Uniqueness of shrinking gradient Kähler-Ricci solitons on non-compact toric manifolds

I would like to thank Song Sun for suggesting this problem, as well as for his continued support and many useful discussions. I would like to thank Ronan Conlon for his interest, useful discussions, and detailed commentary on the preliminary versions of this article. I would also like to thank Vestislav Apostolov for his invaluable suggestions and Akito Futaki for his interest and comments. I am also grateful for the many conversations with Chris Eur in which I learned much of what I know about the algebraic geometry of toric varieties, and to the referee for their insightful suggestions. This work was partially supported by the National Science Foundation RTG grant DMS-1344991.

We begin by recalling some basics from algebraic toric geometry that we will use later on. The main reference here is [15]. Fix an algebraic torus $(\mathbb{C}^{*})^{n}$ and let $\mathfrak{t}$ be the Lie algebra of the real torus $T^{n}\subset(\mathbb{C}^{*})^{n}$. Fix an integer lattice $\Gamma\subset\mathfrak{t}$ so that $(\mathbb{C}^{*})^{n}\cong\mathfrak{t}\oplus i\mathfrak{t}/\Gamma$ acting only in the second factor. Let $\Gamma^{*}$ denote the corresponding dual lattice in $\mathfrak{t}^{*}$.

We emphasize that, contrary to the definitions presented in the introduction, a toric variety $M$ is always assumed to be algebraic. As we will see, the fixed point set of the $(\mathbb{C}^{*})^{n}$-action associated to a toric variety is necessarily finite. In particular, the underlying complex manifold of a smooth toric variety is always a complex toric manifold as defined in the introduction.

The algebraic geometry of toric varieties has a rich interplay with combinatorics, which is integral to many of the constructions that follow. We begin by introducing the relevant combinatorial objects.

We will often not distinguish between a polyhedron $P$ and its interior, but where confusion may arise we will denote by $\overline{P}$ the closed object and $P$ the interior. The intersection of $P$ with the plane $\langle\nu,x\rangle=a$ is a polyhedron $F_{\nu}$ of one less dimension and is called a facet of $P$. The intersections of any number of the $F_{\nu}$’s form the collection of faces of $P$.

Note that (the interior of) $C^{*}$ is necessarily an open cone in $\mathfrak{t}$, even when $C$ is not full-dimensional.

Rational polyhedra play an important role in the algebraic geometry of toric varieties, in that each such $P$ determines a unique quasiprojective toric variety $\mathcal{M}_{P}$. This procedure is constructive and can be understood via the introduction of a fan. A rational polyhedral cone $\sigma$ is by definition a convex subset of $\mathfrak{t}$ of the form

where $\nu_{1},\dots,\nu_{k}\in\Gamma$ is a fixed finite collection of lattice points. The recession cone $C$ of a rational polyhedron $P$ is always a rational polyhedral cone [15, Chapter 7].

To construct $\mathcal{M}_{\Sigma}$ from $\Sigma$, one begins by taking each $n$-dimensional cone $\sigma\in\Sigma$ and constructing an affine toric variety $U_{\sigma}$. We define the dual cone $\sigma^{*}$ of $\sigma$ by (2.1):

Let $S_{\sigma}$ be the semigroup of those lattice points which lie in $\sigma^{*}$ under addition. Then one defines the semigroup ring, as a set, as all finite sums of the form

The ring structure is then defined on monomials by $\lambda_{s_{1}}s_{1}\cdot\lambda_{s_{2}}s_{2}=(\lambda_{s_{1}}\lambda_{s_{2}})(s_{1}+s_{2})$ and extended in the natural way. The basic example is $\sigma=\mathbb{R}^{n}_{+}$, where $\mathbb{C}[S_{\sigma}]$ is naturally isomorphic to $\mathbb{C}[z_{1},\dots,z_{n}]$. Then the affine variety $U_{\sigma}$ is defined to be $\text{Spec}(\mathbb{C}[S_{\sigma}])$. This is automatically endowed with a $(\mathbb{C}^{*})^{n}$-action with an open dense orbit. This construction of course can be implemented on the lower-dimensional cones $\tau\in\Sigma$. If $\sigma_{1}\cap\sigma_{2}=\tau$, then there is a natural way to map $U_{\tau}$ into $U_{\sigma_{1}}$ and $U_{\sigma_{2}}$ isomorphically. Thus one constructs $\mathcal{M}_{\Sigma}$ by declaring the collection of all $U_{\sigma}$ to be an open affine cover with transition data determined by $U_{\tau}$. An important property of this construction is the Orbit-Cone correspondence.

