\mathrm{SF}} + f^{*}\omega_{Y}) \leq \omega(t) \leq C(e^{-t}\omega_{\mathrm{SF}} + f^{*}\omega_{Y}),$$

where we used that on $U$ we have that $\omega_{\mathrm{SF}} + f^{*}\omega_{Y}$ is a Kähler metric. For $t \geq 0$ let $\lambda_t : B \times \mathbb{C}^{n-m} \to B \times \mathbb{C}^{n-m}$ be given by

$$\lambda_t(y, z) = (y, z e^{t/2}),$$

which is a “stretching in the fiber directions” (compare this with the maps $F_t$ in Theorem 5.20 which were instead shrinking the base directions). Then the metrics

$$\omega_t(s) := \lambda_t^* p^* \omega(s+t), \quad -1 \leq s \leq 0,$$

on $B \times \mathbb{C}^{n-m}$ satisfy

$$C^{-1}(e^{-t}\lambda_t^* p^* \omega_{\mathrm{SF}} + \lambda_t^* p^* f^* \omega_Y) \leq \omega_t(s) \leq C(e^{-t}\lambda_t^* p^* \omega_{\mathrm{SF}} + \lambda_t^* p^* f^* \omega_Y),$$

for all $t \geq 0, -1 \leq s \leq 0$, and

$$\frac{\partial}{\partial s}\omega_t(s) = -\mathrm{Ric}(\omega_t(s)) - \omega_t(s), \quad -1 \leq s \leq 0.$$

But we have that $f \circ p \circ \lambda_t = f \circ p$, so $\lambda_t^* p^* f^* \omega_Y = p^* f^* \omega_Y$, and

$$\lambda_t^* p^* \omega_{\mathrm{SF}} = \lambda_t^* \sqrt{-1} \partial \bar{\partial} \eta = \sqrt{-1} \partial \bar{\partial} (\eta \circ \lambda_t) = e^t \sqrt{-1} \partial \bar{\partial} \eta = e^t p^* \omega_{\mathrm{SF}},$$

since

$$\eta \circ \lambda_t(y, z) = \eta(y, z e^{t/2}) = e^t \eta(y, z),$$

thanks to (5.72). Therefore we conclude that on $B \times \mathbb{C}^{n-m}$ we have

$$C^{-1} p^* (\omega_{\mathrm{SF}} + f^* \omega_Y) \leq \omega_t(s) \leq C p^* (\omega_{\mathrm{SF}} + f^* \omega_Y),$$

and so for each given compact set $K \subset B \times \mathbb{C}^{n-m}$ there is a constant $C_K$ such that on $K$ we have

$$C_K^{-1} \omega_E \leq \omega_t(s) \leq C_K \omega_E,$$

for all $t \geq 0, -1 \leq s \leq 0$, where $\omega_E$ is a Euclidean metric on $B \times \mathbb{C}^{n-m}$. We can therefore apply the local higher order estimates in Theorem 3.11 and obtain that for every compact set $K \subset B \times \mathbb{C}^{n-m}$ there are constants $C_{K,k}$ such that

$$\|\omega_t(s)\|_{C^k(K,g_E)} \leq C_{K,k},$$

for all $t \geq 0, -\frac{1}{2} \leq s \leq 0$. Setting $s = 0$ we obtain

$$\|\lambda_t^* p^* \omega(t)\|_{C^k(K,g_E)} \leq C_{K,k}, \qquad (5.73)$$
and we still have

$$ \lambda_t^* p^* \omega(t) \geq C^{-1} \omega_E, $$

on $K \times [0, \infty)$. In particular, this gives

$$ \sup_K |\mathrm{Rm}(\lambda_t^* p^* \omega(t))|_{\lambda_t^* p^* \omega(t)} \leq C, \qquad (5.74) $$

for all $t \geq 0$. If now $K' \subset U \subset X \setminus S$ is a compact set which is small enough so that $K = p^{-1}(K') \subset B \times \mathbb{C}^{n-m}$ is compact and $p$ is a biholomorphism on $K$ (note that such compact sets $K'$ cover $U$) then we have

