Volume and Stability of singularities

Definition Examples and applications Results toward the main conjecture

Volume and Stability of singularities Chenyang Xu Beijing International Center of Mathematics Research

New York, 2017 August

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Local vs global: KLT singularities correspond to Fano varieties. From global to local: If X is a Fano variety, the cone C = C(X , −rKX ) is KLT; From local to global: Given a klt singularity x ∈ X , there exists f : Y → X such that Ex(f ) = f −1 (x) is an irreducible divisor S. Furthermore, if we write (KY + S)|S = KS + D(= KS orb ), then (S, D) is log Fano. This gives a degeneration of X to an orbifold cone C(S, −(KS + D)). Such an S is called a Kollár component. Usually there are more than one choices of Kollár components. How canonical can it be? Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Local vs global: KLT singularities correspond to Fano varieties. From global to local: If X is a Fano variety, the cone C = C(X , −rKX ) is KLT; From local to global: Given a klt singularity x ∈ X , there exists f : Y → X such that Ex(f ) = f −1 (x) is an irreducible divisor S. Furthermore, if we write (KY + S)|S = KS + D(= KS orb ), then (S, D) is log Fano. This gives a degeneration of X to an orbifold cone C(S, −(KS + D)). Such an S is called a Kollár component. Usually there are more than one choices of Kollár components. How canonical can it be? Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Local vs global: KLT singularities correspond to Fano varieties. From global to local: If X is a Fano variety, the cone C = C(X , −rKX ) is KLT; From local to global: Given a klt singularity x ∈ X , there exists f : Y → X such that Ex(f ) = f −1 (x) is an irreducible divisor S. Furthermore, if we write (KY + S)|S = KS + D(= KS orb ), then (S, D) is log Fano. This gives a degeneration of X to an orbifold cone C(S, −(KS + D)). Such an S is called a Kollár component. Usually there are more than one choices of Kollár components. How canonical can it be? Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Local vs global: KLT singularities correspond to Fano varieties. From global to local: If X is a Fano variety, the cone C = C(X , −rKX ) is KLT; From local to global: Given a klt singularity x ∈ X , there exists f : Y → X such that Ex(f ) = f −1 (x) is an irreducible divisor S. Furthermore, if we write (KY + S)|S = KS + D(= KS orb ), then (S, D) is log Fano. This gives a degeneration of X to an orbifold cone C(S, −(KS + D)). Such an S is called a Kollár component. Usually there are more than one choices of Kollár components. How canonical can it be? Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Local vs global: KLT singularities correspond to Fano varieties. From global to local: If X is a Fano variety, the cone C = C(X , −rKX ) is KLT; From local to global: Given a klt singularity x ∈ X , there exists f : Y → X such that Ex(f ) = f −1 (x) is an irreducible divisor S. Furthermore, if we write (KY + S)|S = KS + D(= KS orb ), then (S, D) is log Fano. This gives a degeneration of X to an orbifold cone C(S, −(KS + D)). Such an S is called a Kollár component. Usually there are more than one choices of Kollár components. How canonical can it be? Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Local vs global: KLT singularities correspond to Fano varieties. From global to local: If X is a Fano variety, the cone C = C(X , −rKX ) is KLT; From local to global: Given a klt singularity x ∈ X , there exists f : Y → X such that Ex(f ) = f −1 (x) is an irreducible divisor S. Furthermore, if we write (KY + S)|S = KS + D(= KS orb ), then (S, D) is log Fano. This gives a degeneration of X to an orbifold cone C(S, −(KS + D)). Such an S is called a Kollár component. Usually there are more than one choices of Kollár components. How canonical can it be? Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Definition of volumes

Fix x ∈ X = Spec(R), let ValX ,x denote all valuations of K (R) centered on x. (Ein-Lazarsfeld-Smith): For v ∈ ValX ,x , k) vol(v ) = limk →∞ length(R/a , where ak := {f |v (f ) ≥ k }. k n /n! ˇ Define log discrepancy A(v ) (Jonsson- Musta¸ta): extending A(ordE ). (Chi Li): Define the normalized volume of v to be c ) = An (v ) · vol(v ). Then define the volume vol(v c ). vol(x, X ) = infv ∈ValX ,x vol(v

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Definition of volumes

Fix x ∈ X = Spec(R), let ValX ,x denote all valuations of K (R) centered on x. (Ein-Lazarsfeld-Smith): For v ∈ ValX ,x , k) vol(v ) = limk →∞ length(R/a , where ak := {f |v (f ) ≥ k }. k n /n! ˇ Define log discrepancy A(v ) (Jonsson- Musta¸ta): extending A(ordE ). (Chi Li): Define the normalized volume of v to be c ) = An (v ) · vol(v ). Then define the volume vol(v c ). vol(x, X ) = infv ∈ValX ,x vol(v

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Definition of volumes

Fix x ∈ X = Spec(R), let ValX ,x denote all valuations of K (R) centered on x. (Ein-Lazarsfeld-Smith): For v ∈ ValX ,x , k) vol(v ) = limk →∞ length(R/a , where ak := {f |v (f ) ≥ k }. k n /n! ˇ Define log discrepancy A(v ) (Jonsson- Musta¸ta): extending A(ordE ). (Chi Li): Define the normalized volume of v to be c ) = An (v ) · vol(v ). Then define the volume vol(v c ). vol(x, X ) = infv ∈ValX ,x vol(v

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Definition of volumes

Fix x ∈ X = Spec(R), let ValX ,x denote all valuations of K (R) centered on x. (Ein-Lazarsfeld-Smith): For v ∈ ValX ,x , k) vol(v ) = limk →∞ length(R/a , where ak := {f |v (f ) ≥ k }. k n /n! ˇ Define log discrepancy A(v ) (Jonsson- Musta¸ta): extending A(ordE ). (Chi Li): Define the normalized volume of v to be c ) = An (v ) · vol(v ). Then define the volume vol(v c ). vol(x, X ) = infv ∈ValX ,x vol(v

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Definition of volumes

Fix x ∈ X = Spec(R), let ValX ,x denote all valuations of K (R) centered on x. (Ein-Lazarsfeld-Smith): For v ∈ ValX ,x , k) vol(v ) = limk →∞ length(R/a , where ak := {f |v (f ) ≥ k }. k n /n! ˇ Define log discrepancy A(v ) (Jonsson- Musta¸ta): extending A(ordE ). (Chi Li): Define the normalized volume of v to be c ) = An (v ) · vol(v ). Then define the volume vol(v c ). vol(x, X ) = infv ∈ValX ,x vol(v

