K-stability of Casagrande–Druel varieties
-
Ivan Cheltsov
,
Tiago Duarte Guerreiro
,
Kento Fujita
,
Igor Krylov
and
Jesus Martinez-Garcia
Abstract
We introduce a new subclass of Fano varieties (Casagrande–Druel varieties) that are 𝑛-dimensional varieties constructed from Fano double covers of dimension
Throughout this paper, all varieties are defined over ℂ.
1 Introduction
Let 𝑉 be a Fano variety with Kawamata log terminal singularities, and let 𝐿 be a line bundle on 𝑉 such that the divisor
Using these isomorphisms, fix sections
We allow 𝑅 to be singular, so 𝐵 can be very singular (and even reducible).
However, if the log pair
Let
Moreover, the variety 𝑋 is also a Fano variety (see Section 2).
If the Fano variety 𝑋 has at most Kawamata log terminal singularities, then 𝑋 is called the Casagrande–Druel variety constructed from
The group
and 𝜄 swaps the proper transforms of the sections
and the composition 𝜃 is a 𝐺-equivariant conic bundle such that 𝐺 acts trivially on 𝑉.
Our construction of Casagrande–Druel varieties is inspired by the paper [12]. See [12, Lemma 3.1 (iii)]. But it goes back to the construction of de Jonquieres involutions using hyperelliptic curves instead of Fano double covers. See also [33, 11, 43, 20].
The del Pezzo surface of degree 6 (blow up of
Let
Let
Let
All smooth Casagrande–Druel threefolds are K-polystable; see [27, Theorem 6.1] and [7]. In fact, K-polystable Casagrande–Druel varieties exist in every dimension.
Example 1.7 ([16, 17])
Suppose that
In this paper, we prove the following theorem.
Suppose that
We obtain this result as an application of the following K-polystability criteria.
Suppose that both 𝑉 and 𝑅 are smooth (or equivalently 𝑋 is smooth), and
and set
where
In the notation of Theorem 1.9, if
In this paper, we also prove the following two theorems about K-polystability of several singular Casagrande–Druel 3-folds.
Suppose
Suppose
a singular reduced curve in
To present their applications, let
Let
where
Let
where
If 𝐵 is the smooth del Pezzo surface from Examples 1.4, 1.5, 1.6, then 𝐵 is K-polystable. If 𝐵 is the Fano manifold from Theorem 1.8, then 𝐵 is K-polystable [19, Theorem 1.1]. If 𝐵 is the singular del Pezzo surface from Theorems 1.11 and 1.12 such that 𝑅 is reduced, then 𝐵 is also K-polystable [35]. Inspired by this, we pose the following conjecture.
If 𝑉 and 𝐵 are K-polystable Fano varieties, then 𝑋 is K-polystable.
If 𝐵 is a K-polystable Fano variety, the log Fano pair
Theorem 1.16 ([32])
Suppose that
where
The K-polystability of 𝑉 in Conjecture 1.15 is necessary.
Example 1.17 (Yuchen Liu)
Let
so that the singular del Pezzo surface 𝐵 is K-polystable, but
Let us say few words about the proofs of Theorems 1.9 and 1.12.
In Section 2, we will show that
where 𝜌 is the quotient map, which is a double cover ramified over our divisor
In Section 3, we will prove the following result, which implies Theorem 1.9.
Suppose that 𝑉 and 𝑅 are smooth (so 𝐵 is smooth), and
where
Proof of Theorem 1.9
Indeed, notice that the right-hand side of the inequality in Theorem 1.18 is precisely 𝛾 as defined in Theorem 1.9.
By assumption,
We refer the reader to the excellent survey [46] for an overview on K-stability and to [7, 22] for extensive applications of the celebrated Abban–Zhuang theory introduced in [2]. In these applications (especially in Sections 4 and 5, we will make extensive use of Zhuang’s result [49] that equivariant K-polystability for reductive groups implies K-polystability. We will also make frequent use of the result in [31] to determine K-stability of branched covers.
