# On toric geometry and K-stability of Fano varieties

## Abstract

We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano -fold while in the other they are non-reduced near the closed point associated to the toric Fano -fold, Second, we study K-stability of the general members of two deformation families of smooth Fano -fold. by building degenerations to K-polystable toric Fano -folds -folds.

## 1. Introduction

In this paper, we present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. Working with toric varieties enables us to run many computations explicitly, and to analyse the local structure of some K-moduli spaces and stacks of Fano varieties. -dimensional

In the first part of the paper, we study two examples of K-polystable toric Fano with obstructed deformations. These define non-smooth points of both the K-moduli stack of K-semistable Fano -folds and of the K-moduli space of K-polystable Fano -folds In one case, the K-moduli stack has -folds. branches and the K-moduli space has branches. In the other case, the K-moduli space is a fat point. These are, to the best of our knowledge, the first examples of such behaviour.

Second, we establish the K-polystability of the general member of two families of smooth Fano by showing that they arise as smoothings of K-polystable toric Fano -folds explicitly. In one of these cases, K-semistability was not known, while in the other, our argument provides an alternative proof. -folds

We now state the main results of the paper and present its organisation.

### 1.1. Non-smooth K-moduli

An immediate consequence of Kodaira–Nakano vanishing is that deformations of smooth Fano varieties are unobstructed. It follows that moduli stacks of smooth Fano varieties are smooth. It is also known that deformations (i.e. those satisfying Kollár’s condition) of del Pezzo surfaces with cyclic quotient singularities are unobstructed -GorensteinReference 28, Proposition 3.1 Reference 2, Lemma 6. As in the smooth case, this implies that moduli of del Pezzo surfaces with cyclic quotient singularities are smooth Reference 48 (see also Proposition 2.3).

In dimension there are examples of Fano varieties with obstructed deformations and isolated (canonical) singularities ,Reference 33Reference 50Reference 51. Note however that Fano with terminal singularities have unobstructed deformations -foldsReference 46Reference 58.

In light of recent developments in the moduli theory of Fano varieties, it is natural to ask whether deformations of K-semistable or of K-polystable Fano are obstructed. For example, -foldsReference 40 have shown that the K-moduli stack of K-semistable cubic coincides with the GIT stack, and is therefore smooth. -folds

We show that the naive hope that deformations of K-polystable Fano would be unobstructed is not validated. -folds

In what follows, for and , denotes the moduli stack of families of K-semistable Fano varieties of dimension -Gorenstein with anticanonical degree and denotes its good moduli space (see §2.2).

The local structure of and of near are studied in detail in §3.4 (see Theorem 3.13); we show that the base of the miniversal deformation (Kuranishi family) of has irreducible components, and:

- •
one component parametrises deformations of to a smooth Fano in the deformation family -fold (Picard rank degree , and );

- •
one component parametrises deformations of to a smooth Fano in the deformation family -fold (Picard rank degree , );

- •
the remaining two components parametrise deformations of to smooth Fano in the deformation family -folds (Picard rank degree , ).

This implies that has branches at Two of these branches are identified when passing to the good moduli space, hence . has branches at .

A second example of a K-polystable toric Fano with canonical singularities and obstructed deformations gives: -fold

In general, this shows that K-moduli stacks and spaces can be both reducible and non-reduced.

### 1.2. K-(semi/poly)stability of smooth Fano by degenerations to toric varieties -folds

Recent works have shown that K-semistability is an open property Reference 12Reference 63 (see §2.2). In particular, if a smooth Fano is a general fibre in a -fold smoothing of a K-polystable Fano -Gorenstein then it is automatically K-semistable. In § -fold,5, we construct smoothings of two K-polystable toric Fano -Gorenstein and conclude that the general member of the deformation of each smoothing is K-semistable. Using the local structure of K-moduli described in -folds,Reference 6, we show:

### 1.3. Notation and conventions

All varieties, schemes and stacks considered in this paper are defined over A normal projective variety is .*Fano* if its anticanonical divisor is and ample. A -Cartier*del Pezzo surface* is a Fano variety. We only consider normal toric varieties. -dimensional

The symbol denotes the deformation family of smooth Fano of Picard rank -folds Fano index , and degree The symbol . denotes the entry in the Mori–Mukai list thReference 44Reference 45 of smooth Fano of Picard rank -folds with the exception of the case , where we place the , entry in Mori and Mukai’s rank-4 list in between the first and the second elements of that list. This reordering ensures that, for each th the sequence , , , …is in order of increasing degree. ,

If is a normal scheme of finite type over then , denotes the sheaf of Kähler differentials of over For . we write , for the space -vector and , for the coherent -module The . versions of these are described in § -Gorenstein2.1.

If is a topological space and is an integer, then denotes the singular cohomology group of th with coefficients in and denotes the Betti number of th i.e. the dimension of , The topological Euler–Poincaré characteristic of a topological space . is denoted by .

