Log Fano Research Articles

Boundedness of Log Fano Pairs with Certain K-stability

Abstract We prove several boundedness results for log Fano pairs with certain K-stability. In particular, we prove that K-semistable log Fano pairs of Maeda type in each dimension form a log bounded family. We also compute K-semistable domains for some examples.

Wall crossing for K‐moduli spaces of plane curves

AbstractWe construct proper good moduli spaces parametrizing K‐polystable ‐Gorenstein smoothable log Fano pairs , where is a Fano variety and is a rational multiple of the anticanonical divisor. We then establish a wall‐crossing framework of these K‐moduli spaces as varies. The main application in this paper is the case of plane curves of degree as boundary divisors of . In this case, we show that when the coefficient is small, the K‐moduli space of these pairs is isomorphic to the GIT moduli space. We then show that the first wall crossing of these K‐moduli spaces are weighted blow‐ups of Kirwan type. We also describe all wall crossings for degree 4,5,6 and relate the final K‐moduli spaces to Hacking's compactification and the moduli of K3 surfaces.

Twisted Kähler–Einstein metrics in big classes

AbstractWe prove existence of twisted Kähler–Einstein metrics in big cohomology classes, using a divisorial stability condition. In particular, when is big, we obtain a uniform Yau–Tian–Donaldson (YTD) existence theorem for Kähler–Einstein (KE) metrics. To achieve this, we build up from scratch the theory of Fujita–Odaka type delta invariants in the transcendental big setting, using pluripotential theory. We do not use the K‐energy in our arguments, and our techniques provide a simple roadmap to prove YTD existence theorems for KE type metrics, that only needs convexity of the appropriate Ding energy. As an application, we give a simplified proof of Li–Tian–Wang's existence theorem in the log Fano setting.

Moduli of genus six curves and K-stability

The K-moduli theory provides different compactifications of various moduli spaces, including moduli of curves. As a general genus six curve can be canonically embedded into the smooth quintic del Pezzo surface, we study in this paper the K-moduli spaces M ¯ K ( c ) \overline {M}^K(c) of the quintic log Fano pairs. We classify the strata of genus six curves C C appearing in the K-moduli by explicitly describing the wall-crossing structure. The K-moduli spaces interpolate between two birational moduli spaces constructed by Geometric Invariant Theory (GIT) and moduli of K3 surfaces via Hodge theory.

On the volume of some Fano K-moduli spaces

We compute the CM volume, that is the degree of the descended CM line bundle on the coarse moduli space in two cases: on the Fano K-moduli space of quartic del Pezzo in any dimension, and on the K-moduli space of the log Fano hyperplane arrangements of dimension one and two. Furthermore, we relate these volumes to the Weil–Petersson volumes by extending the notion of Weil–Petersson metric in the log case.

Cox rings of blow-ups of multiprojective spaces

AbstractLet $$X^{1,n}_r$$ X r 1 , n be the blow-up of $$\mathbb {P}^1\times \mathbb {P}^n$$ P 1 × P n in r general points. We describe the Mori cone of $$X^{1,n}_r$$ X r 1 , n for $$r\le n+2$$ r ≤ n + 2 and for $$r = n+3$$ r = n + 3 when $$n\le 4$$ n ≤ 4 . Furthermore, we prove that $$X^{1,n}_{n+1}$$ X n + 1 1 , n is log Fano and give an explicit presentation for its Cox ring.

Minimizing CM Degree and Specially K-stable Varieties

Abstract We prove that the degree of the CM line bundle for a normal family over a curve with fixed general fibers is strictly minimized if the special fiber is either a smooth projective manifold with a unique cscK metric or “specially K-stable”, which is a new class we introduce in this paper. This phenomenon, as conjectured by Odaka (cf., [ 46]), is a quantitative strengthening of the separatedness conjecture of moduli spaces of polarized K-stable varieties. The above-mentioned special K-stability implies the original K-stability and a lot of cases satisfy it, for example, K-stable log Fano, klt Calabi-Yau (i.e., $K_{X}\equiv 0$), lc varieties with the ample canonical divisor and uniformly adiabatically K-stable klt-trivial fibrations over curves (cf., [ 27]).

K-polystability of 3-dimensional log Fano pairs of Maeda type

Using the Abban–Zhuang theory and the classification of 3-dimensional log smooth log Fano pairs due to Maeda, we prove that almost all threefold log Fano pairs [Formula: see text] of Maeda type with reducible boundary [Formula: see text] are K-unstable for any values of the boundary coefficients. We also correct several inaccuracies in Maeda’s classification.

The existence of the Kähler–Ricci soliton degeneration

Abstract We prove an algebraic version of the Hamilton–Tian conjecture for all log Fano pairs. More precisely, we show that any log Fano pair admits a canonical two-step degeneration to a reduced uniformly Ding stable triple, which admits a Kähler–Ricci soliton when the ground field .

