We establish an algebraic approach to prove the properness of moduli spaces of K-polystable Fano varieties and reduce the problem to a conjecture on destabilizations of K-unstable Fano varieties. Specifically, we prove that if the stability threshold of every K-unstable Fano variety is computed by a divisorial valuation, then such K-moduli spaces are proper. The argument relies on studying certain optimal destabilizing test configurations and constructing a Theta-stratification on the moduli stack of Fano varieties.
We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with [Jia17,BL18], the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable Q-Fano variety is finite.
In this Thesis, I investigate how Fano manifolds equipped with a Kahler-Einstein metric can degenerate as metric spaces (in the Gromov-Hausdorff topology) and some of the relations of this question with Algebraic Geometry, in particular in the direction of the study of moduli spaces and their compactifications.
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 consider the space KR(n,F) of Kahler-Ricci solitons on n-dimensional Fano manifolds with Futaki invariant bounded by F. We prove a partial C^0 estimate for KR(n,F) as a generalization of the recent work of Donaldson-Sun for Fano Kahler-Einstein manifolds. In particular, any sequence in KR(n,F) has a convergent subsequence in the Gromov-Hausdorff topology to a Kahler- Ricci soliton on a Q-Fano variety with log terminal singularities.
James Ax conjectured that every pseudo algebraically closed field is $C_1$. We prove this conjecture in characteristic 0 by relating it to degenerations of Fano varieties.
We show that every rank one smooth Fano threefold has a weak Landau-Ginzburg model coming from a toric degeneration. The fibers of these Landau-Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of Cartier divisors in weighted projective space has a very weak Landau-Ginzburg model coming from a toric degeneration.
For fixed degree $d\leq 12$, we study the Hilbert scheme of degree $d$ smooth Fano threefolds in their anticanonical embeddings. We use this to classify all possible degenerations of these varieties to toric Fano varieties with at most canonical Gorenstein singularities.
We construct degenerations of Mukai varieties and linear sections thereof to special unobstructed Fano Stanley-Reisner schemes corresponding to convex deltahedra. This can be used to find toric degenerations of rank one index one Fano threefolds. In the second part we find many higher dimensional unobstructed Fano and Calabi-Yau Stanley-Reisner schemes. The main result is that the Stanley-Reisner ring of the boundary complex of the dual polytope of the associahedron has trivial $T^2$.
In this paper, we give a tropical method for computing Gromov-Witten type invariants of Fano manifolds of special type. This method applies to those Fano manifolds which admit toric degenerations to toric Fano varieties with singularities allowing small resolutions. Examples include (generalized) flag manifolds of type A, and some moduli space of rank two bundles on a genus two curve.
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 finiteness of the number of smooth blow-downs on Fano manifolds and boundedness results for the geometry of non projective Fano-like manifolds. Our proofs use properness of Hilbert schemes and Mori theory.
We prove that on Fano manifolds, the Kähler-Ricci flow produces a "most destabilising" degeneration, with respect to a new stability notion related to the H-functional. This answers questions of Chen-Sun-Wang and He. We give two applications of this result. Firstly, we give a purely algebro-geometric formula for the supremum of Perelman's μ-functional on Fano manifolds, resolving a conjecture of Tian-Zhang-Zhang-Zhu as a special case. Secondly, we use this to prove that if a Fano manifold admits a Kähler-Ricci soliton, then the Kähler-Ricci flow converges to it modulo the action of automorphis…
In the paper "Birational geometry via moduli spaces" by I. Cheltsov, L. Katzarkov, and V. Przyjalkowski a new structure connecting toric degenerations of smooth Fano threefolds by projections was introduced; using Mirror Symmetry these connections were transferred to the side of Landau--Ginzburg models. In the paper mentioned above a nice way to connect of Picard rank one Fano threefolds was found. We apply this approach to all smooth Fano threefolds, connecting their degenerations by toric basic links. In particular, we find a lot of Gorenstein toric degenerations of smooth Fano threefolds we…
Consider a smooth variety $X$ and a smooth divisor $D\subset X$. Kim and Sato (arXiv:0806.3819) define a natural compactification of $(X\setminus D)^n$, denoted $X_D^{[n]}$, which is a moduli space of stable configurations of $n$ points lying on expansions of $(X,D)$ in the sense of Jun Li (arXiv:math/0009097, arXiv:math/0110113). The purpose of this note is to generalize Kim and Sato's construction to the case where $X$ is an algebraic stack; and to construct an analogous projective moduli space $W_π^{[n]}$ for a degeneration $π:W \to B$. We construct $X^n_D$ and $W_π^{[n]}$ and prove their p…
For an appropriate class of Fano complete intersections in toric varieties, we prove that there is a concrete relationship between degenerations to specific toric subvarieties and expressions for Givental's Landau-Ginzburg models as Laurent polynomials. As a result, we show that Fano varieties presented as complete intersections in partial flag manifolds admit degenerations to Gorenstein toric weak Fano varieties, and their Givental Landau-Ginzburg models can be expressed as corresponding Laurent polynomials. We also use this to show that all of the Laurent polynomials obtained by Coates, Kasp…
For an arbitrary smooth n-dimensional Fano variety $X$ we introduce the notion of a small toric degeneration. Using small toric degenerations of Fano n-folds $X$, we propose a general method for constructing mirrors of Calabi-Yau complete intersections in $X$. Our mirror construction is based on a generalized monomial-divisor mirror correspondence which can be used for computing Gromov-Witten invariants of rational curves via specializations of GKZ-hypergeometric series.