Let us describe the structure of this paper.
First, in Section 2, we will prove a few basic properties of Casagrande–Druel varieties.
Then, in Section 3, we will prove Theorem 1.18.
In Sections 4 and 5, we will give proofs of Theorem 1.11 and Theorem 1.12, respectively.
Finally, in Section 6, we will prove Corollary 1.13, and we will show that
2 Preliminaries
Let 𝑉 be a (possibly non-projective) variety, let
Now, let
Using these isomorphisms, fix sections
that correspond to the section
For
Now, we set
so that we have the commutative Cartesian diagram
where
Set
and
Therefore, there is a natural closed embedding
where we identified
Let us identify 𝑋 with its image in 𝑄.
Set
Then
where we use isomorphisms
Set
This is an elementary transformation of the
Let
Using these isomorphisms, fix sections
which correspond to the section
so that we can identify 𝑋 with a Cartier divisor on 𝑈 such that
Suppose that 𝑉 is normal and projective, and
Proof
The normality of the variety 𝑋 follows from Remark 2.1 and [25, Proposition 5.24]. Similarly, using notation introduced in Remark 2.1, we see that
so
To prove the remaining assertion, suppose that
Moreover, applying the adjunction formula to the sections
where we used
This also shows that both divisors
Hence, using Kleiman’s criterion again, we see that
From now on, we assume, in addition, that 𝑉 is normal and projective.
Suppose
Remark 2.4 ([21, Lemma 9.8])
Suppose that 𝑉 is a smooth Fano variety, and assume
where
Therefore, if
From now on, we also assume that
let
fix
There is a double cover
Proof
Let
Using these isomorphisms, fix sections
which corresponds to
Now, we consider the homomorphism
defined by the composition of
and the surjection
obtained by the tensor product of the pullbacks of the following natural surjections:
Then (2.2) is surjective.
This gives the morphism
where we identified
Using the local criterion for flatness, we see that 𝜌 is flat.
Further, 𝜌 is finite of degree 2.
Now, using [23, I (6.11)] and [23, I (6.12)], we see that the morphism 𝜌 is branched over the divisor
Let
we get an isomorphism
Set
which implies the remaining assertions of the proposition. ∎
Let
Suppose that 𝑉 is smooth, 𝐿 is nef, 𝑋 has Kawamata log terminal singularities, and
Proof
By Remark 2.1, 𝑋 can be embedded into
has a semi-universal formal element in the sense of [42, Definition 2.2.6], where 𝒜 is the category of local ℂ-algebras with the residue field ℂ.
Thus, by [41, Proposition 2.4] and [41, Proposition 2.6], the deformations of 𝑋 are unobstructed if
Let us show that
Since
is ample, and
are ample. On the other hand, using the exact sequence of sheaves
we get the following exact sequence:
Moreover, using the Kawamata–Viehweg vanishing theorem, we get
Furthermore, using the exact sequence of sheaves
we get the exact sequence
Since both
This gives
3 K-polystability criteria
The goal of this section is to prove Theorem 1.18. To do so, we will apply the theory of Abban–Zhuang [2], as applied in [7, §1.7] and [22], consisting on bounding delta-invariants below by picking a specific flag.
Fix a positive integer
Unlike Section 1, we do not assume that 𝑉 is a Fano variety.
Fix a positive rational number
Lemma 3.2 (cf. [48])
Let 𝑃 be a point in
where
Proof
where
Indeed, take
Thus, since
Using
Let 𝐹 be any prime divisor over
This gives
which implies the first part of the assertion.
We now assume
This inequality is equivalent to
We must show that the right-hand side of the inequality above is at least 𝑎. But
which is clearly true. ∎
Now, fix a smooth divisor
Let 𝑃 be a point in
where
This result together with Lemma 3.2 implies Theorem 1.18.
Proof of Theorem 1.18.
Note that 𝑉 is a Fano variety and
which gives
so that
where
In the remaining part of the section, we will prove Proposition 3.3 by induction on 𝑛.
