Given any field $k$ (not necessarily perfect), we study the smoothing of a semistable Fano variety over $k$. In characteristic 0, the reduced semistable Fano degenerate fibers of Mori fibrations are classified. In positive characteristic, under a suitable $W_2$ lifting assumption, we prove that a semistable Fano variety always appears as a degenerate fiber in a semistable family if it has a global log structure (in the sense of Fontaine-Illusie-Kato) of semistable type. A similar smoothing result over a mixed characteristic base is also obtained.
In this short note, we show that K-semistable Fano manifolds with the smallest alpha invariant are projective spaces. Singular cases are also investigated.
Consider a family of Fano varieties $π: X \longrightarrow B\ni o$ over a curve germ with a smooth total space $X$. Assume that the generic fiber is smooth and the special fiber $F=π^{-1}(o)$ has simple normal crossings. Then $F$ is called a semistable degeneration of Fano varieties. We show that the dual complex of $F$ is a simplex of dimension $\leq \mathrm{dim}\ F$. Simplices of any admissible dimension can be realized for any dimension of the fiber. Using this result and the Minimal Model Program in dimension $3$ we reproduce the classification of the semistable degenerations of del Pezzo s…
We prove that minimal instanton bundles on a Fano threefold $X$ of Picard rank one and index two are semistable objects in the Kuznetsov component $\mathsf{Ku}(X)$, with respect to the stability conditions constructed by Bayer, Lahoz, Macrì and Stellari. When the degree of $X$ is at least $3$, we show torsion free generalizations of minimal instantons are also semistable objects. As a result, we describe the moduli space of semistable objects with same numerical classes as minimal instantons in $\mathsf{Ku}(X)$. We also investigate the stability of acyclic extensions of non-minimal instantons.
We study semistable sheaves of rank $2$ with Chern classes $c_1=0$, $c_2=2$ and $c_3=0$ on the Fano 3-fold $V_5$ of Picard number $1$, degree $5$ and index $2$. We show that the moduli space of such sheaves has a component that is isomorphic to $\mathbb{P}^5$ by identifying it with the moduli space of semistable quiver representations. This provides a natural smooth compactification of the moduli space of minimal instantons, as well as Ulrich bundles of rank $2$ on $V_5$.
Firstly, we see that the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities are quite restricted. In particular, we classify the analytic germs of embedding dimension $\leq 2$ which are the bases of the miniversal deformations of isolated $\mathbb{Q}$-Gorenstein toric singularities. Secondly, we show that the deformation spaces of isolated Gorenstein toric $3$-fold singularities appear, in a weak sense, as singularities of the K-moduli stack of K-semistable Fano varieties of every dimension $\geq 3$. As a consequence, we prove that the number of local …
We generalize the definition of alpha invariant to arbitrary codimension. We also give a lower bound of these alpha invariants for K-semistable Q-Fano varieties and show that we can characterize projective spaces among all K-semistable Fano manifolds in terms of higher codimensional alpha invariants. Our results demonstrate the relation between alpha invariants of any codimension and volumes of Fano manifolds in the characterization of projective spaces.
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 $Θ$-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.
A toric del Pezzo surface $X_P$ with cyclic quotient singularities determines and is determined by a Fano polygon $P$. We construct an affine manifold with singularities that partially smooths the boundary of $P$; this a tropical version of a Q-Gorenstein partial smoothing of $X_P$. We implement a mild generalization of the Gross-Siebert reconstruction algorithm - allowing singularities that are not locally rigid - and thereby construct (a formal version of) this partial smoothing directly from the affine manifold. This has implications for mirror symmetry: roughly speaking, it implements half…
Suppose that X is a Fano manifold that corresponds under Mirror Symmetry to a Laurent polynomial f, and that P is the Newton polytope of f. In this setting it is expected that there is a family of algebraic varieties over the unit disc with general fiber X and special fiber the toric variety defined by the spanning fan of P. Building on recent work and conjectures by Corti--Hacking--Petracci, who construct such families of varieties, we determine the topology of the general fiber from combinatorial data on P. This provides evidence for the Corti--Hacking--Petracci conjectures, and verifies tha…
In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano Kähler-Einstein manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski open condition and establish the uniqueness for the Gromov-Hausdorff limit for a punctured flat family of Fano Kähler-Einstein manifolds. Based on these, we construct a proper scheme parameterizing the S-equivalent classes of $\QQ$-Gorenstein smoothable, K-semistable Fano varieties, and verify various necessary properties to guarantee that it is a good …
We study Bridgeland moduli spaces of semistable objects of $(-1)$-classes and $(-4)$-classes in the Kuznetsov components on index one prime Fano threefold $X_{4d+2}$ of degree $4d+2$ and index two prime Fano threefold $Y_d$ of degree $d$ for $d=3,4,5$. For every Serre-invariant stability condition on the Kuznetsov components, we show that the moduli spaces of stable objects of $(-1)$-classes on $X_{4d+2}$ and $Y_d$ are isomorphic. We show that moduli spaces of stable objects of $(-1)$-classes on $X_{14}$ are realized by Fano surface $\mathcal{C}(X)$ of conics, moduli spaces of semistable sheav…
Let $S$ be a smooth projective variety and $Δ$ a simple normal crossing $\mathbb{Q}$-divisor with coefficients in $(0,1]$. For any ample $\mathbb{Q}$-line bundle $L$ over $S$, we denote by $\mathscr{E}(L)$ the extension sheaf of the orbifold tangent sheaf $T_S(-\log(Δ))$ by the structure sheaf $\mathcal{O}_S$ with the extension class $c_1(L)$. We show the following two results: (i) If $-(K_S+Δ)$ is ample and $(S, Δ)$ is K-semistable, then for any $λ\in \mathbb{Q}_{>0}$, the extension sheaf $\mathscr{E}({λc_1(-(K_S+Δ))})$ is slope semistable with respect to $-(K_S+Δ)$; (ii) If $K_S+Δ\equiv 0$, …
We prove that every birationally superrigid Fano variety whose alpha invariant is greater than (resp. no smaller than) $\frac{1}{2}$ is K-stable (resp. K-semistable). We also prove that the alpha invariant of a birationally superrigid Fano variety of dimension $n$ is at least $\frac{1}{n+1}$ (under mild assumptions) and that the moduli space (if exists) of birationally superrigid Fano varieties is separated.