$$ \begin{aligned} \sup_{K'} |\mathrm{Rm}(\omega(t))|_{\omega(t)} &= \sup_K |\mathrm{Rm}(p^*\omega(t))|_{p^*\omega(t)} \\ &= \sup_{\lambda_{1/t}(K)} |\mathrm{Rm}(\lambda_t^* p^* \omega(t))|_{\lambda_t^* p^* \omega(t)}, \end{aligned} $$

where $\lambda_{1/t}$ is the inverse map of $\lambda_t$. But the compact sets $\lambda_{1/t}(K)$ are all contained in a fixed compact set of $B \times \mathbb{C}^{n-m}$, and so from (5.74) and a covering argument we easily obtain (5.71). Also, (5.73) easily implies that

$$ \|p^*\omega(t)\|_{C^k(K,g_E)} \leq C_{K,k}, $$

for any given compact set $K \subset B \times \mathbb{C}^{n-m}$ (in fact, (5.73) is a much stronger bound). Since $p$ is a local biholomorphism, this (and another covering argument) proves (5.70), and completes the proof of Theorem 5.24 when $X_y$ is a torus.

If now the fiber $X_y$ is just biholomorphic to a finite quotient of a torus, for some $y \in Y \setminus S'$ (and therefore for all such $y$, by the same argument as before using Ehresmann's Theorem), then we choose again a sufficiently small open set $B \subset Y \setminus S'$ such that $f$ is a locally trivial smooth fiber bundle over $B$, and so there is a diffeomorphism $\Psi : B \times F \to f^{-1}(B)$, compatible with the projections to $B$, where $F$ is the smooth manifold underlying $X_y$. We use $\Psi$ to pull back the complex structure on $f^{-1}(B)$ to a complex structure $J$ on $B \times F$, which is in general different from the product complex structure on $B \times X_y$. This way, the map $\Psi$ becomes a biholomorphism (where here and in the following we always use the complex structure $J$ on $B \times F$). If we now let $\tilde{F} \to F$ be a smooth finite covering map with $\tilde{F}$ a torus, then the map $p : B \times \tilde{F} \to B \times F$ is a smooth finite covering (hence a local diffeomorphism), and so we can use it to pull back the complex structure $J$ to a complex structure $\tilde{J}$ on $B \times \tilde{F}$. This way $p$ is also a local biholomorphism, and so pulling back a Kähler metric on $f^{-1}(B)$ via $\Psi \circ p$ we obtain a compatible Kähler metric on $B \times \tilde{F}$. Then the projection $\pi : B \times \tilde{F}$ is by construction equal to $f \circ \Psi \circ p$, and so it is holomorphic, and clearly a proper submersion. This implies that its fibers $\tilde{X}_y = \pi^{-1}(y)$ are all compact complex submanifolds of $B \times \tilde{F}$ (with the complex structure $\tilde{J}$), and so they are also Kähler, and each $\tilde{X}_y$ is diffeomorphic to the torus $\tilde{F}$. As remarked earlier, this implies that
all fibers $\tilde{X}_y$ are in fact biholomorphic to complex tori $\mathbb{C}^{n-m}/\Lambda_y$. Therefore the family $\pi$ over $B$ has torus fibers, and pulling back the solution $\omega(t)$ of the Kähler-Ricci flow via the holomorphic finite covering map $\Psi \circ p$ we obtain a solution $p^*\Psi^*\omega(t)$ of the Kähler-Ricci flow on $B \times \tilde{F}$. We can then apply Proposition 5.25 to $B \times \tilde{F}$ and get a semi-flat form $\omega_{SF}$ with the same properties, and from Proposition 5.15 we again have

$$C^{-1}(e^{-t}\omega_{SF} + \pi^*\omega_Y) \le p^*\Psi^*\omega(t) \le C(e^{-t}\omega_{SF} + \pi^*\omega_Y),$$

on $B \times \tilde{F}$ for all $t \ge 0$. Then the rest of the argument above goes through, and we obtain (5.70) and (5.71) on $B \times \tilde{F}$ for the metrics $p^*\Psi^*\omega(t)$. Since $\Psi \circ p$ is a holomorphic finite covering, these estimates immediately imply those for $\omega(t)$ on $f^{-1}(B)$. $\square$

# 6. Some open problems

In this closing section, we collect some well-known open problems on the Kähler-Ricci flow (in addition to the conjectures that we have already discussed in Section 4), related to the material discussed in these notes.