If is a scheme of finite type over when considering topological properties we always consider the analytic topology on the set of the , of -points .

## 2. Preliminaries

In this section, we collect some results that will be used throughout the paper.

### 2.1. deformations -Gorenstein

An important insight, originally due to Kollár and Shepherd-Barron Reference 38, is that one should only consider deformations when studying the moduli theory of higher dimensional singular varieties. Roughly speaking, -Gorenstein families are flat families for which the canonical classes of fibres fit together well. More formally, if -Gorenstein is then a -Gorenstein, deformation is one that is induced by a deformation of the canonical cover stack of -Gorenstein (see Reference 1Reference 11). The canonical cover is a Deligne–Mumford stack with coarse moduli space and such that is an isomorphism over the Gorenstein locus of If . is Gorenstein, then and any deformation is a deformation. We denote by -Gorenstein (resp. the ) Ext group (resp. the pushforward to th of the Ext sheaf) of the cotangent complex of There is a spectral sequence .

As usual, is the tangent space and is an obstruction space for the deformation functor -Gorenstein of .

### 2.2. K-stability of Fano varieties

The notion of K-stability was introduced by Tian Reference 60 in an attempt to characterise the existence of Kähler–Einstein metrics on Fano manifolds; it was later reformulated in purely algebraic terms in Reference 26. We will not define the notions of K-semistability, K-polystability or K-stability here; we refer the reader to the survey Reference 62 and to the references therein. The notion of K-stability has received significant interest from algebraic geometers in recent years, as it has become clear that it provides the right framework to construct well behaved moduli stacks and spaces for Fano varieties Reference 5Reference 12Reference 39Reference 48.

Many proofs of K-semistability of Fano varieties rely on the following:

In particular if a K-polystable toric Fano -Gorenstein admits a -fold smoothing to a Fano -Gorenstein in a given deformation family, then the general member of that deformation family is K-semistable. -fold

For every integer and every rational number let , denote the category fibred in groupoids over the category of defined as follows: for every -schemes -scheme , is the groupoid of flat proper finitely presented families with base -Gorenstein of K-semistable klt Fano varieties of dimension and anticanonical volume .

The notion of good moduli space is defined in Reference 4. The stack is called the K-moduli stack, and the algebraic space is called the K-moduli space.

Now we describe the local structure of K-moduli explicitly. Let be a K-polystable Fano variety of dimension and anticanonical degree Let . be the noetherian complete local with residue field -algebra which is the hull of the functor of deformations of -Gorenstein i.e. the formal spectrum of , is the base of the miniversal deformation of -Gorenstein Let . be the automorphism group of then ; is reductive by Reference 5. The group acts on and the Luna étale slice theorem for algebraic stacks Reference 6 gives in this case a cartesian square

where the horizontal arrows are formally étale and map the closed point into the point corresponding to .

In the case of del Pezzo surfaces, this description of K-moduli implies the following properties of the moduli stack and space.

### 2.3. Topology of smoothings

We recall the formalism of vanishing and nearby cycles and show how they relate the topology of the central fibre to that of a smoothing.

Let be a complex analytic space and a flat projective morphism to a complex disc. Denote by the central fibre and set for There is a diagram of spaces and maps: .

where is chosen so that restricts to a topologically trivial fibration where , Let . be the tube about the fibre , the universal covering map, and let be the map making the right hand square of the diagram Cartesian. The nearby cycle complex of with is: -coefficients

where

As

and the vanishing cycle complex is

For any

The vanishing cycles complex of

associated to the cone of the specialisation map

The long exact sequence of hypercohomology associated to Equation 2.1 is:

and since the fibres of

### 2.4. Toric varieties

Toric varieties are a useful source of examples because they have an explicit combinatorial description; a comprehensive reference is Reference 23. Here, we recall some of the terms and notions we will use. Given a lattice

Many geometric properties of

We briefly recall the construction of polarised projective toric varieties Reference 23, Chapters 4 and 6 (see also Reference 49, Lemma 2.3). Let *normal fan* of

where

One may construct an ample

where the

Now we present the combinatorial avatars of toric Fano varieties; more details on these notions can be found in Reference 23, §8.3 and Reference 37.

If *face fan* (or *spanning fan*) of *Fano polytope*, i.e.

If *polar* is the rational polytope

The normal fan of

The polytope *reflexive* if the vertices of

The K-polystability of Gorenstein toric Fano varieties is easy to check combinatorially.

Lastly, as we are interested in K-moduli and hence in automorphism groups, we mention a result determining the automorphism group of a toric Fano variety:

The semidirect product structure is given by the embedding

## 3. An obstructed K-polystable toric Fano -fold

In this section, we construct a K-polystable toric Fano

### 3.1. Definition and first properties of

Let