We show that certain Galois covers of K-semistable Fano varieties are K-stable. We use this to give some new examples of Fano manifolds admitting Kähler-Einstein metrics, including hypersurfaces, double solids and threefolds.
We show that $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees bounded from below form a bounded family.
Let $X$ be a $\mathbb Q$-Fano variety admitting a Kähler-Einstein metric. We prove that up to a finite quasi-étale cover, $X$ splits isometrically as a product of Kähler-Einstein $\mathbb Q$-Fano varieties whose tangent sheaf is stable with respect to the anticanonical polarization. This relies among other things on a very general splitting theorem for algebraically integrable foliations. We also prove that the canonical extension of $T_X$ by $\mathscr O_X$ is semistable with respect to the anticanonical polarization.
We show that the anti-canonical volume of an $n$-dimensional Kähler-Einstein $\mathbb{Q}$-Fano variety is bounded from above by certain invariants of the local singularities, namely $\mathrm{lct}^n\cdot\mathrm{mult}$ for ideals and the normalized volume function for real valuations. This refines a recent result by Fujita. As an application, we get sharp volume upper bounds for Kähler-Einstein Fano varieties with quotient singularities. Based on very recent results by Li and the author, we show that a Fano manifold is K-semistable if and only if a de Fernex-Ein-Mustaţă type inequality holds on …
We prove that for any $\mathbb{Q}$-Fano variety $X$, the special $\mathbb{R}$-test configuration that minimizes the $H$-functional is unique and has a K-semistable $\mathbb{Q}$-Fano central fibre $(W, ξ)$. Moreover there is a unique K-polystable degeneration of $(W, ξ)$. As an application, we confirm the conjecture of Chen-Sun-Wang about the algebraic-uniqueness for Kähler-Ricci flow limits on Fano manifolds, which implies that the Gromov-Hausdorff limit of the flow does not depend on the choice of initial Kähler metrics. The results are achieved by studying algebraic optimal degeneration prob…
We introduce the notion of cracked polytope, and - making use of joint work with Coates and Kasprzyk - construct the associated toric variety $X$ as a subvariety of a non-singular toric variety $Y$ under certain conditions. Restricting to the case in which this subvariety is a complete intersection, we present a sufficient condition for a smoothing of $X$ to exist inside $Y$. We exhibit a relative anti-canonical divisor for this smoothing of $X$, and show that the general member is simple normal crossings.
We overview our recent work defining and studying normal crossings varieties and subvarieties in symplectic topology. This work answers a question of Gromov on the feasibility of introducing singular (sub)varieties into symplectic topology in the case of normal crossings singularities. It also provides a necessary and sufficient condition for smoothing normal crossings symplectic varieties. In addition, we explain some connections with other areas of mathematics and discuss a few directions for further research.
推广复曲面几何，我们研究紧凑品种承认低维环面行动。特别是，我们描述了他们的凸几何方面的因子，并给出了他们的丰富性的标准。这些结果可用于研究具有小环面作用的Fano品种。作为第一个结果，我们分类的Picard数1和Gorenstein指数小于4的log del Pezzo C*-表面。在进一步的例子中，我们展示了如何分类可能会在更高的维度，我们给出了明确的描述一些等变平滑的法诺三倍。
We describe a practical and effective method for reconstructing the deformation class of a Fano manifold X from a Laurent polynomial f that corresponds to X under Mirror Symmetry. We explore connections to nef partitions, the smoothing of singular toric varieties, and the construction of embeddings of one (possibly-singular) toric variety in another. In particular, we construct degenerations from Fano manifolds to singular toric varieties; in the toric complete intersection case, these degenerations were constructed previously by Doran--Harder. We use our method to find models of orbifold del …