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Stable degeneration conjecture c ). (Blum) There always exists a minimizer of vol(v Conjecture (Stable degeneration conjecture of KLT by Li, Li-X.) For a klt singularity x ∈ X , there is a unique minimizing valuation v (up to scaling) which is quasi-monomial (Abhyankar). Its associated graded ring grv (R) = ⊕k ∈R>0 ak /a>k is finitely generated, and gives the unique degeneration to a K-semistable Fano cone singularity.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Stable degeneration conjecture c ). (Blum) There always exists a minimizer of vol(v Conjecture (Stable degeneration conjecture of KLT by Li, Li-X.) For a klt singularity x ∈ X , there is a unique minimizing valuation v (up to scaling) which is quasi-monomial (Abhyankar). Its associated graded ring grv (R) = ⊕k ∈R>0 ak /a>k is finitely generated, and gives the unique degeneration to a K-semistable Fano cone singularity.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Stable degeneration conjecture c ). (Blum) There always exists a minimizer of vol(v Conjecture (Stable degeneration conjecture of KLT by Li, Li-X.) For a klt singularity x ∈ X , there is a unique minimizing valuation v (up to scaling) which is quasi-monomial (Abhyankar). Its associated graded ring grv (R) = ⊕k ∈R>0 ak /a>k is finitely generated, and gives the unique degeneration to a K-semistable Fano cone singularity.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Stable degeneration conjecture c ). (Blum) There always exists a minimizer of vol(v Conjecture (Stable degeneration conjecture of KLT by Li, Li-X.) For a klt singularity x ∈ X , there is a unique minimizing valuation v (up to scaling) which is quasi-monomial (Abhyankar). Its associated graded ring grv (R) = ⊕k ∈R>0 ak /a>k is finitely generated, and gives the unique degeneration to a K-semistable Fano cone singularity.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Fano cone singularity

Consider a torus T = (C∗ )r acting on a singularity X = Spec(R), then we can write R = ⊕α∈Γ Rα , where Γ ⊂ M = Hom(T , C∗ ). The Reeb cone (or weight cone) is t+ = {v ∈ NR := M ∗ ⊗ R|hv , αi > 0 ∀α, Rα 6= 0}. R0 = C, X is log terminal, if v ∈ NQ , then V = X /hC · v i is a Fano variety. In general, for any v ∈ t+ we call (X , v ) a Fano cone singularity. Sasakian, Martelli-Sparks-Yau, Collins-Szekélyhidi.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Fano cone singularity

Consider a torus T = (C∗ )r acting on a singularity X = Spec(R), then we can write R = ⊕α∈Γ Rα , where Γ ⊂ M = Hom(T , C∗ ). The Reeb cone (or weight cone) is t+ = {v ∈ NR := M ∗ ⊗ R|hv , αi > 0 ∀α, Rα 6= 0}. R0 = C, X is log terminal, if v ∈ NQ , then V = X /hC · v i is a Fano variety. In general, for any v ∈ t+ we call (X , v ) a Fano cone singularity. Sasakian, Martelli-Sparks-Yau, Collins-Szekélyhidi.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Fano cone singularity

Consider a torus T = (C∗ )r acting on a singularity X = Spec(R), then we can write R = ⊕α∈Γ Rα , where Γ ⊂ M = Hom(T , C∗ ). The Reeb cone (or weight cone) is t+ = {v ∈ NR := M ∗ ⊗ R|hv , αi > 0 ∀α, Rα 6= 0}. R0 = C, X is log terminal, if v ∈ NQ , then V = X /hC · v i is a Fano variety. In general, for any v ∈ t+ we call (X , v ) a Fano cone singularity. Sasakian, Martelli-Sparks-Yau, Collins-Szekélyhidi.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Fano cone singularity

Consider a torus T = (C∗ )r acting on a singularity X = Spec(R), then we can write R = ⊕α∈Γ Rα , where Γ ⊂ M = Hom(T , C∗ ). The Reeb cone (or weight cone) is t+ = {v ∈ NR := M ∗ ⊗ R|hv , αi > 0 ∀α, Rα 6= 0}. R0 = C, X is log terminal, if v ∈ NQ , then V = X /hC · v i is a Fano variety. In general, for any v ∈ t+ we call (X , v ) a Fano cone singularity. Sasakian, Martelli-Sparks-Yau, Collins-Szekélyhidi.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Fano cone singularity

Consider a torus T = (C∗ )r acting on a singularity X = Spec(R), then we can write R = ⊕α∈Γ Rα , where Γ ⊂ M = Hom(T , C∗ ). The Reeb cone (or weight cone) is t+ = {v ∈ NR := M ∗ ⊗ R|hv , αi > 0 ∀α, Rα 6= 0}. R0 = C, X is log terminal, if v ∈ NQ , then V = X /hC · v i is a Fano variety. In general, for any v ∈ t+ we call (X , v ) a Fano cone singularity. Sasakian, Martelli-Sparks-Yau, Collins-Szekélyhidi.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Log Fano cone singularity We can define special test configutations, generalized Futaki invariants, and K-(semi,poly)stability of (X , v ). If v ∈ NQ , this is the same as the notions of V = X /hC · v i. Fano cone singularties naturally appear on degenerations in the following way: If R0 = grv (R) is finitely generated, then there is a torus T action with dim(T ) equal to the rational rank of v by the grading. In fact, the valuative group h : Φ ∼ = M. A vector v0 on T is by given v0 (f ) = min{α|fα 6= 0}, i.e., hv , h−1 (α)i = α ∈ R. As v0 generates T , it is rational if and only if the rational rank of v is 1, this case is called quasi-regular. Otherwise, it’s called irregular. Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Log Fano cone singularity We can define special test configutations, generalized Futaki invariants, and K-(semi,poly)stability of (X , v ). If v ∈ NQ , this is the same as the notions of V = X /hC · v i. Fano cone singularties naturally appear on degenerations in the following way: If R0 = grv (R) is finitely generated, then there is a torus T action with dim(T ) equal to the rational rank of v by the grading. In fact, the valuative group h : Φ ∼ = M. A vector v0 on T is by given v0 (f ) = min{α|fα 6= 0}, i.e., hv , h−1 (α)i = α ∈ R. As v0 generates T , it is rational if and only if the rational rank of v is 1, this case is called quasi-regular. Otherwise, it’s called irregular. Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Log Fano cone singularity We can define special test configutations, generalized Futaki invariants, and K-(semi,poly)stability of (X , v ). If v ∈ NQ , this is the same as the notions of V = X /hC · v i. Fano cone singularties naturally appear on degenerations in the following way: If R0 = grv (R) is finitely generated, then there is a torus T action with dim(T ) equal to the rational rank of v by the grading. In fact, the valuative group h : Φ ∼ = M. A vector v0 on T is by given v0 (f ) = min{α|fα 6= 0}, i.e., hv , h−1 (α)i = α ∈ R. As v0 generates T , it is rational if and only if the rational rank of v is 1, this case is called quasi-regular. Otherwise, it’s called irregular. Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Log Fano cone singularity We can define special test configutations, generalized Futaki invariants, and K-(semi,poly)stability of (X , v ). If v ∈ NQ , this is the same as the notions of V = X /hC · v i. Fano cone singularties naturally appear on degenerations in the following way: If R0 = grv (R) is finitely generated, then there is a torus T action with dim(T ) equal to the rational rank of v by the grading. In fact, the valuative group h : Φ ∼ = M. A vector v0 on T is by given v0 (f ) = min{α|fα 6= 0}, i.e., hv , h−1 (α)i = α ∈ R. As v0 generates T , it is rational if and only if the rational rank of v is 1, this case is called quasi-regular. Otherwise, it’s called irregular. Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Background Normalized local volume

