Section 59

Comparing a GIT-stable filling with a K-polystable filling

Part IX · Family 2.16: conics on intersections of two quadricsPDF page 211

We recall precisely the two K-moduli inputs. A general smooth member of Family 2.16 is K-stable; this is already known from [3, 28]. Properness of K-moduli gives a K-polystable filling, after finite base change, for a punctured family of K-stable members [21]. In this family, [34, Proposition A.13] reconstructs that filling over the base, not just on its closed fibre, as the blow-up of a relative intersection of two quadrics along a relative plane conic. Choosing projective frames consequently gives a morphism to $W$ .

The other input is CM-degree minimization: among $\mathbb Q$ -Gorenstein Fano fillings of a fixed punctured family, a filling with K-semistable closed fibre minimizes the CM degree. We use its local version over a discrete valuation ring. It follows from the filtration comparison in [35, Remark 7.9]; the local difference is the same as the degree difference after gluing identical families away from the marked point.

Definition 59.1. Let $R$ be a discrete valuation ring with parameter $t$ and fraction field $K$ . For two line bundles $M,M'$ on $\operatorname{Spec}R$ with an identification of their generic fibres, define the relative degree $\deg_R(M'-M)$ as follows. If $e,e'$ are generators and the generic identification sends $e$ to $t^r v e'$ , where $v\in R^*$ , put $\deg_R(M'-M)=r$ . This does not depend on the generators. It is the degree of the divisor of the rational section induced by the generic identification. For rational line bundles one first takes a common integral multiple and divides by that multiple.

Lemma 59.2. Let $G$ be reductive, let $W$ be projective with an ample $G$ -linearized line bundle $\mathcal L$ , and let $w,w':\operatorname{Spec}R\to W$ have $G(K)$ -equivalent generic fibres. Suppose that $w(0)$ is GIT semistable. Then $\deg_R(w'^*\mathcal L-w^*\mathcal L)\ge0.$ If equality holds, $w'(0)$ is also semistable and the two closed points have the same image in $W/\!/_{\mathcal L}G$ . If $w(0)$ is stable, they belong to the same $G$ -orbit.

Proof. Choose an invariant section $s\in H^0(W,\mathcal L^{\otimes m})^G$ which is nonzero at $w(0)$ . Its pullback along $w$ is a unit times a generator. Invariance identifies it generically with $w'^*s$ . The latter is a regular section, so its vanishing order is nonnegative. By the definition of relative degree this order is $m\deg_R(w'^*\mathcal L-w^*\mathcal L)$ .

If the relative degree is zero, $s(w'(0))\ne0$ , proving semistability. Both morphisms now lie in the semistable locus, and their images in the separated projective GIT quotient agree on the generic point. Hence they agree on the closed point. Each fibre of the quotient contains a unique closed orbit. If it contains a stable point, that orbit has dimension $\dim G$ , and no strictly larger orbit can have it in its closure. Thus the whole fibre is that stable orbit. ◻

Lemma 59.3. Let $w\in W^\mathrm{sm}$ be GIT stable for the CM polarization $\mathcal L$ . Suppose a curve through $w$ has K-stable general blow-up and its K-polystable filling admits the relative presentation of [34, Proposition A.13]. Then the blow-up represented by $w$ is K-polystable.

Proof. Pass to a finite cover of the curve and then its completed local discrete valuation ring. Denote the original smooth blow-up family by $\mathcal X$ and a K-polystable filling by $\mathcal X'$ . The relative presentation and a choice of frames give morphisms $w_R,w'_R:\operatorname{Spec}R\to W$ . Their generic fibres differ by an element of $\operatorname{PGL}(6,K)$ : the contraction is determined by the pullback of the hyperplane class, and on a smooth member it is the unique birational extremal contraction. After another finite extension this projective transformation lifts to $\operatorname{SL}(6,K)$ . Such base changes multiply all relative degrees by the same positive integer.

The canonical generic identification of the two CM lines agrees with that induced by these frames and the linearization in Lemma 57.3. Lemma 59.2 therefore gives $\deg_R(\lambda_{\mathrm{CM},\mathcal X'}- \lambda_{\mathrm{CM},\mathcal X})\ge0.$ Since the closed fibre of $\mathcal X'$ is K-polystable, CM-degree minimization gives the reverse inequality. Equality follows. Lemma 59.2 now shows that $w'_R(0)$ belongs to the same orbit as $w_R(0)=w$ . Their blow-ups are isomorphic. Hence the original closed fibre is K-polystable. ◻

Remark 59.4. Only the two actual Fano fillings enter this comparison. An arbitrary one-parameter subgroup may have a limit in $W$ whose blow-up is not a Fano variety. We neither use that limit as a test configuration nor deduce the converse between GIT and K-stability by identifying its weight with a Donaldson–Futaki invariant. Instead, a nonvanishing invariant section gives one inequality for relative CM degrees, and CM minimization gives the other. Thus no classification of the entire GIT boundary, and no unproved global GIT–K-moduli comparison, is required.