3.1 Base of induction
Let 𝑉 be a smooth projective surface, let 𝐿 be an ample Cartier divisor on 𝑉, let 𝜇 be the smallest rational number such that
and let
Since
Moreover, if
Suppose, in addition, that
Let 𝑃 be a point in 𝑌 such that
One has
In the remaining part of this subsection, we will prove this result.
We will only consider the case
Let
where we use isomorphism
and
where
Note that
Let 𝑓 be the fiber of the
where
For instance, if ℎ is the ordinary blow up at the point 𝑃, then
Thus, to prove Proposition 3.5, it is enough to bound each term in (3.1) by
We set
We denote by
Suppose 𝐵 intersects 𝑓 transversally.
Then
Proof
Let
We have the following intersection numbers:
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This shows that
If
and the negative part is
Similarly, if
and the negative part is
Now, using results from [7, Section 1.7], we compute
Moreover, we have
Let 𝑄 be a point in 𝐸. Then, using results from [7, Section 1.7], we compute
where
because
only when
and
Recall from Remark 3.4 that
so we can discard the first term in (3.2).
Moreover, since
which gives
Now, we deal with the case when 𝑓 is tangent to 𝐵 at the point 𝑃.
Suppose 𝐵 and 𝑓 are tangent at 𝑃.
Then
Proof
Now, we let
Moreover, we have the following intersection numbers:
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𝐸 |
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| 𝐸 | 0 | 1 |
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Thus, the divisor
If
and the negative part is
Similarly, if
and the negative part is
Now, using results from [7, Section 1.7], we compute
Moreover, since
Let 𝑄 be a point in 𝐸. Using results from [7, Section 1.7], we get
where
There are three cases to consider.
and
Now, arguing as in the end of the proof of Lemma 3.6, we find
and the result follows. ∎
3.2 The induction
Let us use all assumptions and notation introduced in Section 3.
Recall that 𝜇 is the smallest rational number for which
Therefore, we suppose that Proposition 3.3 holds for varieties of dimension
where
Let
For simplicity, set
so
and
Recall that
Now, we compute
Set
One has
Proof
The equality follows from the formulas for
which gives
Set
for every prime divisor 𝐸 over the variety
Suppose that
and
Hence, combining, we obtain
Let us compute these two integrals separately. We have
and
Adding these two integrals, we get
This gives
3.3 Applications
The only application of Theorem 1.9 we could find is Theorem 1.8.
Let us use assumptions and notation of Theorem 1.9.
Let
One has
Proof
One has
Thus, it is enough to show that
Substituting
which holds since
One has
Proof
The inequality is equivalent to
Substituting
One has
Proof
We have
Substituting
Theorem 1.8 follows from Lemmas 3.9, 3.10, 3.11 and Theorem 1.9.
4 Proof of Theorem 1.11
The goal of this section is to prove Theorem 1.11 and describe singular K-polystable limits of smooth Fano 3-folds in the deformation family № 4.2. We start with the following (probably well-known) result, which we fail to find in the literature.
Let 𝐶 be a
GIT stable for
GIT strictly polystable if and only if it is one of the curves in Theorem 1.11.
Proof
Choose homogeneous coordinates
Observe that any one-parameter subgroup
for some integers
Clearly, if
Suppose that 𝐶 is singular at the point
Conversely, suppose there exists a one-parameter subgroup 𝜆 for which
Now, let us describe the unstable locus.
Suppose
which is negative if and only if
This describes all possibilities for the curve 𝐶 to be GIT-semistable, which easily implies the description of GIT-polystable
Now, we set
let
Finally, we set
Suppose that
The barycenter of the moment polytope is the origin, so 𝑋 is K-polystable. See also [24].
Our next step is the following simple lemma.
Suppose
Proof
Here, the morphism 𝜙 is a weighted blow up at the intersection
Set
such that
𝛼 is the blow up along the intersection curves
𝜓 contracts the proper transform of the surface
Indeed, we have that
Observe that
We let
Now, we take
and
Hence, we have
which implies that 𝑋 is K-polystable. ∎
To complete the proof of Theorem 1.11, let us present 𝑋 as a codimension two complete intersection in a toric variety.