Log Fano cone singularity We can define special test configutations, generalized Futaki invariants, and K-(semi,poly)stability of (X , v ). If v ∈ NQ , this is the same as the notions of V = X /hC · v i. Fano cone singularties naturally appear on degenerations in the following way: If R0 = grv (R) is finitely generated, then there is a torus T action with dim(T ) equal to the rational rank of v by the grading. In fact, the valuative group h : Φ ∼ = M. A vector v0 on T is by given v0 (f ) = min{α|fα 6= 0}, i.e., hv , h−1 (α)i = α ∈ R. As v0 generates T , it is rational if and only if the rational rank of v is 1, this case is called quasi-regular. Otherwise, it’s called irregular. Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

(Y. Liu) vol(x, X ) = infa mult(a) · lctn (X ; a), where a runs over all m-primary ideals. (dFEM) For x ∈ Cn , infa mult(a) · lctn (X ; a) = nn . (Liu-X.) vol(x, X ) ≤ nn and ‘=’ implies smoothness. (Li-X.) For x ∈ Cn /G, then vol(x, X ) = nn /|G|. (Li, Li-Liu, Li-X.) Let V be a Fano variety. For x the vertex of the cone C(V , −rKV ), the valuation v obtained by blowing up x is a minimizer if and only if V is K-semistable. (Li, Li-Liu, Li-X) The minimizer of all ADE singularities in all dimensions, except 4-dimensional D4 , is given by some weighted blow up. (Blum) There exist examples for which the minimizers are given by non-divisorial valuations, e.g., a cone over F1 . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

(Y. Liu) vol(x, X ) = infa mult(a) · lctn (X ; a), where a runs over all m-primary ideals. (dFEM) For x ∈ Cn , infa mult(a) · lctn (X ; a) = nn . (Liu-X.) vol(x, X ) ≤ nn and ‘=’ implies smoothness. (Li-X.) For x ∈ Cn /G, then vol(x, X ) = nn /|G|. (Li, Li-Liu, Li-X.) Let V be a Fano variety. For x the vertex of the cone C(V , −rKV ), the valuation v obtained by blowing up x is a minimizer if and only if V is K-semistable. (Li, Li-Liu, Li-X) The minimizer of all ADE singularities in all dimensions, except 4-dimensional D4 , is given by some weighted blow up. (Blum) There exist examples for which the minimizers are given by non-divisorial valuations, e.g., a cone over F1 . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

(Y. Liu) vol(x, X ) = infa mult(a) · lctn (X ; a), where a runs over all m-primary ideals. (dFEM) For x ∈ Cn , infa mult(a) · lctn (X ; a) = nn . (Liu-X.) vol(x, X ) ≤ nn and ‘=’ implies smoothness. (Li-X.) For x ∈ Cn /G, then vol(x, X ) = nn /|G|. (Li, Li-Liu, Li-X.) Let V be a Fano variety. For x the vertex of the cone C(V , −rKV ), the valuation v obtained by blowing up x is a minimizer if and only if V is K-semistable. (Li, Li-Liu, Li-X) The minimizer of all ADE singularities in all dimensions, except 4-dimensional D4 , is given by some weighted blow up. (Blum) There exist examples for which the minimizers are given by non-divisorial valuations, e.g., a cone over F1 . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

(Y. Liu) vol(x, X ) = infa mult(a) · lctn (X ; a), where a runs over all m-primary ideals. (dFEM) For x ∈ Cn , infa mult(a) · lctn (X ; a) = nn . (Liu-X.) vol(x, X ) ≤ nn and ‘=’ implies smoothness. (Li-X.) For x ∈ Cn /G, then vol(x, X ) = nn /|G|. (Li, Li-Liu, Li-X.) Let V be a Fano variety. For x the vertex of the cone C(V , −rKV ), the valuation v obtained by blowing up x is a minimizer if and only if V is K-semistable. (Li, Li-Liu, Li-X) The minimizer of all ADE singularities in all dimensions, except 4-dimensional D4 , is given by some weighted blow up. (Blum) There exist examples for which the minimizers are given by non-divisorial valuations, e.g., a cone over F1 . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

(Y. Liu) vol(x, X ) = infa mult(a) · lctn (X ; a), where a runs over all m-primary ideals. (dFEM) For x ∈ Cn , infa mult(a) · lctn (X ; a) = nn . (Liu-X.) vol(x, X ) ≤ nn and ‘=’ implies smoothness. (Li-X.) For x ∈ Cn /G, then vol(x, X ) = nn /|G|. (Li, Li-Liu, Li-X.) Let V be a Fano variety. For x the vertex of the cone C(V , −rKV ), the valuation v obtained by blowing up x is a minimizer if and only if V is K-semistable. (Li, Li-Liu, Li-X) The minimizer of all ADE singularities in all dimensions, except 4-dimensional D4 , is given by some weighted blow up. (Blum) There exist examples for which the minimizers are given by non-divisorial valuations, e.g., a cone over F1 . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

(Y. Liu) vol(x, X ) = infa mult(a) · lctn (X ; a), where a runs over all m-primary ideals. (dFEM) For x ∈ Cn , infa mult(a) · lctn (X ; a) = nn . (Liu-X.) vol(x, X ) ≤ nn and ‘=’ implies smoothness. (Li-X.) For x ∈ Cn /G, then vol(x, X ) = nn /|G|. (Li, Li-Liu, Li-X.) Let V be a Fano variety. For x the vertex of the cone C(V , −rKV ), the valuation v obtained by blowing up x is a minimizer if and only if V is K-semistable. (Li, Li-Liu, Li-X) The minimizer of all ADE singularities in all dimensions, except 4-dimensional D4 , is given by some weighted blow up. (Blum) There exist examples for which the minimizers are given by non-divisorial valuations, e.g., a cone over F1 . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