In particular, the fixed point set of the $(\mathbb{C}^{*})^{n}$-action is in natural bijection with the full-dimensional cones in $\Sigma$, and is therefore always finite. At the other extreme, each ray $\sigma\in\Sigma$ determines a unique torus-invariant divisor $D_{\sigma}$. As a consequence, a torus-invariant Weil divisor $D$ on $\mathcal{M}_{\Sigma}$ naturally determines a polyhedron $P_{D}\subset\mathfrak{t}^{*}$ as follows. We can decompose $D$ uniquely as $D=\sum_{i=1}^{N}a_{i}D_{\sigma_{i}}$, where $\sigma_{i}\in\Sigma$, $i=1,\dots,N$ is the collection of rays. By assumption, there exists a unique minimal $\nu_{i}\in\sigma_{i}\cap\Gamma$. Then set

The importance of polyhedra for our purposes lies in the fact that this procedure is partially reversible. That is, given a suitable polyhedron $P$, one can determine a unique toric variety $\mathcal{M}_{P}$ through its normal fan $\Sigma_{P}$. To form $\Sigma_{P}$, one starts with a vertex $v\in P$ and considers those facets $F$ containing $v$. This determines a cone $\sigma_{v}$ spanned by the inner normals $\nu_{F}$ corresponding to each such $F$. Then there is a unique fan $\Sigma_{P}$ which consists of the collection of $\sigma_{v}$ along with all of each of their faces. Finally, $\mathcal{M}_{P}$ is defined to be the toric variety associated to $\Sigma_{P}$. As we will see, the variety $\mathcal{M}_{P}$ comes naturally equipped with a divisor $D$ whose corresponding polyhedron is precisely $P$. Moreover,

Let $M$ be a complex manifold together with an effective holomorphic $(\mathbb{C}^{*})^{n}$-action. Such an action always has an open and dense orbit. Indeed, let $T^{n}\subset(\mathbb{C}^{*})^{n}$ be the real torus with Lie algebra $\mathfrak{t}$. Choose a basis $(X_{1},\dots,X_{n})$ for $\mathfrak{t}$. Then each $X_{i}$ is a holomorphic vector field on $M$, and thus vanishes along an analytic subvariety. In particular, there is a fixed analytic subvariety $V\subset M$ such that on $U=M-V$, none of the vector fields $X_{i}$ vanish. Clearly $X_{i}$ and $JX_{i}$ are complete and commute, and so the vector fields $(X_{1},JX_{1},\dots,X_{n},JX_{n})$ can be integrated to determine an isomorphism $U\cong(\mathbb{C}^{*})^{n}$. Throughout the remainder of the paper we will make heavy use of this natural coordinate system, which we usually just denote by $(\mathbb{C}^{*})^{n}\subset M$. In particular, we fix once and for all such a basis $(X_{1},\dots,X_{n})$ for $\mathfrak{t}$. This induces a background coordinate system $(\xi^{1},\dots,\xi^{n})$ on $\mathfrak{t}$. We use the natural inner product on $\mathfrak{t}$ to identify $\mathfrak{t}\cong\mathfrak{t}^{*}$ and thus can also identify $\mathfrak{t}^{*}\cong\mathbb{R}^{n}$. For clarity, we will denote the induced coordinates on $\mathfrak{t}^{*}$ by $(x^{1},\dots,x^{n})$. Let $(z_{1},\dots,z_{n})$ be the natural coordinates on $(\mathbb{C}^{*})^{n}$ as an open subset of $\mathbb{C}^{n}$. There is a natural diffeomorphism $\text{Log}:(\mathbb{C}^{*})^{n}\to\mathfrak{t}\times T^{n}$, which provides a one-to-one correspondence between $T^{n}$-invariant smooth functions on $(\mathbb{C}^{*})^{n}$ and smooth functions on $\mathfrak{t}$. Explicitly, $\text{Log}(z_{1},\dots,z_{n})=(\log(r_{1}),\dots,\log(r_{n}),\theta_{1},\dots,\theta_{n})$, where $z_{j}=r_{j}e^{i\theta_{j}}$. Given a function $H(\xi)$ on $\mathfrak{t}$, we can extend $H$ trivially to $\mathfrak{t}\times T^{n}$ and pull back by Log to obtain a $T^{n}$-invariant function on $(\mathbb{C}^{*})^{n}$. Clearly, any $T^{n}$-invariant function on $(\mathbb{C}^{*})^{n}$ can be written in this form.

