Log Fano Research Articles

K-MODULI OF CURVES ON A QUADRIC SURFACE AND K3 SURFACES

AbstractWe show that the K-moduli spaces of log Fano pairs $\left(\mathbb {P}^1\times \mathbb {P}^1, cC\right)$ , where C is a $(4,4)$ curve and their wall crossings coincide with the VGIT quotients of $(2,4)$ , complete intersection curves in $\mathbb {P}^3$ . This, together with recent results by Laza and O’Grady, implies that these K-moduli spaces form a natural interpolation between the GIT moduli space of $(4,4)$ curves on $\mathbb {P}^1\times \mathbb {P}^1$ and the Baily–Borel compactification of moduli of quartic hyperelliptic K3 surfaces.

Homological mirror symmetry for Milnor fibers of simple singularities

We prove homological mirror symmetry for Milnor fibers of simple singularities in dimensions greater than one, which are among the log Fano cases of Conjecture 1.5 in arXiv:1806.04345. The proof is based on a relation between matrix factorizations and Calabi--Yau completions. As an application, we give an explicit computation of the Hochschild cohomology group of the derived $n$-preprojective algebra of a Dynkin quiver for any $n \geq 1$, and the symplectic cohomology group of the Milnor fiber of any simple singularity in any dimension greater than one.

Approximating delta invariants in the sense of plt complements

We show that delta invariant of a log Fano pair can be approximated by lc places of plt complements if it is no greater than one. Under the assumption that delta invariant (no greater than one) of a log Fano pair can be approximated by lc places of bounded plt complements, we show the existence of divisorial valuation computing delta invariant of this log Fano pair.

Bounding non-rationality of divisors on 3-fold Fano fibrations

Abstract In this paper we investigate non-rationality of divisors on 3-fold log Fano fibrations ( X , B ) → Z {(X,B)\to Z} under mild conditions. We show that if D is a component of B with coefficient ≥ t > 0 {\geq t>0} which is contracted to a point on Z, then D is birational to ℙ 1 × C {\mathbb{P}^{1}\times C} , where C is a smooth projective curve with gonality bounded depending only on t. Moreover, if t > 1 2 {t>\frac{1}{2}} , then genus of C is bounded depending only on t.

K-stability of log Fano hyperplane arrangements

In this article, we completely determine which log Fano hyperplane arrangements are uniformly K-stable, K-stable, K-polystable, K-semistable, or not.

Optimal destabilizing centers and equivariant K-stability

We give an algebraic proof of the equivalence of equivariant K-semistability (resp. equivariant K-polystability) with geometric K-semistability (resp. geometric K-polystability). Along the way we also prove the existence and uniqueness of minimal optimal destabilizing centers on K-unstable log Fano pairs.

Algebraicity of the metric tangent cones and equivariant K-stability

We prove two new results on the K K -polystability of Q \mathbb {Q} -Fano varieties based on purely algebro-geometric arguments. The first one says that any K K -semistable log Fano cone has a special degeneration to a uniquely determined K K -polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to complete the proof of Donaldson-Sun’s conjecture which says that the metric tangent cone of any point appearing on a Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds depends only on the algebraic structure of the singularity. The second result says that for any log Fano variety with the torus action, K K -polystability is equivalent to equivariant K K -polystability, that is, to check K K -polystability, it is sufficient to check special test configurations which are equivariant under the torus action.

Towards finite generation of higher rational rank valuations

We propose a finite generation conjecture for the valuation which computes the stability threshold of a log Fano pair. We also initiate a degeneration strategy for attacking the conjecture. Bibliography: 17 titles.

Some criteria for uniform $K$-stability

We prove some criteria for uniform K-stability of log Fano pairs. In particular, we show that uniform K-stability is equivalent to $\beta$-invariant having a positive lower bound. Then we study the relation between optimal destabilization conjecture and the conjectural equivalence between uniform K-stability and K-stability in the twisted setting.

Boundedness of anti-canonical volumes of singular $\operatorname{log}$ Fano threefolds

We prove Weak Borisov--Alexeev--Borisov Conjecture in dimension three which states that the anti-canonical volume of an $\epsilon$-klt log Fano pair of dimension three is bounded from above.