(Y. Liu) vol(x, X ) = infa mult(a) · lctn (X ; a), where a runs over all m-primary ideals. (dFEM) For x ∈ Cn , infa mult(a) · lctn (X ; a) = nn . (Liu-X.) vol(x, X ) ≤ nn and ‘=’ implies smoothness. (Li-X.) For x ∈ Cn /G, then vol(x, X ) = nn /|G|. (Li, Li-Liu, Li-X.) Let V be a Fano variety. For x the vertex of the cone C(V , −rKV ), the valuation v obtained by blowing up x is a minimizer if and only if V is K-semistable. (Li, Li-Liu, Li-X) The minimizer of all ADE singularities in all dimensions, except 4-dimensional D4 , is given by some weighted blow up. (Blum) There exist examples for which the minimizers are given by non-divisorial valuations, e.g., a cone over F1 . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

Theorem (Liu-X.) If x ∈ X is a three dimensional klt singularity. 1

2

vol(x, X ) ≤ 16 if it is singular, and the equality holds only for rational double point. vol(x, X ) ≤ 9, if Picloc (x, X ) has a torsion, and x ∈ X is not a quotient singularity.

Corollary (Liu-X.) A cubic threefold X ⊂ P4 is GIT-(semi,poly)stable if and only if it is K-(semi,poly)stable. In particular, we have a precise list of cubic threefolds with KE metric, including all smooth ones. (Allcock) GIT-stability of cubic threefolds is classified. Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

Theorem (Liu-X.) If x ∈ X is a three dimensional klt singularity. 1

2

vol(x, X ) ≤ 16 if it is singular, and the equality holds only for rational double point. vol(x, X ) ≤ 9, if Picloc (x, X ) has a torsion, and x ∈ X is not a quotient singularity.

Corollary (Liu-X.) A cubic threefold X ⊂ P4 is GIT-(semi,poly)stable if and only if it is K-(semi,poly)stable. In particular, we have a precise list of cubic threefolds with KE metric, including all smooth ones. (Allcock) GIT-stability of cubic threefolds is classified. Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

Theorem (Liu-X.) If x ∈ X is a three dimensional klt singularity. 1

2

vol(x, X ) ≤ 16 if it is singular, and the equality holds only for rational double point. vol(x, X ) ≤ 9, if Picloc (x, X ) has a torsion, and x ∈ X is not a quotient singularity.

Corollary (Liu-X.) A cubic threefold X ⊂ P4 is GIT-(semi,poly)stable if and only if it is K-(semi,poly)stable. In particular, we have a precise list of cubic threefolds with KE metric, including all smooth ones. (Allcock) GIT-stability of cubic threefolds is classified. Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

Theorem (Liu-X.) If x ∈ X is a three dimensional klt singularity. 1

2

vol(x, X ) ≤ 16 if it is singular, and the equality holds only for rational double point. vol(x, X ) ≤ 9, if Picloc (x, X ) has a torsion, and x ∈ X is not a quotient singularity.

Corollary (Liu-X.) A cubic threefold X ⊂ P4 is GIT-(semi,poly)stable if and only if it is K-(semi,poly)stable. In particular, we have a precise list of cubic threefolds with KE metric, including all smooth ones. (Allcock) GIT-stability of cubic threefolds is classified. Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

The space of K-polystable smoothable Fano varieties is compact. KE cubic Xt degenerates to KE Q-Fano X∞ . (K. Fujita, Y. Liu) Connect global and local: if X is nn n K-semistable, for ∀x ∈ X , vol(x, X ) ≥ (n+1) n (−KX ) . Locally, x ∈ X∞ is smoothable with vol(x, X∞ ) ≥ 81/8. Theorem implies Picloc (x, X∞ ) is torsion free. Then −KX∞ = 2L and L is Cartier. (T. Fujita) Then X∞ ⊂ P4 . (Paul-Tian) K-stability of hypersurface implies GIT-stability.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

The space of K-polystable smoothable Fano varieties is compact. KE cubic Xt degenerates to KE Q-Fano X∞ . (K. Fujita, Y. Liu) Connect global and local: if X is nn n K-semistable, for ∀x ∈ X , vol(x, X ) ≥ (n+1) n (−KX ) . Locally, x ∈ X∞ is smoothable with vol(x, X∞ ) ≥ 81/8. Theorem implies Picloc (x, X∞ ) is torsion free. Then −KX∞ = 2L and L is Cartier. (T. Fujita) Then X∞ ⊂ P4 . (Paul-Tian) K-stability of hypersurface implies GIT-stability.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

The space of K-polystable smoothable Fano varieties is compact. KE cubic Xt degenerates to KE Q-Fano X∞ . (K. Fujita, Y. Liu) Connect global and local: if X is nn n K-semistable, for ∀x ∈ X , vol(x, X ) ≥ (n+1) n (−KX ) . Locally, x ∈ X∞ is smoothable with vol(x, X∞ ) ≥ 81/8. Theorem implies Picloc (x, X∞ ) is torsion free. Then −KX∞ = 2L and L is Cartier. (T. Fujita) Then X∞ ⊂ P4 . (Paul-Tian) K-stability of hypersurface implies GIT-stability.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

The space of K-polystable smoothable Fano varieties is compact. KE cubic Xt degenerates to KE Q-Fano X∞ . (K. Fujita, Y. Liu) Connect global and local: if X is nn n K-semistable, for ∀x ∈ X , vol(x, X ) ≥ (n+1) n (−KX ) . Locally, x ∈ X∞ is smoothable with vol(x, X∞ ) ≥ 81/8. Theorem implies Picloc (x, X∞ ) is torsion free. Then −KX∞ = 2L and L is Cartier. (T. Fujita) Then X∞ ⊂ P4 . (Paul-Tian) K-stability of hypersurface implies GIT-stability.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

The space of K-polystable smoothable Fano varieties is compact. KE cubic Xt degenerates to KE Q-Fano X∞ . (K. Fujita, Y. Liu) Connect global and local: if X is nn n K-semistable, for ∀x ∈ X , vol(x, X ) ≥ (n+1) n (−KX ) . Locally, x ∈ X∞ is smoothable with vol(x, X∞ ) ≥ 81/8. Theorem implies Picloc (x, X∞ ) is torsion free. Then −KX∞ = 2L and L is Cartier. (T. Fujita) Then X∞ ⊂ P4 . (Paul-Tian) K-stability of hypersurface implies GIT-stability.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

The space of K-polystable smoothable Fano varieties is compact. KE cubic Xt degenerates to KE Q-Fano X∞ . (K. Fujita, Y. Liu) Connect global and local: if X is nn n K-semistable, for ∀x ∈ X , vol(x, X ) ≥ (n+1) n (−KX ) . Locally, x ∈ X∞ is smoothable with vol(x, X∞ ) ≥ 81/8. Theorem implies Picloc (x, X∞ ) is torsion free. Then −KX∞ = 2L and L is Cartier. (T. Fujita) Then X∞ ⊂ P4 . (Paul-Tian) K-stability of hypersurface implies GIT-stability.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