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

We show that the K-moduli spaces of log Fano pairs ({mathbb {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 {mathbb {P}}^3.

Some examples of log Fano structures on blow-ups along subvarieties in products of two projective spaces

We give a series of examples of log Fano manifolds in any dimension greater than or equal to three by using successive blow-ups along subvarieties in products of two projective spaces. More precisely, we first blow-up a product of two projective spaces along a smooth hypersurface contained in a fiber of one of the two projections and then along the strict transform of a fiber (intersecting the center of the first blow-up) of the other projection. By computing the nef cone and giving an explicit boundary divisor, we show that the resulting variety is always a log Fano manifold. Moreover, the effective cone and the extremal contractions are described. Note that this variety depends on three integral parameters: the dimensions of the two projective spaces and the degree of the hypersurface, which is the center of the first blow-up. Hence, we obtain a series of examples of log Fano manifolds. We also determine which ones are weak Fano.

Finite generation for valuations computing stability thresholds and applications to K-stability

We prove that on any log Fano pair of dimension $n$ whose stability threshold is less than $\frac{n+1}{n}$, any valuation computing the stability threshold has a finitely generated associated graded ring. Together with earlier works, this implies that (a) a log Fano pair is uniformly K-stable (resp. reduced uniformly K-stable) if and only if it is K-stable (resp. K-polystable); (b) the K-moduli spaces are proper and projective; and combining with the previously known equivalence between the existence of Kähler-Einstein metric and reduced uniform K-stability proved by the variational approach, (c) the Yau-Tian-Donaldson conjecture holds for general (possibly singular) log Fano pairs.

Maximal log Fano manifolds are generalized Bott towers

We prove that maximal log Fano manifolds are generalized Bott towers. As an application, we prove that in each dimension, there is a unique maximal snc Fano variety satisfying Friedman's d-semistability condition.

Log Fano blowups of mixed products of projective spaces and their effective cones

We compute the cones of effective divisors on blowups of mathbb P^1 times mathbb P^2 and mathbb P^1 times mathbb P^3 in up to 6 points. We also show that all these varieties are log Fano, giving a conceptual explanation for the fact that all the cones we compute are rational polyhedral.

Positivity of the CM line bundle for K-stable log Fanos

We prove the bigness of the Chow–Mumford line bundle associated to a Q \mathbb {Q} -Gorenstein family of log Fano varieties of maximal variation with uniformly K-stable general geometric fibers. This result generalizes a theorem of Codogni and Patakfalvi to the logarithmic setting.

On Uniform K-Stability for Some Asymptotically log del Pezzo Surfaces

Motivated by the problem of the existence of Kähler–Einstein edge metrics, Cheltsov and Rubinstein conjectured the K-polystability of asymptotically log Fano varieties with small cone angles when the anti-log-canonical divisors are not big. Cheltsov, Rubinstein and Zhang proved it affirmatively in dimension 2 with irreducible boundaries except for the type (I.9B.$n$) with $1 \leq n \leq 6$. Unfortunately, Fujita, Liu, Süß, Zhang and Zhuang recently showed the non-K-polystability for some members of type (I.9B.1) and for some members of type (I.9B.2). In this article we show that Cheltsov–Rubinstein’s problem is true for all of the remaining cases. More precisely, we explicitly compute the deltainvariant for asymptotically log del Pezzo surfaces of type (I.9B.$n$) for all $n \geq 1$ with small cone angles. As a consequence, we finish Cheltsov–Rubinstein’s problem in dimension 2 with irreducible boundaries.

Integral Points of Bounded Height on a Log Fano Threefold

AbstractWe determine an asymptotic formula for the number of integral points of bounded height on a blow-up of ${\mathbb {P}}^3$ outside certain planes using universal torsors.

On the body of ample angles of asymptotically log Fano varieties

In dimension two, we reduce the classification problem for asymptotically log Fano pairs to the problem of determining generality conditions on certain blow-ups. In any dimension, we prove the rationality of the body of ample angles of an asymptotically log Fano pair, i.e., these convex bodies are always rational polytopes.

Openness of K-semistability for Fano varieties

In this paper, we prove the openness of K-semistability in families of log Fano pairs by showing that the stability threshold is a constructible function on the fibers. We also prove that any special test configuration arises from a log canonical place of a bounded complement and establish properties of any minimizer of the stability threshold.

Equivariant K-stability under finite group action

We show that [Formula: see text]-equivariant K-semistability (respectively, [Formula: see text]-equivariant K-polystability) implies K-semistability (respectively, K-polystability) for log Fano pairs with klt singularities when [Formula: see text] is a finite group.