## 6.1. Diameter bounds

Diameter bounds for solutions of the Kähler-Ricci flow as we approach a singularity are not easy to get. In general we expect:

**CONJECTURE 6.1.** — Let $X$ be a compact Kähler manifold and $\omega(t)$ a solution of the Kähler-Ricci flow (1.1) defined on a maximal time interval $[0, T)$ with $T < \infty$. Then there is a constant $C > 0$ such that

$$\mathrm{diam}(X, \omega(t)) \le C,$$

for all $t \in [0, T)$.

This conjecture is known in the case when the limiting class $[\alpha] = [\omega_0] - 2\pi c_1(X)$ is equal to $\pi^*[\omega_Y]$ where $\pi: X \to Y$ is the blowup of a compact Kähler manifold $Y$ at finitely many distinct point and $\omega_Y$ is a Kähler metric on $Y$, thanks to [67], and this is in fact the general case when $n=2$ and the singularity is noncollapsed. The conjecture is also known when $X$ is Fano and $[\omega_0] = \lambda c_1(X)$ for some $\lambda > 0$, since as we mentioned earlier Perelman proved that in this case $\mathrm{diam}(X, \omega(t)) \le C(T-t)^{\frac{1}{2}}$ (see [56]), and it is also proved in [62] for some special Fano fibrations (also discussed earlier).

In the case of infinite time solutions, we expect:
**CONJECTURE 6.2.** — Let $X$ be a compact Kähler manifold and $\omega(t)$ a solution of the Kähler-Ricci flow (1.1) defined on $[0, \infty)$. Then there is a constant $C > 0$ such that

$$ \mathrm{diam} \left( X, \frac{\omega(t)}{t} \right) \le C, $$

for all $t \ge 0$.

Recall that the existence of an infinite time solution is equivalent to $K_X$ being nef. As mentioned earlier, the Abundance Conjecture for Kähler manifolds would imply that $K_X$ is semiample, so in particular $\kappa(X) \ge 0$. Assuming $K_X$ is semiample, if $\kappa(X) = 0$ then $X$ is Calabi-Yau (by Lemma 5.4) and we even have that $\mathrm{diam}(X, \omega(t)) \le C$, thanks to Theorem 5.1. If $\kappa(X) = n$ then $K_X$ is nef and big, and in this case Conjecture 6.2 is proved in [34] when $n=2$ and in [74] when $n \le 3$ (see also [33] for further progress). Lastly, when $0 < \kappa(X) < n$ (this is the setup of Theorem 5.5), Conjecture 6.2 seems to be open even in the case when $n=2, \kappa(X) = 1$.

## 6.2. Volume growth

The growth of the total volume of $X$ as $t \to \infty$ for an infinite time solution is a delicate issue as well. Indeed, the following conjecture is equivalent to the Abundance Conjecture in the general Kähler case:

**CONJECTURE 6.3.** — Let $X$ be a compact Kähler manifold and $\omega(t)$ a solution of the Kähler-Ricci flow (1.1) defined on $[0, \infty)$. Then $\kappa(X) \ge 0$ and there is a constant $C > 0$ such that

$$ C^{-1}t^{\kappa(X)} \le \mathrm{Vol}(X, \omega(t)) \le Ct^{\kappa(X)}, \quad (6.1) $$

for all $t \ge 0$.

Since the Abundance Conjecture in the Kähler case is now known for $n \le 3$ by [5], so is this conjecture. Indeed, by the Abundance Conjecture we have that $K_X$ is semiample, and then as explained in Section 5 we get a fiber space $f: X \to Y$ onto a normal projective variety of dimension $\kappa(X) \ge 0$, such that $K_X^{\otimes \ell} = f^*L$ for an ample line bundle $L$ on $Y$. This implies that $c_1(K_X)^p = 0$ for all $p > \kappa(X)$, and so

$$ \begin{aligned} \mathrm{Vol}(X, \omega(t)) &= \int_X (\omega_0 + 2\pi t c_1(K_X))^n \\ &= ct^{\kappa(X)} \int_X \omega_0^{n-\kappa(X)} \wedge c_1(K_X)^{\kappa(X)} + O(t^{\kappa(X)-1}), \end{aligned} $$
where $c > 0$ and

$$ \int_X \omega_0^{n-\kappa(X)} \wedge c_1(K_X)^{\kappa(X)} = \frac{1}{\ell^{\kappa(X)}} \int_X \omega_0^{n-\kappa(X)} \wedge f^*c_1(L)^{\kappa(X)} > 0. $$

