Showing papers on "Minimal model program published in 2016"

Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties

Abstract: Given a quasiprojective variety X with only Kawamata log terminal singularities, we study the obstructions to extending finite etale covers from the smooth locus Xreg of X to X itself. A simplified version of our main results states that there exists a Galois cover Y→X, ramified only over the singularities of X, such that the etale fundamental groups of Y and of Yreg agree. In particular, every etale cover of Yreg extends to an etale cover of Y. As a first major application, we show that every flat holomorphic bundle defined on Yreg extends to a flat bundle that is defined on all of Y. As a consequence, we generalize a classical result of Yau to the singular case: every variety with at worst terminal singularities and with vanishing first and second Chern class is a finite quotient of an abelian variety. As a further application, we verify a conjecture of Nakayama and Zhang describing the structure of varieties that admit polarized endomorphisms.

The essential skeleton of a degeneration of algebraic varieties

Abstract: In this paper, we explore the connections between the Minimal Model Program and the theory of Berkovich spaces. Let k be a field of characteristic zero and let X be a smooth and proper k((t))-variety with semiample canonical divisor. We prove that the essential skeleton of X coincides with the skeleton of any minimal dlt-model and that it is a strong deformation retract of the Berkovich analytification of X. As an application, we show that the essential skeleton of a Calabi-Yau variety over k((t)) is a pseudo-manifold.

Terminal valuations and the Nash problem

Abstract: Let \(X\) be an algebraic variety of characteristic zero. Terminal valuations are defined in the sense of the minimal model program, as those valuations given by the exceptional divisors on a minimal model over \(X\). We prove that every terminal valuation over \(X\) is in the image of the Nash map, and thus it corresponds to a maximal family of arcs through the singular locus of \(X\). In dimension two, this result gives a new proof of the theorem of Fernandez de Bobadilla and Pe Pereira stating that, for surfaces, the Nash map is a bijection.

The KSBA compactification for the moduli space of degree two $K$3 pairs

Abstract: Inspired by the ideas of the minimal model program, ShepherdBarron, Kollar, and Alexeev have constructed a geometric compactification for the moduli space of surfaces of log general type. In this paper, we discuss one of the simplest examples that fits into this framework: the case of pairs (X,H) consisting of a degree two K3 surface X and an ample divisor H. Specifically, we construct and describe explicitly a geometric compactification P2 for the moduli of degree two K3 pairs. This compactification has a natural forgetful map to the Baily–Borel compactification of the moduli space F2 of degree two K3 surfaces. Using this map and the modular meaning of P2, we obtain a better understanding of the geometry of the standard compactifications of F2.

Projectivity and birational geometry of Bridgeland moduli spaces on an Enriques surface

Abstract: We construct moduli spaces of semistable objects on an Enriques surface for generic Bridgeland stability condition and prove their projectivity. We further generalize classical results about moduli spaces of semistable sheaves on an Enriques surface to their Bridgeland counterparts. Using Bayer and Macri's construction of a natural nef divisor varying with the stability condition, we begin a systematic exploration of the relation between wall-crossing on the Bridgeland stability manifold and the minimal model program for these moduli spaces. We give three applications of our machinery to obtain new information about the classical moduli spaces of Gieseker-stable sheaves: 1) We obtain a region in the ample cone of the moduli space of Gieseker-stable sheaves which works for all unnodal Enriques surfaces. 2) We determine the nef cone of the Hilbert scheme of $n$ points on an unnodal Enriques surface in terms of the classical geometry of its half-pencils and the Cossec-Dolgachev $\phi$-function. 3) We recover some classical results on linear systems on Enriques surfaces and obtain some new ones about $n$-very ample line bundles.

Numerical analogues of the Kodaira dimension and the Abundance Conjecture

Abstract: We add further notions to Lehmann’s list of numerical analogues to the Kodaira dimension of pseudo-effective divisors on smooth complex projective varieties, and show new relations between them. Then we use these notions and relations to fill in a gap in Lehmann’s arguments, thus proving that most of these notions are equal. Finally, we show that the Abundance Conjecture, as formulated in the context of the Minimal Model Program, and the Generalized Abundance Conjecture using these numerical analogues to the Kodaira dimension, are equivalent for non-uniruled complex projective varieties.

Fano Varieties in Mori Fibre Spaces

Abstract: We show that being a general fibre of a Mori fibre space (MFS) is a rather restrictive condition for a Fano variety. More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with terminal singularities to be realised as a fibre of a Mori fibre space, which turn into a characterisation in the rigid case. We apply our criteria to figure out this property up to dimension 3 and on rational homogeneous spaces. The smooth toric case is studied and an interesting connection with K-semistability is also investigated.

Deformations of singular symplectic varieties and termination of the log minimal model program

Abstract: We generalize Huybrechts’ theorem on deformation equivalence of birational irreducible symplectic manifolds to the singular setting. More precisely, under suitable natural hypotheses, we show that two birational symplectic varieties are locally trivial deformations of one another. As an application we show the termination of any log-minimal model program for a pair (X, ∆) of a projective irreducible symplectic manifold X and an effective R-divisor ∆. To prove this result we follow Shokurov’s strategy and show that LSC and ACC for mlds hold for all the models appearing along any log-MMP of the initial pair.

Berkovich Skeleta and Birational Geometry

Abstract: We give a survey of joint work with Mircea Mustaţa and Chenyang Xu on the connections between the geometry of Berkovich spaces over the field of Laurent series and the birational geometry of one-parameter degenerations of smooth projective varieties. The central objects in our theory are the weight function and the essential skeleton of the degeneration. We tried to keep the text self-contained, so that it can serve as an introduction to Berkovich geometry for birational geometers.

Non-rigid quartic 3-folds

Abstract: Let X⊂P4 be a terminal factorial quartic 3-fold If X is non-singular, X is birationally rigid, ie the classical minimal model program on any terminal Q-factorial projective variety Z birational to X always terminates with X This no longer holds when X is singular, but very few examples of non-rigid factorial quartics are known In this article, we first bound the local analytic type of singularities that may occur on a terminal factorial quartic hypersurface X⊂P4 A singular point on such a hypersurface is of type cAn (n ≥ 1), or of type cDm (m ≥ 4) or of type cE6, cE7 or cE8 We first show that if (P 2 X) is of type cAn, n is at most 7 and, if (PϵX) is of type cDm, m is at most 8 We then construct examples of non-rigid factorial quartic hypersurfaces whose singular loci consist (a) of a single point of type cAn for 2≤n≤7, (b) of a single point of type cDm for m = 4 or 5 and (c) of a single point of type cEk for k = 6, 7 or 8

Abundance for varieties with many differential forms

Abstract: We prove that the abundance conjecture holds on a variety $X$ with mild singularities if $X$ has many reflexive differential forms with coefficients in pluricanonical bundles, assuming the Minimal Model Program in lower dimensions. This implies, for instance, that under this condition, hermitian semipositive canonical divisors are almost always semiample, and that klt pairs whose underlying variety is uniruled have good models in many circumstances. When the numerical dimension of $K_X$ is $1$, our results hold unconditionally in every dimension. We also treat a related problem on the semiampleness of nef line bundles on Calabi-Yau varieties.

$\mathbb{Q}$-Gorenstein smoothings of surfaces and degenerations of curves

Abstract: In this paper we mainly describe $\mathbb{Q}$-Gorenstein smoothings of projective surfaces with only Wahl singularities which have birational fibers. For instance, these degenerations appear in normal degenerations of the projective plane, and in boundary divisors of the KSBA compactification of the moduli space of surfaces of general type [KSB88]. We give an explicit description of them as smooth deformations plus 3-fold birational operations, through the flips and divisorial contractions in [HTU13]. We interpret the continuous part (smooth deformations) as degenerations of certain curves in the general fiber. At the end, we work out examples happening in the KSBA boundary for invariants $K^2=1$, $p_g=0$, and $\pi_1=0$ using plane curves.

Finite generation and geography of models

Abstract: There are two main examples where a version of the Minimal Model Program can, at least conjecturally, be performed successfully: the first is the classical MMP associated to the canonical divisor, and the other is Mori Dream Spaces. In this paper we formulate a framework which generalises both of these examples. Starting from divisorial rings which are finitely generated, we determine precisely when we can run the MMP, and we show why finite generation alone is not sufficient to make the MMP work.

The LMMP for log canonical 3-folds in char $p$

Abstract: We prove that one can run the log minimal model program for log canonical $3$-fold pairs in characteristic $p>5$. In particular we prove the Cone Theorem, Contraction Theorem, the existence of flips and the existence of log minimal models for pairs with log divisor numerically equivalent to an effective divisor. These follow from our main results, which are that certain log minimal models are good.

Smoothly embedded rational homology balls

Abstract: In this paper we prove the existence of rational homology balls smoothly embedded in regular neighborhoods of certain linear chains of smooth $2$-spheres by using techniques from minimal model program for 3-dimensional complex algebraic variety.

On log canonical rational singularities

Abstract: We prove that the class of log canonical rational singularities is closed under the basic operations of the minimal model program. We also give some supplementary results on the minimal model program for log canonical surfaces.

On (2, 4) complete intersection threefolds that contain an Enriques surface

Abstract: We study nodal complete intersection threefolds of type (2, 4) in $${\mathbb {P}}^5$$ which contain an Enriques surface in its Fano embedding. We completely determine Calabi–Yau birational models of a generic such threefold. These models have Hodge numbers $$h^{11}=2,h^{12}=32$$ . We also describe Calabi–Yau varieties with Hodge numbers $$(h^{11},h^{12})$$ equal to (2, 26), (23, 5) and (31, 1). The last two pairs of Hodge numbers are, to the best of our knowledge, new.

Aspects of the Geometry of Varieties with Canonical Singularities

Abstract: This survey reports on recent developments regarding the global structure of complex varieties which occur in the minimal model program.

Local fano-mori contractions of high nef-value

Abstract: Let X be a variety with at most terminal Q-factorial singularities of dimension n. We study local contractions f : X ! Z supported by a Q- Cartier divisor of the type KX + �L, where L is an f-ample Cartier divisor and � � 0 is a rational number. Equivalently, f is a Fano-Mori contraction associated to an extremal face in NE(X) KX+�L=0 ; these maps naturally arise in the context of the minimal model program. We prove that, if � > (n 3) > 0, the general element X ' 2 |L| is a variety with at most terminal singularities. We apply this to characterize, via an inductive argument, some birational contractions as above with � > (n 3) � 0.

Remarks on the abundance conjecture

Abstract: We prove the abundance theorem for log canonical $n$-folds such that the boundary divisor is big assuming the abundance conjecture for log canonical $(n-1)$-folds. We also discuss the log minimal model program for log canonical 4-folds.

Building blocks of polarized endomorphisms of normal projective varieties

Abstract: An endomorphism $f$ of a projective variety X is polarized (resp. quasi-polarized) if $f^*H$ is linearly equivalent to $qH$ for some ample (resp. nef and big) Cartier divisor $H$ and integer $q > 1$. First, we use cone analysis to show that a quasi-polarized endomorphism is always polarized, and the polarized property descends via any equivariant dominant rational map. Next, we show that a suitable maximal rationally connected fibration (MRC) can be made $f$-equivariant using a construction of N. Nakayama, that $f$ descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi-etale quotient of an abelian variety). Finally, we show that we can run the minimal model program (MMP) $f$-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one. As a consequence, the building blocks of polarized endomorphisms are those of Q-abelian varieties and those of Fano varieties of Picard number one. Along the way, we show that $f$ always descends to a polarized endomorphism of the Albanese variety Alb(X) of X, and that the pullback of a power of $f$ acts as a scalar multiplication on the Neron-Severi group of X (modulo torsion) when X is smooth and rationally connected. Partial answers about X being of Calabi-Yau type, or Fano type are also given with an extra primitivity assumption on $f$ which seems necessary by an example.

Notes on Kähler-Ricci Flow

Abstract: In these notes some aspects of the Analytic Minimal Model Program through Kahler-Ricci flow which was initiated by J. Song and the author are discussed. Some open problems will be also presented.

A bound on embedding dimensions of geometric generic fibers

Abstract: The author finds a limit on the singularities that arise in geometric generic fibers of morphisms between smooth varieties of positive characteristic by studying changes in embedding dimension under inseparable field extensions. This result is then used in the context of the minimal model program to rule out the existence of smooth varieties fibered by certain nonnormal del Pezzo surfaces over bases of small dimension.

The Log Minimal Model Program for horospherical varieties via moment polytopes

Abstract: In a previous work, we described the Minimal Model Program in the family of $\Qbb$-Gorenstein projective horospherical varieties, by studying certain continuous changes of moment polytopes of polarized horospherical varieties. Here, we summarize the results of the previous work and we explain how to generalize them in order to describe the Log Minimal Model Program for pairs $(X,\D)$ when $X$ is a projective horospherical variety.

Some remarks on log surfaces

Abstract: Fujino and Tanaka established the minimal model theory for $\mathbb Q$-factorial log surfaces in characteristic $0$ and $p$, respectively. We prove that every intermediate surface has only log terminal singularities if we run the minimal model program starting with a pair consisting of a smooth surface and a boundary $\mathbb R$-divisor. We further show that such a property does not hold if the initial surface is singular.

Betti numbers in three dimensional minimal model program

Abstract: Let $X$ be a smooth projective threefold. We prove that, among the process of minimal model program, the variance of the third Betti number can be bounded by some integer depends only on the Picard number of $X$.

Log canonical models of elliptic surfaces

Abstract: We give a classification of the log canonical models of elliptic surface pairs consisting of an elliptic fibration, a section, and a weighted sum of marked fibers In particular, we show how the log canonical models depend on the choice of the weights We describe a wall and chamber decomposition of the space of weights based on how the log canonical model changes In addition, we give a generalized formula for the canonical bundle of an elliptic surface with section and marked fibers This is the first step in constructing compactifcations of moduli spaces of elliptic surfaces using the minimal model program

Analytical log minimal model program via conical Kähler Ricci flow : Song-Tian program

Abstract: Existence of canonical metric on a projective variety was a long standing conjecture and the major part of this conjecture is about varieties which do not have definite first Chern class(most of the manifolds do not have definite first Chern class). Thereis a program which is known as SongTian program for finding canonical metric on canonical model of a projective variety by using Minimal Model Program. The main aim of this thesis is better undrestanding of SongTian program on pair (X;D). In this thesis, we apply SongTian program for pair (X;D) via Log Minimal Model Program where D is a simple normal crossing divisor on X with conic singularities. We investigate conical Kahler Ricci flow on holomorphic fiber spaces (X;D) -→B whose generic fibers are log Calabi Yau pairs (Xs;Ds), c1(KB) 0 also). We show that there is a unique conical Kahler Einstein metric on (X;D) which is twisted by logarithmic Weil Petersson metric and an additional term which we will find it explicitly. We consider the semipositivity of fiberwise singular Kahler Einstein metric via SongTian program. We consider a twisted Kahler Einstein metric along Mori fibre space. Moreover, we give an analogue version of SongTian program for Sasakian manifolds. We give an arithmetic version of SongTian program for arithmetic varieties. Also we give a short proof of Tian’s formula for Kahler potential of logarithmic WeilPetersson metric on moduli space of log CalabiYau varieties (if such moduli space exists!).