The Kähler metrics on the complex torus $(\mathbb{C}^{*})^{n}$ itself with respect to which the standard $T^{n}$-action is Hamiltonian have a natural characterization due to Guillemin.

Suppose that $(M,J,\omega)$ admits an effective and holomorphic $(\mathbb{C}^{*})^{n}$-action and that $\omega$ is the Kähler form of a $T^{n}$-invariant compatible Kähler metric. In this context, Proposition 2.4 implies that if the $T^{n}$-action on $M$ is Hamiltonian with respect to $\omega$, then restriction of $\omega$ to the dense orbit is $\partial\bar{\partial}$-exact. As before, let $(z_{1},\dots,z_{n})$ denote the standard coordinates on $(\mathbb{C}^{*})^{n}$. Choose any branch of $\log$ and write $w=\log(z)$. Then clearly $w=\xi+i\theta$ (or, more precisely, there is a corresponding lift of $\theta$ to the universal cover with respect to which the equality holds), and so if $\phi$ is $T^{n}$-invariant and $\omega=2i\partial\bar{\partial}\phi$, we have that

In this setting, the metric $g$ corresponding to $\omega$ is given on $\mathfrak{t}\times T^{n}$ by

The moment map $\mu$ as a map $\mu:\mathfrak{t}\times T^{n}\to\mathfrak{t}^{*}$ is defined by the relation

for all $b\in\mathfrak{t}$, and where $\nabla\phi$ is the Euclidean gradient of $\phi$. Since the Hessian of $\phi$ is positive-definite, it follows that $\phi$ is strictly convex on $\mathfrak{t}$. In particular, $\nabla\phi$ is a diffeomorphism onto its image. Using the identifications mentioned at the beginning of this section, we view $\nabla\phi$ as a map from $\mathfrak{t}$ into an open subset of $\mathfrak{t}^{*}$.

Suppose now that $(g,X)$ is a shrinking gradient Kähler-Ricci soliton on a complex toric manifold $M$ and that $g$ is $T^{n}$-invariant. Restricting to the dense orbit, we see that $g$ is determined by a convex function $\phi$ on $\mathfrak{t}$. We wish therefore to write equation (1.4) as an equation for $\phi$. From (1.4), we can assume by averaging that the soliton potential $f$, and therefore the vector field $X$, must also be $T^{n}$-invariant. Writing $f=f(\xi,\theta)$ in the real coordinate system $(\xi,\theta)$ above, it follows that $f$ is independent of $\theta$. Therefore we have that

In fact, the coefficients $\phi^{ij}\frac{\partial f}{\partial\xi^{i}}$ must be constant. Indeed, let $w=\log(z)$ as above, where $z$ is the standard coordinate on $(\mathbb{C}^{*})^{n}$, so that $w=\xi+i\theta$. In these coordinates we can write

where the coefficients $\phi^{ij}\frac{\partial f}{\partial\xi^{i}}$ depend only on the real part $\xi$ of $w$. Since $X$ is holomorphic, it follows that

In particular, it follows that $JX\in\mathfrak{t}$. We will denote the coefficients $\phi^{ij}\frac{\partial f}{\partial\xi^{j}}=b_{X}^{i}$, so that $JX=b_{X}^{i}\frac{\partial}{\partial\theta^{i}}$ is determined by the constant $b_{X}\in\mathfrak{t}$.

On the dense orbit then, the term $\mathcal{L}_{X}\omega$ in (1.4) is given by

Hence, up to a constant, the soliton potential $f$ is given in real logarithmic coordinates on the dense orbit by

Since the Ricci form of $\omega$ is given by

we can succinctly rewrite (1.4) in terms of $\phi$ alone.

As we have seen, the metric $g$ depends only on the Hessian of $\phi$. Part of the content of Proposition 2.6 therefore is a normalization for the potential $\phi$, and we will make use of this later on.

This says that each vertex of a Delzant polyhedron, when translated to the origin, can be made to look locally like standard $\mathbb{R}^{n}_{+}$ via an element of $\text{GL}(n,\mathbb{Z})$. It follows from the definition that there is a well-defined normal fan $\Sigma$ associated to any Delzant polyhedron $P$. One only needs to check that the relevant cones are rational polyhedral cones. This can be shown by induction, for example, since any face of a Delzant polyhedron must itself be Delzant. Therefore, given any Delzant polyhedron $P$, there is an associated toric variety $\mathcal{M}_{P}=\mathcal{M}_{\Sigma}$. The condition on the vertices of $P$ is precisely what is required to ensure that $\mathcal{M}_{P}$ is smooth; see [15, Theorem 3.1.19] and the preceding statements there.