There exist exactly 166 4-dimensional reflexive polytopes such that the corresponding 4-dimensional Gorenstein toric Fano varieties have at worst terminal singularities in codimension 3 and their anticanonical divisor is divisible by 2. For every such a polytope, one naturally obtains a family of Fano hypersurfaces X with at worst conifold singularities. A generic 3-dimensional Fano hypersurface X can be interpreted as a flat conifold degeneration of some smooth Fano 3-folds Y whose classification up to deformation was obtained by Iskovskikh, Mori and Mukai. In this case, both Fano varieties X…
In this paper we study the geometry of toric degeneration of a Bott-Samelson-Demazure-Hansen (BSDH) variety. 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 arXiv:1604.01998, by toric methods.
In this article we exhibit certain projective degenerations of smooth $K3$ surfaces of degree $2g-2$ in $\Bbb P^g$ (whose Picard group is generated by the hyperplane class), to a union of two rational normal scrolls, and also to a union of planes. As a consequence we prove that the general hyperplane section of such $K3$ surfaces has a corank one Gaussian map, if $g=11$ or $g\geq 13$. We also prove that the general such hyperplane section lies on a unique $K3$ surface, up to projectivities. Finally we present a new approach to the classification of prime Fano threefolds of index one, which doe…
We study the birational boundedness of special fibers of log Calabi-Yau fibrations and Fano fibrations. We show that for a locally stable family of Fano varieties or polarised log Calabi-Yau pairs over a curve, if the general fiber satisfies some natural boundedness conditions, then every irreducible component of the special fiber is birationally bounded.
We propose a refined but natural notion of toric degenerations that respect a given embedding and show that within this framework a Gorenstein Fano variety can only be degenerated to a Gorenstein Fano toric variety if it is embedded via its anticanonical embedding. This also gives a precise criterion for reflexive polytopes to appear, which might be required for applications in mirror symmetry. For the proof of this statement we will study polytopes whose polar dual is a lattice polytope. As a byproduct we generalize a connection between the number of lattice points in a rational convex polyto…
The interpretation, due to T. Mabuchi, of the classical Futaki invariant of Fano toric manifolds is extended to the case of the Generalized Futaki invariant, introduced by W. Ding and G. Tian, of almost Fano toric varieties. As an application it is shown that the real part of the Generalized Futaki invariant is positive for all degenerations of the Fano manifold V_{38}, obtained by intersection of the Veronese embedding of ${\bf P}^3\times{\bf P}^2 \subset {\bf P}^{11}$ with codimension-two hyperplanes.
This is a short companion paper to arXiv:0810.3470. We construct an integrable system on an open subset of a Fano manifold equipped with a toric degeneration, and compute the potential function for its Lagrangian torus fibers if the central fiber is a toric Fano variety admitting a small resolution.
Using ideas from the theory of tropical curves and degeneration, we prove that any Fano hypersurface (and more generally Fano complete intersections) is swept by at most quadratic rational curves.
A perfect PAC field containing an algebraically closed field is known to be $C_1$, i.e., every degeneration of a Fano complete intersection has a point. We prove that also every degeneration of a separably rationally connected variety has a point.
In this paper, we study the K-stability of polarized spherical varieties. After reduction, it can be treated as a variational problem of the reduced functional of the Futaki invariant on the associated moment polytope. With the convexity constraint of the problem, the minimizers are shown to satisfy the homogeneous Monge-Ampère equation (HMA). When the spherical variety has rank two, a simpler characterization can be established through properties of the HMA. As an application, we determine the strict semistability and polystable degenerations for Fano spherical varieties of rank two.