The space of K-polystable smoothable Fano varieties is compact. KE cubic Xt degenerates to KE Q-Fano X∞ . (K. Fujita, Y. Liu) Connect global and local: if X is nn n K-semistable, for ∀x ∈ X , vol(x, X ) ≥ (n+1) n (−KX ) . Locally, x ∈ X∞ is smoothable with vol(x, X∞ ) ≥ 81/8. Theorem implies Picloc (x, X∞ ) is torsion free. Then −KX∞ = 2L and L is Cartier. (T. Fujita) Then X∞ ⊂ P4 . (Paul-Tian) K-stability of hypersurface implies GIT-stability.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Singularities in general dimension Dimension three and cubics

The space of K-polystable smoothable Fano varieties is compact. KE cubic Xt degenerates to KE Q-Fano X∞ . (K. Fujita, Y. Liu) Connect global and local: if X is nn n K-semistable, for ∀x ∈ X , vol(x, X ) ≥ (n+1) n (−KX ) . Locally, x ∈ X∞ is smoothable with vol(x, X∞ ) ≥ 81/8. Theorem implies Picloc (x, X∞ ) is torsion free. Then −KX∞ = 2L and L is Cartier. (T. Fujita) Then X∞ ⊂ P4 . (Paul-Tian) K-stability of hypersurface implies GIT-stability.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Theorem (Li-X.) Let x ∈ X be a klt singularity. 1

If v = ordS is a divisorial valuation which is a minimizer of c x,X , then S is a K-semistable Kollár component. vol c c Furthermore, vol(ord S ) < vol(ordT ) if S 6= T .

2

Conversely, if S is a K-semistable Kollár component, then ordS is a minimizer.

This answers the question on searching for ‘the canonical’ Kollár component for quasi-regular singularities.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Theorem (Li-X.) Let x ∈ X be a klt singularity. 1

If v = ordS is a divisorial valuation which is a minimizer of c x,X , then S is a K-semistable Kollár component. vol c c Furthermore, vol(ord S ) < vol(ordT ) if S 6= T .

2

Conversely, if S is a K-semistable Kollár component, then ordS is a minimizer.

This answers the question on searching for ‘the canonical’ Kollár component for quasi-regular singularities.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Theorem (Li-X.) Let x ∈ X be a klt singularity. 1

If v = ordS is a divisorial valuation which is a minimizer of c x,X , then S is a K-semistable Kollár component. vol c c Furthermore, vol(ord S ) < vol(ordT ) if S 6= T .

2

Conversely, if S is a K-semistable Kollár component, then ordS is a minimizer.

This answers the question on searching for ‘the canonical’ Kollár component for quasi-regular singularities.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Theorem (Li-X.) Let x ∈ X be a klt singularity. 1

If v = ordS is a divisorial valuation which is a minimizer of c x,X , then S is a K-semistable Kollár component. vol c c Furthermore, vol(ord S ) < vol(ordT ) if S 6= T .

2

Conversely, if S is a K-semistable Kollár component, then ordS is a minimizer.

This answers the question on searching for ‘the canonical’ Kollár component for quasi-regular singularities.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Theorem (Li-X.) Let x ∈ X be a klt singularity. Assume v is a quasi-monomial. c and gr is finitely generated, then 1 If v is a minimizer of vol v X0 = Spec(grv ) is klt and (X0 , v ) is a K-semistable Fano cone. Furthermore, v is the unique quasi-monomial minimizer up to rescaling. 2

Conversely, if (X0 = Spec(grv ), v ) is a K-semistable Fano cone, then v is a minimizer.

Corollary (Li-X.+ work in progress by Li-Wang-X.) Donaldson-Sun’s conjecture on the algebracity of the metric tangent cone holds.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Theorem (Li-X.) Let x ∈ X be a klt singularity. Assume v is a quasi-monomial. c and gr is finitely generated, then 1 If v is a minimizer of vol v X0 = Spec(grv ) is klt and (X0 , v ) is a K-semistable Fano cone. Furthermore, v is the unique quasi-monomial minimizer up to rescaling. 2

Conversely, if (X0 = Spec(grv ), v ) is a K-semistable Fano cone, then v is a minimizer.

Corollary (Li-X.+ work in progress by Li-Wang-X.) Donaldson-Sun’s conjecture on the algebracity of the metric tangent cone holds.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Theorem (Li-X.) Let x ∈ X be a klt singularity. Assume v is a quasi-monomial. c and gr is finitely generated, then 1 If v is a minimizer of vol v X0 = Spec(grv ) is klt and (X0 , v ) is a K-semistable Fano cone. Furthermore, v is the unique quasi-monomial minimizer up to rescaling. 2

Conversely, if (X0 = Spec(grv ), v ) is a K-semistable Fano cone, then v is a minimizer.

Corollary (Li-X.+ work in progress by Li-Wang-X.) Donaldson-Sun’s conjecture on the algebracity of the metric tangent cone holds.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Theorem (Li-X.) Let x ∈ X be a klt singularity. Assume v is a quasi-monomial. c and gr is finitely generated, then 1 If v is a minimizer of vol v X0 = Spec(grv ) is klt and (X0 , v ) is a K-semistable Fano cone. Furthermore, v is the unique quasi-monomial minimizer up to rescaling. 2

Conversely, if (X0 = Spec(grv ), v ) is a K-semistable Fano cone, then v is a minimizer.

Corollary (Li-X.+ work in progress by Li-Wang-X.) Donaldson-Sun’s conjecture on the algebracity of the metric tangent cone holds.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Theorem (Li-X.) Let x ∈ X be a klt singularity. Assume v is a quasi-monomial. c and gr is finitely generated, then 1 If v is a minimizer of vol v X0 = Spec(grv ) is klt and (X0 , v ) is a K-semistable Fano cone. Furthermore, v is the unique quasi-monomial minimizer up to rescaling. 2

Conversely, if (X0 = Spec(grv ), v ) is a K-semistable Fano cone, then v is a minimizer.

Corollary (Li-X.+ work in progress by Li-Wang-X.) Donaldson-Sun’s conjecture on the algebracity of the metric tangent cone holds.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Theorem (Li-X.) Let x ∈ X be a klt singularity. Assume v is a quasi-monomial. c and gr is finitely generated, then 1 If v is a minimizer of vol v X0 = Spec(grv ) is klt and (X0 , v ) is a K-semistable Fano cone. Furthermore, v is the unique quasi-monomial minimizer up to rescaling. 2

Conversely, if (X0 = Spec(grv ), v ) is a K-semistable Fano cone, then v is a minimizer.