The fact that conversely Conjecture 6.3 implies the Abundance Conjecture follows easily from [53, Theorem 5.5] (which is the extension of [43, Corollary 6.1.13] to the Kähler case), since (6.1) implies that $K_X$ is abundant (i.e. its numerical dimension is equal to $\kappa(X)$).

We also have the following simple observation, related to Conjecture 6.3:

**PROPOSITION 6.4.** — Let $X$ be a compact Kähler manifold and $\omega(t)$ a solution of the Kähler–Ricci flow (1.1) defined on $[0, \infty)$. Then there is a constant $C > 0$ such that

$$ \mathrm{Vol}(X, \omega(t)) \le C, \qquad (6.2) $$

if and only if $X$ is Calabi–Yau.

*Proof.* — If $X$ is Calabi–Yau then $\mathrm{Vol}(X, \omega(t))$ is clearly constant. Conversely, if (6.2) holds then expanding

$$ \mathrm{Vol}(X, \omega(t)) = \int_X (\omega_0 + 2\pi t c_1(K_X))^n, $$

we see that we must have

$$ \int_X \omega_0^{n-1} \wedge c_1(K_X) = 0. $$

Since $c_1(K_X)$ is nef, the Khovanskii–Teissier inequality for nef classes (see e.g. [23])

$$ \int_X \omega_0^{n-1} \wedge c_1(K_X) \ge \left( \int_X \omega_0^{n-2} \wedge c_1(K_X)^2 \right)^{\frac{1}{2}} \left( \int_X \omega_0^n \right)^{\frac{1}{2}}, $$

implies that $\int_X \omega_0^{n-2} \wedge c_1(K_X)^2 = 0$. The result now follows from the Hodge–Riemann bilinear relations on Kähler manifolds, proved in [14]. Indeed, following their notation, we set $\omega_1 = \cdots = \omega_{n-1} := \omega_0$, so that the condition $\int_X \omega_0^{n-1} \wedge c_1(K_X) = 0$ says that $c_1(K_X) \in P^{1,1}(X)$, while the condition $\int_X \omega_0^{n-2} \wedge c_1(K_X)^2 = 0$ says that $Q(c_1(K_X), c_1(K_X)) = 0$. Since by [14, Theorem A] the bilinear form $Q$ is positive definite on $P^{1,1}(X)$, this implies that $c_1(K_X) = 0$, and so $X$ is Calabi–Yau. $\square$

## 6.3. Singularity types

Following [37] we say that a solution $\omega(t)$ of the Kähler–Ricci flow (1.1) on a compact Kähler manifold $X$, defined on a maximal time interval $[0, T)$, $T <
$\infty$, develops a type I singularity at time $T$ if we have

$$ \sup_{X \times [0, T)} (T-t)|\mathrm{Rm}(\omega(t))|_{\omega(t)} < +\infty, $$

and a type IIa singularity if

$$ \sup_{X \times [0, T)} (T-t)|\mathrm{Rm}(\omega(t))|_{\omega(t)} = +\infty. $$

While type I singularities are easy to construct, this is not the case for type IIa singularities. The first compact examples, for the Ricci flow on Riemannian manifolds, were constructed in [32]. Since these examples are not Kähler, this leaves open the following:

**PROBLEM 6.5.** — Construct a type IIa finite time singularity of the Kähler-Ricci flow on a compact Kähler manifold.

For example, when $X$ is Fano and $[\omega_0] = \lambda c_1(X)$ for some $\lambda > 0$, then the singularity being type I is equivalent to the curvature remaining uniformly bounded for all $t \ge 0$, after we renormalize the flow to have constant volume (the normalized flow exists for all $t \ge 0$). It seems very likely that there exist Fano manifolds where the normalized flow does not have uniformly bounded curvature, but no examples have been found yet.