Let
and 𝐼 is the irrelevant ideal
Then the projection
Let
and the irrelevant ideal
Then 𝜑 induces the blow up of 𝑌 along 𝑅.
Thus, we can identify 𝑋 with the complete intersection in the toric variety
Now, the subgroup
Note that 𝜄 is not canonically defined, since we can conjugate it with an element in Γ.
Suppose that
where
We can see from the equation
It follows that
where
Let 𝐺 be the subgroup in
The following assertions hold:
𝑋 does not contain 𝐺-fixed points,
𝑋 does not contain 𝐺-invariant irreducible curves,
𝑋 contains two 𝐺-invariant irreducible surfaces – they are cut out by
Proof
Left to the reader. ∎
Now, we can complete the proof of Theorem 1.11.
Suppose that 𝑋 is not K-polystable.
Using [49], we see that there is a 𝐺-invariant prime divisor 𝐅 over 𝑋 such that
5 Proof of Theorem 1.12
In this section, we prove Theorem 1.12. This result describes all singular K-polystable limits of smooth Fano 3-folds in the family № 3.9. To show this, we need the following theorem.
Theorem 5.1 ([26, Theorem 2], [34, Example 7.13], [3])
Let 𝐶 be a quartic curve in
GIT stable for
GIT strictly polystable if and only if it is one of the remaining curves in Theorem 1.12.
Let us prove Theorem 1.12.
Set
a reduced quartic curve with at most
Suppose that
Proof
In this case, the threefold 𝑋 has non-isolated singularities along a smooth curve, and the proof is very similar to the proof of Lemma 4.3. Namely, we have
and there exists exactly one
As in the proof of Lemma 4.3, we set
such that
𝛼 is a blow up along the curves
𝜓 is a contraction of the proper transform of
Let
We let
Take
This shows that
Hence, we have
which implies that 𝑋 is K-polystable. ∎
Similarly, we can show that 𝑋 is K-polystable if
However, this approach fails in the case when 𝑅 has a singular point of type
where 𝜌 is the quotient map.
Moreover, we also proved that the double cover 𝜌 is ramified over a divisor
However, everything in life comes with a price: the action of the group
To explicitly describe
with irrelevant ideal
where
In the remaining part of the section, we will prove that the pair
We will split the proof in several lemmas and propositions. We start with the following lemma.
Let 𝑃 be a point in
Proof
Let us apply Lemma 3.2. We have
where
But
so that
Similarly, applying Proposition 3.5, we obtain the following lemma.
Let 𝑃 be a point 𝑌 such that
Proof
By Lemma 5.3, we may assume that
where
The two most difficult parts of the proof that
Let 𝑃 be a point in 𝐵 such that 𝐵 has singular point of type
Let 𝑃 be a point in 𝐵 such that 𝐵 has singular point of type
By Lemmas 5.3 and 5.4 and Propositions 5.5 and 5.6, the log pair
where one of the following three cases holds:
Suppose that
Proof
Suppose
Suppose
Proof
The surface 𝐵 has a singular point of type
Proof
In this case, we have
We already know from Lemma 5.2 that the threefold 𝑋 is K-polystable in this case, so that
Let
such that 𝛼 is a blow up along the curve
Note that the commutative diagram above is
Let 𝐹 be 𝛼-exceptional surface, let
First, we compute
Let 𝑢 be a non-negative real number. Then
and this divisor is pseudo-effective if and only if
and
This gives
Suppose that
Let
because
where
Let us compute
which is easy to compute, because
Let
and
Thus, we see that
which is a contradiction. ∎
In the remaining part of this sections, we will prove Proposition 5.5 and 5.6.
5.1 Proof of Proposition 5.5
Let us use notation introduced earlier in this section before Proposition 5.5, and let 𝑃 be an isolated ordinary double point of the surface 𝐵.