We investigate a potential relationship between mirror symmetry for Calabi-Yau manifolds and the mirror duality between quasi-Fano varieties and Landau-Ginzburg models. More precisely, we show that if a Calabi-Yau admits a so-called Tyurin degeneration to a union of two Fano varieties, then one should be able to construct a mirror to that Calabi-Yau by gluing together the Landau-Ginzburg models of those two Fano varieties. We provide evidence for this correspondence in a number of different settings, including Batyrev-Borisov mirror symmetry for K3 surfaces and Calabi-Yau threefolds, Dolgachev…
We give a method to estimate Seshadri constants on toric varieties at any point. By using the estimations and toric degenerations, we can obtain some new computations or estimations of Seshadri constants on non-toric varieties. In particular, we investigate Seshadri constants on hypersurfaces in projective spaces and Fano 3-folds with Picard number one in detail.
We analize the semistable degeneration of the Fano surface F when the cubic threefold becomes the Segre primal. This gives an explicit topological decomposition for F. The decomposition is used to decide that the Fano surface is not an an Eilenberg Mac-Lane K(π,1) space, this was the question that prompted us to look into the matter.
We study rational curves on general Fano hypersurfaces in projective space, mostly by degenerating the hypersurface along with its ambient projective space to reducible varieties. We prove results on existence of low-degree rational curves with balanced normal bundle, and reprove some results on irreducibility of spaces of rational curves of low degree.
The moduli space of (1,7)-polarized abelian surfaces with a level structure was shown by Manolache and Schreyer to be rational with compactification a Fano 3-fold of genus 12 which is also a compactification of the variety of powersum presentations of the Klein quartic curve. In this paper the possible degenerations of the abelian surfaces are classified in terms of degenerations of powersum presentations. In particular, the surface degenerations into fourteen planes are recovered. These were independently discovered by Alfio Marini using degenerations of twisted cubic curves.
We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by Mirror Symmetry, we give conditions for the limit toric variety to be a Gorenstein Fano, and provide many examples. We also provide an explanation for the limits as boundary points of the moduli space of stable pairs whose existence is predicted by the Minimal Model Program.
We present some applications of the deformation theory of toric Fano varieties to K-(semi/poly)stability of Fano varieties. First, we present two examples of K-polystable toric Fano 3-fold with obstructed deformations. In one case, the K-moduli spaces and stacks are reducible near the closed point associated to the toric Fano 3-fold, while in the other they are non-reduced near the closed point associated to the toric Fano 3-fold. Second, we study K-stability of the general members of two deformation families of smooth Fano 3-folds by building degenerations to K-polystable toric Fano 3-folds.
In this paper we construct a degeneration of Bott-Samelson-Demazure-Hansen varieties to toric varieties in an algebraic family and study the geometry of the resulting toric varieties. We give a natural set of torus invariant curves that generate the Chow group of $1$-cycles of the limiting toric variety and express the ample cone of this toric variety as a sub-cone of the ample cone of the corresponding Bott-Samelson-Demazure-Hansen variety. We also give a description of Extremal and Mori rays and determine when this toric variety is Fano.
We introduce a variety $\hat{G}_2$ parameterizing isotropic five-spaces of a general degenerate four-form in a seven dimensional vector space. It is in a natural way a degeneration of the variety $G_2$, the adjoint variety of the simple Lie group $\mathbb{G}_2$. It occurs that it is also the image of $\mathbb{P}^5$ by a system of quadrics containing a twisted cubic. Degenerations of this twisted cubic to three lines give rise to degenerations of $G_2$ which are toric Gorenstein Fano fivefolds. We use these two degenerations to construct geometric transitions between Calabi--Yau threefolds. We …
The 'moduli continuity method' permits an explicit algebraisation of the Gromov-Hausdorff compactification of Kähler-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, …
We prove a type of the Lefschetz hyperplane section theorem on Q-Fano 3-folds with Picard number one and $1/2(1,1,1)$-singularities by using some degeneration method. As a byproduct, we obtain a new example of a Calabi-Yau 3-fold $X$ with Picard number one whose invariants are $$(H_X^3, c_2 (X) \cdot H_X, {e} (X)) = (8, 44, -88),$$ where $H_X$, $e(X)$ and $c_2(X)$ are an ample generator of $\Pic(X)$, the topological Euler characteristic number and the second Chern class of $X$ respectively.
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 …
We show that complex Fano hypersurfaces can have arbitrarily large degrees of irrationality. More precisely, if we fix a Fano index e, then the degree of irrationality of a very general complex Fano hypersurface of index e and dimension n is bounded from below by a constant times $\sqrt{n}$. To our knowledge this gives the first examples of rationally connected varieties with degrees of irrationality greater than 3. The proof follows a degeneration to characteristic p argument which Kollár used to prove nonrationality of Fano hypersurfaces. Along the way we show that in a family of varieties, …