Corollary (Li-X.+ work in progress by Li-Wang-X.) Donaldson-Sun’s conjecture on the algebracity of the metric tangent cone holds.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit

A family of Kähler-Einstein Fano manifolds Xt . Donaldson-Sun (DS), Tian showed a GH limit metric space X∞ of Xti is a Fano variety with klt singularities. For a singularity x ∈ X∞ , it is important to known its local metric structure captured by its metric tangent cone C. C is defined to be the metric limit of a sequence d limri →0 (X∞ ∩ Bri (x), Xr∞ ). i C admits a Ricci-flat cone metric. Remark: vol(x, X∞ ) =

nn π n /n!

Chenyang Xu

· limr →0

vol(Br ,x ) . r 2n

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit

A family of Kähler-Einstein Fano manifolds Xt . Donaldson-Sun (DS), Tian showed a GH limit metric space X∞ of Xti is a Fano variety with klt singularities. For a singularity x ∈ X∞ , it is important to known its local metric structure captured by its metric tangent cone C. C is defined to be the metric limit of a sequence d limri →0 (X∞ ∩ Bri (x), Xr∞ ). i C admits a Ricci-flat cone metric. Remark: vol(x, X∞ ) =

nn π n /n!

Chenyang Xu

· limr →0

vol(Br ,x ) . r 2n

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit

A family of Kähler-Einstein Fano manifolds Xt . Donaldson-Sun (DS), Tian showed a GH limit metric space X∞ of Xti is a Fano variety with klt singularities. For a singularity x ∈ X∞ , it is important to known its local metric structure captured by its metric tangent cone C. C is defined to be the metric limit of a sequence d limri →0 (X∞ ∩ Bri (x), Xr∞ ). i C admits a Ricci-flat cone metric. Remark: vol(x, X∞ ) =

nn π n /n!

Chenyang Xu

· limr →0

vol(Br ,x ) . r 2n

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit

A family of Kähler-Einstein Fano manifolds Xt . Donaldson-Sun (DS), Tian showed a GH limit metric space X∞ of Xti is a Fano variety with klt singularities. For a singularity x ∈ X∞ , it is important to known its local metric structure captured by its metric tangent cone C. C is defined to be the metric limit of a sequence d limri →0 (X∞ ∩ Bri (x), Xr∞ ). i C admits a Ricci-flat cone metric. Remark: vol(x, X∞ ) =

nn π n /n!

Chenyang Xu

· limr →0

vol(Br ,x ) . r 2n

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit

A family of Kähler-Einstein Fano manifolds Xt . Donaldson-Sun (DS), Tian showed a GH limit metric space X∞ of Xti is a Fano variety with klt singularities. For a singularity x ∈ X∞ , it is important to known its local metric structure captured by its metric tangent cone C. C is defined to be the metric limit of a sequence d limri →0 (X∞ ∩ Bri (x), Xr∞ ). i C admits a Ricci-flat cone metric. Remark: vol(x, X∞ ) =

nn π n /n!

Chenyang Xu

· limr →0

vol(Br ,x ) . r 2n

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit

A family of Kähler-Einstein Fano manifolds Xt . Donaldson-Sun (DS), Tian showed a GH limit metric space X∞ of Xti is a Fano variety with klt singularities. For a singularity x ∈ X∞ , it is important to known its local metric structure captured by its metric tangent cone C. C is defined to be the metric limit of a sequence d limri →0 (X∞ ∩ Bri (x), Xr∞ ). i C admits a Ricci-flat cone metric. Remark: vol(x, X∞ ) =

nn π n /n!

Chenyang Xu

· limr →0

vol(Br ,x ) . r 2n

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit

A family of Kähler-Einstein Fano manifolds Xt . Donaldson-Sun (DS), Tian showed a GH limit metric space X∞ of Xti is a Fano variety with klt singularities. For a singularity x ∈ X∞ , it is important to known its local metric structure captured by its metric tangent cone C. C is defined to be the metric limit of a sequence d limri →0 (X∞ ∩ Bri (x), Xr∞ ). i C admits a Ricci-flat cone metric. Remark: vol(x, X∞ ) =

nn π n /n!