Then, up to a change of coordinates, we may assume that
Let
with irrelevant ideal
To describe its fan, denote the vector generating the ray corresponding to
The cone structure can be derived from the irrelevant ideal
Let
Let
And for pullback, we have
Divisors
Moreover, we have
In particular, a divisor 𝐷 with
Intersections of divisors
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Proof
Recall that, for distinct torus-invariant divisors
This fact together with the linear equivalences implies the required assertion. ∎
Using Lemma 5.10, we obtain the following intersection table:
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1 | −1 | 1 |
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−1 | 1 | 0 |
Now, we set
Then
The divisor
where the torus action is the same (given by the matrix 𝑀) and the irrelevant ideal
which is obtained from
Abusing our previous notation, we denote the divisor
Moreover, intersections of divisors on
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Using these intersections, we obtain the following intersection table:
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The proper transform on
where
One has
Proof
The equality
Thus, we compute
as claimed. ∎
Now, we construct a common toric resolution
Set
Let
The Zariski decomposition of the divisor
and
where
Let
and
We are ready to apply [2, 7, 22].
Set
where the infimum is taken over all prime divisors 𝐸 over
Let 𝐅 be a prime divisor over 𝑌 such that
It follows from [22, Theorem 4.8] and [22, Corollary 4.9] that
Suppose
Let 𝑍 be the center of the divisor 𝐸 on the surface
One has
Proof
Note that
For every
Observe that
Moreover, we have
Furthermore, we have
which gives
Integrating, we get
Suppose that
where
Integrating, we get
Thus, we see that 𝑍 is disjoint from 𝐞. In particular, we see that
for every
Let
the map 𝜃 is an ordinary blow up in the case when
the map 𝜃 is a weighted blow up at the point
If 𝜃 is a usual blow up, then
where 𝐨 is the singular point of the surface
Now, for
For every
by (5.4) and [22, Corollary 4.18], where the infimum is taken by all points
for
Denote by
Suppose that 𝜃 is an ordinary blow up. Let 𝑄 be a point in 𝐶. Then
Proof
One has
This easily implies that
so that
which gives
Thus, integrating, we get
This gives the first required inequality. Similarly, we compute
Observe that
Moreover, if
which implies the second required inequality. ∎
Thus, it follows from (5.5) and Lemma 5.13
that
This gives
and
Then
Now, integrating, we get
so it follows from (5.5) that there is a point
where
Recall that
Thus, summarizing, we get
In particular, we see that
5.2 Proof of Proposition 5.6
Let us use notation introduced earlier in this section before Proposition 5.6, and let 𝑃 be a singular point of type
Let
where the irrelevant ideal is
To describe the fan of the toric threefold
and the cone structure can be visualized with the following diagram:
Let
Intersections of divisors
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This gives the following intersection table:
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Now, we set
Observe that
Abusing our previous notation, we denote the divisor
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Thus, we see that the proper transform on
where the torus action is again given by the matrix 𝑀 and the irrelevant ideal
Then the fan of
We abuse our notation again and denote the divisor
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The proper transform on
while the positive part is
Integrating, we get
Next we construct a partial common toric resolution for
Set
Then we have the following toric diagram:
where toric maps can be described as follows:
| Map | Center | Weights | Exceptional divisor | Relation |
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Here,
Now, we set
Using this, we describe the Zariski decomposition of the divisor
and
where
Now, we describe
where the torus action is given by the matrix
and the irrelevant ideal
To obtain a similar description of the surface
Using this, we see that
and
The fan of the toric surface
with obvious cone structure.
Note that we can also recover this structure by noticing that
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Further, we compute
and
Let
and 𝜃 is a composition of the ordinary blow up at the point
Let
This curve is smooth, it does not contain the singular point of
Note that the surface
Then
and
Now, we are ready to apply [2, 7, 22].