The normalized volume of a singularity is lower semicontinuous

We show that in any \mathbb Q -Gorenstein flat family of klt singularities, normalized volumes are lower semicontinuous with respect to the Zariski topology. A quick consequence is that smooth points have the largest normalized volume among all klt singularities. Using an alternative characterization of K-semistability developed by Li, Liu, and Xu, we show that K-semistability is a very generic or empty condition in any \mathbb Q -Gorenstein flat family of log Fano pairs.

Applications of the moduli continuity method to log K‐stable pairs

The 'moduli continuity method' permits an explicit algebraisation of the Gromov-Hausdorff compactification of K\"ahler-Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the 'log setting' to describe explicitly some compact moduli spaces of K-polystable log Fano pairs. We focus on situations when the angle of singularities is perturbed in an interval sufficiently close to one, by considering constructions arising from Geometric Invariant Theory. More precisely, we discuss the cases of pairs given by cubic surfaces with anticanonical sections, and of projective space with non-Fano hypersurfaces, and we show ampleness of the CM line bundle on their good moduli space (in the sense of Alper). Finally, we introduce a conjecture relating K-stability (and degenerations) of log pairs formed by a fixed Fano variety and pluri-anticanonical sections to certain natural GIT quotients.

On ruled surfaces with big anti-canonical divisor and numerically trivial divisors on weak log Fano surfaces

We investigate the structure of geometrically ruled surfaces whose anti-canonical class is big. As an application we show that the Picard group of a normal projective surface whose anti-canonical class is nef and big is a free abelian group of finite rank.

On the existence of conic Kähler-Einstein metrics

In this paper, we prove the conic version of YTD conjecture on log Fano manifolds.

Reductivity of the automorphism group of K-polystable Fano varieties

We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and $\Theta$-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space.

Dual Complex of Log Fano Pairs and Its Application to Witt Vector Cohomology

AbstractWe prove the contractibility of the dual complexes of weak log Fano pairs. As applications, we obtain a vanishing theorem of Witt vector cohomology of Ambro–Fujino type and a rational point formula in Dimension 3.

A note on toric degeneration of a Bott–Samelson–Demazure–Hansen variety

In this paper, we study the geometry of toric degeneration of a Bott–Samelson–Demazure–Hansen (BSDH) variety, which was algebraically constructed by Pasquier (J Algebra 323(10):2834–2847, 2010). We give some applications to BSDH varieties. Precisely, we classify Fano, weak Fano and log Fano BSDH varieties and their toric limits in Kac–Moody setting. We prove some vanishing theorems for the cohomology of tangent bundle (and line bundles) on BSDH varieties. We also recover the results in (Parameswaran and Karuppuchamy, Toric degeneration of Bott–Samelson–Demazure–Hansen varieties. arXiv:1604.01998 , 2016) and extend them to the Kac-Moody setting, by toric methods.

Rational points on log Fano threefolds over a finite field

We prove the W\mathcal{O} -rationality of klt threefolds and the rational chain connectedness of klt Fano threefolds over a perfect field of characteristic p > 5 . As a consequence, any klt Fano threefold over a finite field has a rational point.

Uniform K-stability and plt blowups of log Fano pairs

We show relationships between uniform K-stability and plt blowups of log Fano pairs. We see that it is enough to evaluate certain invariants defined by volume functions for all plt blowups in order to test the uniform K-stability of log Fano pairs. We also discuss the uniform K-stability of two log Fano pairs under crepant finite covers. Moreover, we give another proof of the K-semistability of the projective plane.

EFFECTIVE DIVISORS ON BOTT–SAMELSON VARIETIES

We compute the cone of effective divisors on a Bott–Samelson variety corresponding to an arbitrary sequence of simple roots. The main tool is a general result concerning effective cones of certain B-equivariant ℙ1 bundles. As an application, we compute the cone of effective codimension-two cycles on Bott–Samelson varieties corresponding to reduced words. We also obtain an auxiliary result giving criteria for a Bott–Samelson variety to contain a dense B-orbit, and we construct desingularizations of intersections of Schubert varieties. An appendix exhibits an explicit divisor showing that any Bott–Samelson variety is log Fano.