Chenyang Xu

· limr →0

vol(Br ,x ) . r 2n

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit DS has a two-step construction of C: Step 1, x ∈ X∞ degenerates to a K-semistable cone W , which is induced by a valuation (dictated by the metric). Step 2, W degenerates to C induced by a valuation on W . W and C are algebraic. DS conjecture both of them only depend on the algebraic structure of x ∈ X∞ . One can show W = (grv (R), v0 ) is K-semistable. So v is a minimizer in our program, which is quasi-monomial. Our theorem says such v is unique. In work in progress (Li-Wang-X.), we show any K-semistable cone has a unique K-polystable cone degeneration, and a cone with Ricci-flat cone metric is K-polystable. Thus C is uniquely determined by W . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit DS has a two-step construction of C: Step 1, x ∈ X∞ degenerates to a K-semistable cone W , which is induced by a valuation (dictated by the metric). Step 2, W degenerates to C induced by a valuation on W . W and C are algebraic. DS conjecture both of them only depend on the algebraic structure of x ∈ X∞ . One can show W = (grv (R), v0 ) is K-semistable. So v is a minimizer in our program, which is quasi-monomial. Our theorem says such v is unique. In work in progress (Li-Wang-X.), we show any K-semistable cone has a unique K-polystable cone degeneration, and a cone with Ricci-flat cone metric is K-polystable. Thus C is uniquely determined by W . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit DS has a two-step construction of C: Step 1, x ∈ X∞ degenerates to a K-semistable cone W , which is induced by a valuation (dictated by the metric). Step 2, W degenerates to C induced by a valuation on W . W and C are algebraic. DS conjecture both of them only depend on the algebraic structure of x ∈ X∞ . One can show W = (grv (R), v0 ) is K-semistable. So v is a minimizer in our program, which is quasi-monomial. Our theorem says such v is unique. In work in progress (Li-Wang-X.), we show any K-semistable cone has a unique K-polystable cone degeneration, and a cone with Ricci-flat cone metric is K-polystable. Thus C is uniquely determined by W . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit DS has a two-step construction of C: Step 1, x ∈ X∞ degenerates to a K-semistable cone W , which is induced by a valuation (dictated by the metric). Step 2, W degenerates to C induced by a valuation on W . W and C are algebraic. DS conjecture both of them only depend on the algebraic structure of x ∈ X∞ . One can show W = (grv (R), v0 ) is K-semistable. So v is a minimizer in our program, which is quasi-monomial. Our theorem says such v is unique. In work in progress (Li-Wang-X.), we show any K-semistable cone has a unique K-polystable cone degeneration, and a cone with Ricci-flat cone metric is K-polystable. Thus C is uniquely determined by W . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit DS has a two-step construction of C: Step 1, x ∈ X∞ degenerates to a K-semistable cone W , which is induced by a valuation (dictated by the metric). Step 2, W degenerates to C induced by a valuation on W . W and C are algebraic. DS conjecture both of them only depend on the algebraic structure of x ∈ X∞ . One can show W = (grv (R), v0 ) is K-semistable. So v is a minimizer in our program, which is quasi-monomial. Our theorem says such v is unique. In work in progress (Li-Wang-X.), we show any K-semistable cone has a unique K-polystable cone degeneration, and a cone with Ricci-flat cone metric is K-polystable. Thus C is uniquely determined by W . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit DS has a two-step construction of C: Step 1, x ∈ X∞ degenerates to a K-semistable cone W , which is induced by a valuation (dictated by the metric). Step 2, W degenerates to C induced by a valuation on W . W and C are algebraic. DS conjecture both of them only depend on the algebraic structure of x ∈ X∞ . One can show W = (grv (R), v0 ) is K-semistable. So v is a minimizer in our program, which is quasi-monomial. Our theorem says such v is unique. In work in progress (Li-Wang-X.), we show any K-semistable cone has a unique K-polystable cone degeneration, and a cone with Ricci-flat cone metric is K-polystable. Thus C is uniquely determined by W . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit DS has a two-step construction of C: Step 1, x ∈ X∞ degenerates to a K-semistable cone W , which is induced by a valuation (dictated by the metric). Step 2, W degenerates to C induced by a valuation on W . W and C are algebraic. DS conjecture both of them only depend on the algebraic structure of x ∈ X∞ . One can show W = (grv (R), v0 ) is K-semistable. So v is a minimizer in our program, which is quasi-monomial. Our theorem says such v is unique. In work in progress (Li-Wang-X.), we show any K-semistable cone has a unique K-polystable cone degeneration, and a cone with Ricci-flat cone metric is K-polystable. Thus C is uniquely determined by W . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit DS has a two-step construction of C: Step 1, x ∈ X∞ degenerates to a K-semistable cone W , which is induced by a valuation (dictated by the metric). Step 2, W degenerates to C induced by a valuation on W . W and C are algebraic. DS conjecture both of them only depend on the algebraic structure of x ∈ X∞ . One can show W = (grv (R), v0 ) is K-semistable. So v is a minimizer in our program, which is quasi-monomial. Our theorem says such v is unique. In work in progress (Li-Wang-X.), we show any K-semistable cone has a unique K-polystable cone degeneration, and a cone with Ricci-flat cone metric is K-polystable. Thus C is uniquely determined by W . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit DS has a two-step construction of C: Step 1, x ∈ X∞ degenerates to a K-semistable cone W , which is induced by a valuation (dictated by the metric). Step 2, W degenerates to C induced by a valuation on W . W and C are algebraic. DS conjecture both of them only depend on the algebraic structure of x ∈ X∞ . One can show W = (grv (R), v0 ) is K-semistable. So v is a minimizer in our program, which is quasi-monomial. Our theorem says such v is unique. In work in progress (Li-Wang-X.), we show any K-semistable cone has a unique K-polystable cone degeneration, and a cone with Ricci-flat cone metric is K-polystable. Thus C is uniquely determined by W . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Metric tangent cone on GH limit DS has a two-step construction of C: Step 1, x ∈ X∞ degenerates to a K-semistable cone W , which is induced by a valuation (dictated by the metric). Step 2, W degenerates to C induced by a valuation on W . W and C are algebraic. DS conjecture both of them only depend on the algebraic structure of x ∈ X∞ . One can show W = (grv (R), v0 ) is K-semistable. So v is a minimizer in our program, which is quasi-monomial. Our theorem says such v is unique. In work in progress (Li-Wang-X.), we show any K-semistable cone has a unique K-polystable cone degeneration, and a cone with Ricci-flat cone metric is K-polystable. Thus C is uniquely determined by W . Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Volumes of the models P Define the volume of a model vol(Y /X ) = vol( i A(Ei )Ei ), ∗ (O(−kD))) . where vol(D) = limk →∞ length(R/f k n /n! If Ex(Y /X ) is an irreducible divisor S, then c vol(Y /X ) = vol(ord S ). P A(Ei )Ei ∼X ,Q KY + i Ei , so MMP decreases the volumes of models. Another characterization: vol(x, X ) = inf vol(Y /X ) = inf vol(ordS ) for all Kollár components S.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Volumes of the models P Define the volume of a model vol(Y /X ) = vol( i A(Ei )Ei ), ∗ (O(−kD))) . where vol(D) = limk →∞ length(R/f k n /n! If Ex(Y /X ) is an irreducible divisor S, then c vol(Y /X ) = vol(ord S ). P A(Ei )Ei ∼X ,Q KY + i Ei , so MMP decreases the volumes of models. Another characterization: vol(x, X ) = inf vol(Y /X ) = inf vol(ordS ) for all Kollár components S.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Volumes of the models P Define the volume of a model vol(Y /X ) = vol( i A(Ei )Ei ), ∗ (O(−kD))) . where vol(D) = limk →∞ length(R/f k n /n! If Ex(Y /X ) is an irreducible divisor S, then c vol(Y /X ) = vol(ord S ). P A(Ei )Ei ∼X ,Q KY + i Ei , so MMP decreases the volumes of models. Another characterization: vol(x, X ) = inf vol(Y /X ) = inf vol(ordS ) for all Kollár components S.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Volumes of the models P Define the volume of a model vol(Y /X ) = vol( i A(Ei )Ei ), ∗ (O(−kD))) . where vol(D) = limk →∞ length(R/f k n /n! If Ex(Y /X ) is an irreducible divisor S, then c vol(Y /X ) = vol(ord S ). P A(Ei )Ei ∼X ,Q KY + i Ei , so MMP decreases the volumes of models. Another characterization: vol(x, X ) = inf vol(Y /X ) = inf vol(ordS ) for all Kollár components S.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Degeneration is a K-semistable Fano cone

If the minimizer v = ordS , X0 = C(S, ∆S ). In higher rank, using diophantine approximation of v , construct a sequence of log canonical models (Wi , Ei ) over x ∈ X , such that v ∈ DMR(Wi+1 , Ei+1 ) ⊂ DMR(Wi , Ei ). Then v can be approximated by Kollár components Si such that vSi ∈ DMR(Wi , Ei ). Finite generation assumption =⇒ grv (R) = grv 0 (R), for a small rational perturbation v 0 of v . Show (X0 , v0 ) is K-semistable.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Degeneration is a K-semistable Fano cone