On the other hand, in the Fano case Perelman has proved a uniform scalar curvature bound (see [56]), which in the unnormalized flow translates to the estimate

$$ R(t) \le \frac{C}{T-t}, \qquad (6.3) $$

on $X \times [0, T)$. It is not known whether (6.3) holds for all finite time singularities of the Kähler-Ricci flow, but see [90] for partial results.

We now discuss infinite time solutions, and their singularity types “at infinity”. Again following [37] we say that a solution $\omega(t)$ of the Kähler-Ricci flow (1.1) on a compact Kähler manifold $X$, defined for all $t \ge 0$, develops a type III singularity at infinity if we have

$$ \sup_{X \times [0, \infty)} t |\mathrm{Rm}(\omega(t))|_{\omega(t)} < +\infty, $$

and a type IIb singularity if

$$ \sup_{X \times [0, \infty)} t |\mathrm{Rm}(\omega(t))|_{\omega(t)} = +\infty. $$

A simple scaling argument shows that type III is equivalent to the solution of the normalized flow (5.1) having uniformly bounded curvature for all $t \ge 0$, and type IIb to its negation. When $n = 1$ it follows from work of Hamilton [36] that all infinite time solutions are type III. In the case of the Ricci flow on real 3-dimensional compact Riemannian manifolds, all infinite
time solutions are type III thanks to [2]. However, in the Kähler case when $n = 2$ there are type IIB solutions. It is enough to take $X$ a $K3$ surface, and $\omega$ a Ricci-flat Kähler metric on $X$, which exists thanks to Yau [88]. Then $\omega$ cannot be flat since $\chi(X) = 24 \neq 0$, so $\sup_X |\mathrm{Rm}(\omega)|_\omega = c > 0$. Then $\omega(t) \equiv \omega$ is a static solution of the Kähler-Ricci flow (1.1), and

$$ \sup_{X \times [0, \infty)} t |\mathrm{Rm}(\omega(t))|_{\omega(t)} = \sup_{t \in [0, \infty)} ct = +\infty, $$

so this solution is type IIB.

The following theorem was proved in [82]:

**THEOREM 6.6.** — Let $X^n$ be a compact Kähler manifold with $K_X$ semiample, and consider a solution of the Kähler-Ricci flow (1.1) (which necessarily exists for all positive time).

(1) Assume $\kappa(X) = 0$. Then the solution is type III if and only if $X$ is a finite unramified quotient of a torus

(2) Assume $\kappa(X) = n$. Then the solution is type III if and only if $K_X$ is ample

(3) Assume $0 < \kappa(X) < n$, and let $X_y$ be any smooth fiber of the fibration $f: X \to Y$ defined by sections of $K_X^{\otimes \ell}$, for $\ell$ large. If $X_y$ is not a finite unramified quotient of a torus then the solution is type IIB, while if $X_y$ is a finite unramified quotient of a torus and $S = \emptyset$ (i.e. $Y$ is smooth and $f$ is a submersion) then the solution is type III.

In particular, in all these cases the singularity type does not depend on the initial metric.

Another proof of (2) was obtained in [33]. This theorem leaves open the case when the generic fiber $X_y$ is a finite unramified quotient of a torus, but $f$ is not a submersion everywhere. In this case the solution can be either type IIB or type III, depending on the singularities and multiplicities of the fibers contained in $S$. A complete classification when $n = 2$ is obtained in [82], where it is also shown that in general dimensions if any component of singular fiber is uniruled then the solution is of type IIB. It remains to understand what happens when no such component is uniruled.

Considering Theorem 6.6, it is then natural to conjecture:

**CONJECTURE 6.7.** — Let $X$ be a compact Kähler manifold with $K_X$ nef, so every solution of the Kähler-Ricci flow (1.1) exists for all positive time. Then the singularity type at infinity does not depend on the choice of the initial metric $\omega_0$.

As mentioned above, this conjecture is only known when $n \le 2$, thanks to [82].
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