Let 𝑄 be a point in
For a real number
Write
where
for
where we consider 𝑄 as a point in
We will choose 𝐶 such that the pair
and the pair
Let 𝐅 be a prime divisor over 𝑌 such that
Suppose
where the choice of 𝐶 in the infimum depends on 𝑄. Thus, since
we have
In fact, since
it follows from [22, Corollary 4.18] and [2, Theorem 3.3] that we have a strict inequality
Let us use this to obtain a contradiction, which would finish the proof of Proposition 5.6.
Namely, we will show that, for every point
if
if
if
Proof
Note that
Moreover, we have
and
which gives
and
Integrating, we get
so (5.6) holds.
Similarly, we compute
where
Observe that
Thus, we have
which implies (5.7). ∎
Proof
For
We compute
and
which gives
and
Thus, integrating we get
so (5.6) holds.
Since
Thus, we have
which implies (5.6). ∎
Let 𝑄 be a point in
Proof
Note that
For every
and
which gives
and
Thus, integrating, we get
so (5.6) holds.
Since
Integrating, we get
which implies (5.6). ∎
Lemmas 5.14, 5.14, 5.16 complete the proof of Proposition 5.6.
6 On the K-moduli spaces
In this section, we prove Corollary 1.13. The proof of Corollary 1.14 is almost identical, so we omit it. To start with, let us present the following well-known assertion.
Let 𝑋 be a smooth Fano threefold. Then
where
Proof
The required assertion immediately follows from the Akizuki–Nakano vanishing theorem and the Hirzebruch–Riemann–Roch theorem, since
Now, let us use notation and assumptions introduced in Corollary 1.13.
Let
Proof
There exists a one-parameter subgroup
is a GIT polystable point in 𝑇.
Let
Now, we are ready to prove Corollary 1.13.
Proof of Corollary 1.13
Since the construction of Casagrande–Druel 3-folds is functorial, there exists a Γ-equivariant flat morphism
Let
Since 𝑀 is the good moduli space of
Observe that 𝑀 is normal.
Take
by Lemma 6.1, since
The variety
and
Proposition 6.3 (Noam Elkies)
One has
Proof
Let
Then it follows from [39, §11.9] or [18, §4.6] that
with
Let us find generators of
of the space 𝑊, let
such that
Since the polynomials
In fact, Proposition 6.3 is a classical result – Peano [38] and Turnbull [44] showed that
The surface
Funding source: Engineering and Physical Sciences Research Council
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Funding source: Japan Society for the Promotion of Science
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Funding source: Institute for Basic Science
Award Identifier / Grant number: IBS-R003-D1
Funding statement: Ivan Cheltsov was supported by EPSRC grant EP/V054597/1, Tiago Duarte Guerreiro was supported by EPSRC grant EP/V055399/1, Kento Fujita was supported by JSPS KAKENHI Grant Number 22K03269, Igor Krylov was supported by IBS-R003-D1 grant, and Jesus Martinez-Garcia was supported by EPSRC grant EP/V055399/1.
Acknowledgements
This paper was written during our visit to the Gökova Geometry Topology Institute in April 2023. We are very grateful to the institute for its hospitality. We would like to thank Yuchen Liu for his help with the proof of Corollary 1.13, and we would like to thank Noam Elkies for his proof of Proposition 6.3.
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© 2024 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Frontmatter
- Geometric main conjectures in function fields
- Noncompact self-shrinkers for mean curvature flow with arbitrary genus
- K-stability of Casagrande–Druel varieties
- Geometry at infinity of the space of Kähler potentials and asymptotic properties of filtrations
- The Nash problem for torus actions of complexity one
- Min-max theory for capillary surfaces
- Floer homology and right-veering monodromy
- The fourth moment of the Hurwitz zeta function
Articles in the same Issue
- Frontmatter
- Geometric main conjectures in function fields
- Noncompact self-shrinkers for mean curvature flow with arbitrary genus
- K-stability of Casagrande–Druel varieties
- Geometry at infinity of the space of Kähler potentials and asymptotic properties of filtrations
- The Nash problem for torus actions of complexity one
- Min-max theory for capillary surfaces
- Floer homology and right-veering monodromy
- The fourth moment of the Hurwitz zeta function
