Log K-stability of GIT-stable divisors on Fano manifolds

Zhou, Changchun

doi:10.48550/arxiv.2102.07458

Cited by 2 publications

(4 citation statements)

References 29 publications

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“…That is, by varying the coefficient c, the K-moduli spaces M K c provide a natural interpolation between the GIT quotient for quartic surfaces and the Baily-Borel compactification, and explicitly resolve the period map. We note that a special case of Theorem 1.1(1) was proved earlier in [36,Theorem 1.2] and [3,Theorem 1.4] (see also [103]). We also note that the two walls which are divisorial contractions are actually weighted blowups of Kirwan type (see Remarks 5.17 and 6.10).…”

Section: Git Moduli Space Mmentioning

confidence: 59%

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K-stability and birational models of moduli of quartic K3 surfaces

2022

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We show that the K-moduli spaces of log Fano pairs $$({\mathbb {P}}^3, cS)$$ ( P 3 , c S ) where S is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily–Borel compactification of moduli of quartic K3 surfaces as c varies in the interval (0, 1). We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza–O’Grady’s prediction on the Hassett–Keel–Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of $${\mathbb {P}}^3$$ P 3 .

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Section: Git Moduli Space Mmentioning

confidence: 59%

“…Next we recall a result connecting K-stability and GIT stability as a special case of [36, Theorem 1.2] and [3, Theorem 1.4] (see also [103]).…”

Section: K-moduli Of Quartic Surfacesmentioning

confidence: 99%

K-stability and birational models of moduli of quartic K3 surfaces

2022

View full text Add to dashboard Cite

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“…That is, by varying the coefficient c, the K-moduli spaces M K c provide a natural interpolation between the GIT quotient for quartic surfaces and the Baily-Borel compactification, and explicitly resolve the period map. We note that a special case of Theorem 1.1(1) was proved earlier in [GMGS21, Theorem 1.2] and [ADL19, Theorem 1.4] (see also [Zho21a]).…”

mentioning

confidence: 66%

“…Next we recall a result connecting K-stability and GIT stability as a special case of [GMGS21, Theorem 1.2] and [ADL19, Theorem 1.4] (see also [Zho21a]).…”

Section: Geometry and Moduli Of Quartic K3 Surfacesmentioning

confidence: 99%

K-stability and birational models of moduli of quartic K3 surfaces

Ascher¹,

DeVleming²,

Liu³

2021

Preprint

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We show that the K-moduli spaces of log Fano pairs (P 3 , cS) where S is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily-Borel compactification of moduli of quartic K3 surfaces as c varies in the interval (0, 1). We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza-O'Grady's prediction on the Hassett-Keel-Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of P 3 .as an anti-canonical divisor. Then using a cone construction, a covering trick, and interpolation (see Section 5.1 for more detalis), we show that K 3c−1 4 admits a closed embedding into M K c for c > 1 3 whose image H h,c is a birational transform of H h (see Theorem 5.7). Then we construct K-polystable replacements of W i by first embedding all of P 3 , X h , P(1, 1, 2, 4) into P(1 4 , 2) as weighted hypersurfaces of degree two, and then finding a particular 1-PS (coming from VGIT in [LO21]) that degenerates (P 3 , cS) to a K-polystable pair in H h,c (see Theorem 5.10). Then we use deformation theory to classify exceptional loci after the walls. In particular, all K-moduli spaces M K c are isomorphic outside of the loci H h,c and H u,c . We give a complete description of all wall-crossings of M K c in Theorem 5.14. Finally, in Section 6 we prove the main theorems. We observe that F M K c is a birational contraction by Theorem 5.14. The upshot to showto use ampleness of log CM line bundles [XZ20], and to compute the variation of log CM line bundles which interpolate between the Hodge line bundle and the absolute CM line bundle (see (6.1)). Then we perform necessary birational modifications on M K c to obtain F(a, b) (see Definition 6.4), and show that the pushforward of λ + a 2 H h + b 2 H u is ample. 13 Theorem 4.19. The K-moduli spaces M K c has a wall at c = 9 13 . Moreover, we have (1) The wall crossing morphism φ − : M K 9 13 −ǫ → M K 9 13 replaces [(P 3 , T )] by [(X u , T 0 )], and is isomorphic near [(X u , T 0 )]. (2) The wall crossing morphism φ + : M K 9 13 +ǫ → M K 9 13 replaces [(X u , T 0 )] by the divisor H u, 9 13 +ǫ . (3) For any c ∈ ( 9 13 , 1), the birational map M K c M K 9 13 +ǫ is an isomorphism over a neighborhood of H u, 9 13 +ǫ . Proof. (1) Let U T := M GIT \ W 8 be an open neighborhood of [T ]. By (3.1), we know that any [S] ∈ U T \ {[T ]} is slc, thus (P 3 , cS) is K-stable for any c ∈ (0, 1). Since kst(P 3 , T ) = 9 13 by Theorem 4.3, there are open immersions U T ֒→ M K c when c ∈ (0, 9 13 ) and U T \ {[T ]} ֒→ M K c when c ∈ [ 9 13 , 1). Thus the map φ − : M K 9 13 −ǫ → M K 9 13is isomorphic on U T \ {[T ]}. On the other hand, we know that φ − ([(P 3 , T )]) = [(X u , T 0 )] by Proposition 4.7. By Corollary 4.17, we know thatis normal near [(X u , T 0 )] by Lemma 2.19. By Zariski's main theorem, we know that (φ − ) −1 ([(X u , T 0 )]) is connected, thus it has to be the singleton {[(P 3 , T )]}. Hence φ − is...

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Log K-stability of GIT-stable divisors on Fano manifolds

Cited by 2 publications

References 29 publications

K-stability and birational models of moduli of quartic K3 surfaces

K-stability and birational models of moduli of quartic K3 surfaces

K-stability and birational models of moduli of quartic K3 surfaces

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