In Section 2.1, we encountered a purely algebraic construction which produced a toric variety, and therefore a complex toric manifold, $\mathcal{M}_{P}$ from the data of a Delzant polyhedron. We now introduce a different construction, this time coming from symplectic geometry, which will produce a symplectic toric manifold from the data of $P$. The idea is to construct a complex symplectic manifold $(M_{P},\omega_{P},J_{P})$ as a Kähler quotient of $\mathbb{C}^{N}$ by a subgroup $G_{\mathbb{C}}$ of the standard torus $(\mathbb{C}^{*})^{N}$. The next proposition is standard for compact symplectic toric manifolds, and in the more general setting of potentially singular and non-compact varieties it is essentially proved in [10, Lemma 2.1], and earlier in [7, Chapter VI, Proposition 3.1.1]. We could not find the precise statement that we use in the literature, and so we briefly outline the proof below.

In particular, given the data of a rational polyhedron $P$, we have two constructions, each associating to $P$ a toric geometric object in the appropriate category. These turn out, after making the relevant identifications, to be equivalent. Let $P$ be a rational polyhedron and $\mathcal{M}_{P}$ be the toric variety constructed in Section 2.1.

We omit the proof here, but this is essentially proven in [10, Lemma 2.1] (c.f. [7, Chapter VI, Proposition 3.2.1]). From the description of $M_{P}$ given there, one simply applies the main theorem in [14] to deduce the proposition.

In sum, given the data of a rational polyhedron $P$, we have two constructions, each associating to $P$ a toric geometric object in the appropriate category, and these constructions are compatible up to an appropriate identification. For the remainder of this section we work with a given Delzant polyhedron $P$ and denote $M=\mathcal{M}_{P}\cong M_{P}$. In particular, we have a canonical Kähler metric $\omega_{P}$ on $M$. The induced $T^{n}$-action is Hamiltonian by construction, so that by Proposition 2.4 there is a Kähler potential $\omega_{P}=2i\partial\bar{\partial}\phi_{P}$ on the dense orbit.

We move on to consider an arbitrary Kähler metric $\omega$ on $M$ with respect to which the $T^{n}$-action is Hamiltonian, not necessarily equal to $\omega_{P}$. We impose the additional assumption that the corresponding moment map $\mu$ also has image equal to $\overline{P}$. Recall from Proposition 2.4 that there then exists a potential $\phi$ on the dense orbit $(\mathbb{C}^{*})^{n}\subset M$. We introduce logarithmic coordinates $(\xi^{j},\theta^{j})$ as in the previous section so that the moment map $\mu$ is determined by the diffeomorphism $\nabla\phi:\mathfrak{t}\to P$. We can then use the moment map to introduce a change of coordinates $\nabla\phi=x$, and thereby view $(\mathbb{C}^{*})^{n}\cong P\times T^{n}$. In these coordinates the Kähler form $\omega$ is standard, i.e.

So the moment map $\mu=\nabla\phi$ induces a natural choice of Darboux coordinates, and for this reason $(x^{j},\theta^{j})$ are typically referred to as symplectic coordinates on $M$. This is only a real coordinate system, and hence the coefficients of the Kähler form do not determine those of the corresponding Riemannian metric. One can still determine the metric $g$ by introducing a smooth function $u$ on $P$ which is related to $\phi$ by the Legendre transform:

Then the metric $g$ is given by

Thus the metric structure is determined by the Hessian of the function $u$, and so by analogy with the complex case this function is sometimes called the symplectic potential for $g$. Although we will not use this here, it is worth noting that it is more natural to view the function $u$ as determining the complex structure $J$, from which the formula (2.7) for the metric is a consequence. The Legendre transform will be used heavily in the remainder of the paper, and so for convenience we collect some basic properties here. For references focusing on aspects most closely related to the situation here; see for example [8, 20, 30].

The third item can be understood to mean that the Euclidean Hessians $\nabla^{2}\phi$ and $\nabla^{2}u$ are inverse to each other, under the appropriate change of coordinates. In most situations, we will use the shorthand $\phi_{u}=L(u)$. One application that will be used throughout the paper is the following. The last item is often taken as the definition of the Legendre transform (the so called Legendre-Fenchel transform), as it can be used to define $L(\phi)$ for $\phi$ merely continuous. Henceforth we will take $(v)$ as the definition of $L(\phi)$ in any case where $\phi$ is not necessarily $C^{1}$.