If the minimizer v = ordS , X0 = C(S, ∆S ). In higher rank, using diophantine approximation of v , construct a sequence of log canonical models (Wi , Ei ) over x ∈ X , such that v ∈ DMR(Wi+1 , Ei+1 ) ⊂ DMR(Wi , Ei ). Then v can be approximated by Kollár components Si such that vSi ∈ DMR(Wi , Ei ). Finite generation assumption =⇒ grv (R) = grv 0 (R), for a small rational perturbation v 0 of v . Show (X0 , v0 ) is K-semistable.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Degeneration is a K-semistable Fano cone

If the minimizer v = ordS , X0 = C(S, ∆S ). In higher rank, using diophantine approximation of v , construct a sequence of log canonical models (Wi , Ei ) over x ∈ X , such that v ∈ DMR(Wi+1 , Ei+1 ) ⊂ DMR(Wi , Ei ). Then v can be approximated by Kollár components Si such that vSi ∈ DMR(Wi , Ei ). Finite generation assumption =⇒ grv (R) = grv 0 (R), for a small rational perturbation v 0 of v . Show (X0 , v0 ) is K-semistable.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Degeneration is a K-semistable Fano cone

If the minimizer v = ordS , X0 = C(S, ∆S ). In higher rank, using diophantine approximation of v , construct a sequence of log canonical models (Wi , Ei ) over x ∈ X , such that v ∈ DMR(Wi+1 , Ei+1 ) ⊂ DMR(Wi , Ei ). Then v can be approximated by Kollár components Si such that vSi ∈ DMR(Wi , Ei ). Finite generation assumption =⇒ grv (R) = grv 0 (R), for a small rational perturbation v 0 of v . Show (X0 , v0 ) is K-semistable.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Degeneration is a K-semistable Fano cone

If the minimizer v = ordS , X0 = C(S, ∆S ). In higher rank, using diophantine approximation of v , construct a sequence of log canonical models (Wi , Ei ) over x ∈ X , such that v ∈ DMR(Wi+1 , Ei+1 ) ⊂ DMR(Wi , Ei ). Then v can be approximated by Kollár components Si such that vSi ∈ DMR(Wi , Ei ). Finite generation assumption =⇒ grv (R) = grv 0 (R), for a small rational perturbation v 0 of v . Show (X0 , v0 ) is K-semistable.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Uniqueness for Fano cone

On (X0 , v0 ), if it is K-semistable, then v0 is the unique minimizer among all quasi-monomial valuations up to rescaling. A Newton-Okounkov body construction for the valuation translates the volume of valuations to the volume of polytopes. Connect v and any quasi-monomial valuations, use the strict convexity of the volume of the affine hyperplane slice of a cone passing through a fixed vector.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Uniqueness for Fano cone

On (X0 , v0 ), if it is K-semistable, then v0 is the unique minimizer among all quasi-monomial valuations up to rescaling. A Newton-Okounkov body construction for the valuation translates the volume of valuations to the volume of polytopes. Connect v and any quasi-monomial valuations, use the strict convexity of the volume of the affine hyperplane slice of a cone passing through a fixed vector.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Uniqueness for Fano cone

On (X0 , v0 ), if it is K-semistable, then v0 is the unique minimizer among all quasi-monomial valuations up to rescaling. A Newton-Okounkov body construction for the valuation translates the volume of valuations to the volume of polytopes. Connect v and any quasi-monomial valuations, use the strict convexity of the volume of the affine hyperplane slice of a cone passing through a fixed vector.

Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Uniqueness in general We can not degenerate valuations, but there is a degeneration of ideals (or graded sequence of ideals): a → in(a). This preserves the colength, and log canonical thresholds stays the same. For a degeneration of ideal sequence associated to another minimizer w, since the normalized multiplicities do not change, the log canonical thresholds stay the same, MMP gives a degeneration of the model Vi → X (for w) to Vi0 → X0 . Using the model Vi0 with its exceptional divisors, one can define a quasi-monomial valuation w0 as the degeneration of w. Then vol(w) = vol(w 0 ) =⇒ w(f ) = w0 (in(f )). Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Uniqueness in general We can not degenerate valuations, but there is a degeneration of ideals (or graded sequence of ideals): a → in(a). This preserves the colength, and log canonical thresholds stays the same. For a degeneration of ideal sequence associated to another minimizer w, since the normalized multiplicities do not change, the log canonical thresholds stay the same, MMP gives a degeneration of the model Vi → X (for w) to Vi0 → X0 . Using the model Vi0 with its exceptional divisors, one can define a quasi-monomial valuation w0 as the degeneration of w. Then vol(w) = vol(w 0 ) =⇒ w(f ) = w0 (in(f )). Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Uniqueness in general We can not degenerate valuations, but there is a degeneration of ideals (or graded sequence of ideals): a → in(a). This preserves the colength, and log canonical thresholds stays the same. For a degeneration of ideal sequence associated to another minimizer w, since the normalized multiplicities do not change, the log canonical thresholds stay the same, MMP gives a degeneration of the model Vi → X (for w) to Vi0 → X0 . Using the model Vi0 with its exceptional divisors, one can define a quasi-monomial valuation w0 as the degeneration of w. Then vol(w) = vol(w 0 ) =⇒ w(f ) = w0 (in(f )). Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Uniqueness in general We can not degenerate valuations, but there is a degeneration of ideals (or graded sequence of ideals): a → in(a). This preserves the colength, and log canonical thresholds stays the same. For a degeneration of ideal sequence associated to another minimizer w, since the normalized multiplicities do not change, the log canonical thresholds stay the same, MMP gives a degeneration of the model Vi → X (for w) to Vi0 → X0 . Using the model Vi0 with its exceptional divisors, one can define a quasi-monomial valuation w0 as the degeneration of w. Then vol(w) = vol(w 0 ) =⇒ w(f ) = w0 (in(f )). Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Uniqueness in general We can not degenerate valuations, but there is a degeneration of ideals (or graded sequence of ideals): a → in(a). This preserves the colength, and log canonical thresholds stays the same. For a degeneration of ideal sequence associated to another minimizer w, since the normalized multiplicities do not change, the log canonical thresholds stay the same, MMP gives a degeneration of the model Vi → X (for w) to Vi0 → X0 . Using the model Vi0 with its exceptional divisors, one can define a quasi-monomial valuation w0 as the degeneration of w. Then vol(w) = vol(w 0 ) =⇒ w(f ) = w0 (in(f )). Chenyang Xu

The Dual Complex

Definition Examples and applications Results toward the main conjecture

Main results Donaldson-Sun’s conjecture on the metric tangent cone Some ideas in the proof

Thank you very much!

Chenyang Xu

The Dual Complex