The estimate (2.9) is an immediate corollary of Lemma 2.9. For smooth functions one can see (2.8) directly; since $0$ is in the domain of $u$, there is some $\xi$ such that $\nabla\phi_{u}(\xi)=0$. Then $\phi_{u}$ is a strictly convex function with a minimum, and hence must grow at least linearly. However in what follows we will need to make use of (2.9) even for smooth functions. Although the notation is suggestive of the situation where $\phi$ is the Kähler potential of a toric metric, it is worth noting, and will be used later on, that this is completely symmetric in $\phi$ and $u$. That is to say, if $0$ lies in the domain $\Omega^{\prime}$ of $\phi$, it follows that $u$ must also satisfy (2.8) (with respect to the coordinate $x$ in $\Omega$). Indeed the entirety of Lemma 2.10 is entirely symmetric in $u$ and $\phi$.

We collect some further elementary properties of the behavior of convex functions under the Legendre transform, all consequences of the properties laid out in Lemma 2.9. As we will see, these in turn give rise to interesting geometric consequences when interpreted in the context of Kähler geometry on complex toric manifolds.

Let $M$ be a complex toric manifold together with a Kähler metric $\omega$ with respect to which the real $T^{n}$ action is Hamiltonian, and let $\phi$ be a strictly convex function on the dense orbit $(\mathbb{C}^{*})^{n}\subset M$ such that $\omega=2i\partial\bar{\partial}\phi$. Let $\mu:M\to\mathfrak{t}^{*}$ denote the corresponding moment map, normalized so that $\langle\mu,b\rangle=\langle\nabla\phi,b\rangle$ on the dense orbit as in Section 2.2, and suppose that the image of $\mu$ is equal to a Delzant polyhedron $P$. Recall also from Section 2.2 that we fix a basis $X_{1},\dots,X_{n}$ for $\mathfrak{t}$. Then the action of $\text{GL}(n,\mathbb{Z})$ on $\phi$ corresponds simply to changing this basis by an automorphism of $(\mathbb{C}^{*})^{n}$. This will be useful to simplify calculations later on, since by the Delzant condition we can use this to assume that $P$ locally coincides with a translate of the positive orthant $\mathbb{R}^{n}_{+}$ near any vertex. The action of $\mathfrak{t}$ on $\phi$ given in $(ii)$ corresponds to composing the $(\mathbb{C}^{*})^{n}$-action on $M$ with an element of the form $e^{-b_{1}}\in(\mathbb{C}^{*})^{n}$. Notice that this is is always induced from the global automorphism $e^{-b_{1}}:M\to M$ of $M$. The $\mathfrak{t}^{*}$-action of $(iii)$ is most naturally viewed as a modification of the moment map $\mu$ by the action of $\mathfrak{t}^{*}$ on itself by translation.

Recall from Proposition 2.6 that we are interested in the case where $\phi$ is a solution to (2.5) on $\mathfrak{t}$. Since $\phi$ uniquely determines and is uniquely determined by its Legendre transform $u=L(\phi)$, we can once again make use of the properties laid out in Lemma 2.9 to rewrite (2.5) as a real Monge-Ampère equation for the convex function $u$, defined on the interior of the image of the moment map. We assume as above that this image is equal to a Delzant polyhedron $P$.

We return now to the canonical metric $\omega_{P}$ defined in Proposition 2.7. We have just seen that there is a corresponding symplectic potential $u_{P}$ on $P$. The main result of [30] is an explicit formula for $u_{P}$, only in terms of the data of $P$, in the case that $\overline{P}$ (and therefore $M$) is compact. This has been generalized in [10, Theorem 5.2] to (essentially) arbitrary polyhedra, the Delzant case included. Let $F_{i}$, $i=1,\dots,d$ denote the $(n-1)$-dimensional facets of $P$ with inward-pointing normal vector $\nu_{i}\in\Gamma$, normalized so that $\nu_{i}$ is the minimal generator of $\sigma_{i}=\mathbb{R}_{+}\cdot\nu_{i}$ in $\Gamma$. Let $\ell_{i}(x)=\langle\nu_{i},x\rangle$, so that $\overline{P}$ is defined by the system of inequalities $\ell_{i}(x)\geq-a_{i}$, $i=1,\dots,N$, $a_{i}\in\mathbb{R}$. Then from [10] we have the following explicit formula for $u_{P}$: