Miguel A. Barja
We study lower bounds for the self-intersection of the canonical divisor of "canonical varieties" (i.e. varieties whose canonical linear system gives a birational map). We give some improvements for the known results in the case of surfaces and new bounds for the case of canonical threefolds.
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We totally classify the projective toric varieties whose canonical divisors are divisible by their dimensions. In Appendix, we show that Reid's toric Mori theory implies Mabuchi's characterization of the projective space for toric varieties.
Shokurov's ACC Conjecture says that the set of all log canonical thresholds on varieties of bounded dimension satisfies the Ascending Chain Condition. This conjecture was proved for log canonical thresholds on smooth varieties in [EM1]. Here we use this result and inversion of adjunction to establish the conjecture for locally complete intersection varieties.
Building on results of Kollár, we prove Shokurov's ACC Conjecture for log canonical thresholds on smooth varieties, and more generally, on varieties with quotient singularities.
In this paper we study certain homogeneous spaces, which we call generalised affine Stiefel algebraic varieties. The main aim is to characterise the canonical divisor of generalised affine Stiefel algebraic varieties in terms of group representations. Affine Stiefel algebraic varieties and in particular $S^{n}$ are two special cases of the generalised affine Stiefel algebraic varieties.
This is an expository article on the recent developments of Hodge theory on moduli spaces of smooth projective varieties with semi-ample canonical line bundles.
In this paper, we generalise the theory of complements to log canonical log fano varieties and prove boundedness of complements for them in dimension less than or equal to 3. We also prove some boundedness results for the canonical index of sdlt log Calabi-Yau varieties in dimension 2.
We prove a more general and precise version of the Noether-Fano inequalities for birational maps between Mori fiber spaces. This is applied to give descriptions of global canonical thresholds on Fano varieties of Picard number one.
In the present paper, we study the (twisted) 3-canonical map of varieties of Albanese fiber dimension one. Based on a theorem about the regularity of direct image of canonical sheaves, we prove that the 3-canonical map is generically birational when the genus of a general fiber of the Albanese map is 2.
We study the anti-canonical ring of a projective variety and we characterise varieties of log Fano type depending on the singularities of these models.
Let A be a central division algebra over a field F with ind A = n. In computing canonical p-dimension of projective PGL_1(A)-homogeneous varieties, for p prime, we can reduce to the case of generalized Severi-Brauer varieties X_e(A) with ind A a power of p divisible by e. We prove that canonical 2-dimension (and hence canonical dimension) equals dimension for all X_e(A) with ind A = 2e a power of 2.
We give a sufficient condition for the termination of flips. Then we discuss a semi-stable minimal model program for varieties with (numerically) trivial canonical divisor as an application. We also treat a slight refinement of dlt blow-ups.
This paper aims to study canonical pencils of higher dimensional projective varieties. It seems that the geometric genus of the general fibre for the derived fibration from the canonical pencil for a 3-fold of general type does not have an upper bound.
In this paper we show that any smoothable complex projective variety, smooth in codimension two, with klt singularities and numerically trivial canonical class admits a finite cover, étale in codimension one, that decomposes as a product of an abelian variety, and singular analogues of irreducible Calabi-Yau and irreducible symplectic varieties.
Shokurov conjectured that the set of all log canonical thresholds on varieties of bounded dimension satisfies the ascending chain condition. In this paper we prove that the conjecture holds for log canonical thresholds on smooth varieties and, more generally, on locally complete intersection varieties and on varieties with quotient singularities.
A cheap method for constructing canonical models and complete moduli for complex projective varieties with a structure called "rational plurifibration" is given. A result about semistable reduction (whose nature is slightly different from alg-geom/9707012) is deduced. The method involves recursive application of moduli stacks of twisted stable maps (math.AG/9908167) and a systematic circumvention of any difficult MMP issues.
In this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle. As one of the consequences of the main theorem, we give bounds on very ampleness and projective normality of pluricanonical linear systems on varieties of general type in dimensions three, four and five. These improve known such results.
This paper gives the all possible global indices of log Calabi-Yau 3-folds with standard coefficients on the boundaries and having lc, non-klt singularities. This follows easily from the discussion in the paper: The indices of log canonical singularities by Fujino (AG/9909035).
We prove that smooth, projective, $K$-trivial, weakly ordinary varieties over a perfect field of characteristic $p>0$ are not geometrically uniruled. We also show a singular version of our theorem, which is sharp in multiple aspects. Our work, together with Langer's results, implies that varieties of the above type have strongly semistable tangent bundles with respect to any polarization.
We answer an open question concerning the boundedness of canonical fiber spaces in high dimensions and prove the following: for any set of integers $n\geq 3$, $0<d<n$ and $N>0$, there exists a nonsingular projective $n$-fold $X$ of general type so that $X$ is canonically fibred by $d$-dimensional varieties $F$ with $p_g(F)\geq N$. This disproves the desired boundedness parallel to Beauville's boundedness theorem in the surface case.
We prove that the canonical ring of a canonical variety in the sense of de Fernex and Hacon is finitely generated. We prove that canonical varieties are klt if and only if R(-K_X) is finitely generated. We introduce a notion of nefness for non-Q-Gorenstein varieties and study some of its properties. We then focus on these properties for non-Q-Gorenstein toric varieties.
We show that if X is a nonsingular projective variety of general type over an algebraically closed field k of positive characteristic and X has maximal Albanese dimension and the Albanese map is separable, then |4K_X| induces a birational map.
On the global and local geometry of quasi-$F$-split varieties with trivial canonical bundle
研究具有平凡典范丛的拟\(F -\)分裂簇，通过多种方法证明正则拟\(F^{\infty}-\)分裂簇性质、解答相关曲面问题、证明正规簇几何正规性，以及在奇点解消假设下证明特定拟\(F^e -\)纯正规簇是对数典范的。
On the canonical bundle formula and effective birationality for Fano varieties in char $p>0$
本文研究了正特征下 Fano 型簇的双有理几何与有界性问题，证明了有界性在正规化下不变，给出了更易用的典范丛公式，并证实了具有良好奇点的弱 Fano 簇的有效双有理性。
In this article we study pluriregular varieties X of general type with base-point-free canonical bundle whose canonical morphism has degree 3 and maps X onto a variety of minimal degree Y. We carry out our study from two different perspectives. First we study in Section 2 the canonical ring of X describing completely the degrees of its minimal generators. We apply this to the study of the projective normality of the images of the pluricanonical morphisms of X. Our study of the canonical ring of $X$ also shows that, if the dimension of X is greater than or equal to 3, there does not exist a con…
We prove that the anti-canonical divisors of weak Fano 3-folds with log canonical singularities are semiample. Moreover, we consider semiampleness of the anti-log canonical divisor of any weak log Fano pair with log canonical singularities. We show semiampleness dose not hold in general by constructing several examples. Based on those examples, we propose sufficient conditions which seem to be the best possible and we prove semiampleness under such conditions. In particular we derive semiampleness of the anti-canonical divisors of log canonical weak Fano 4-folds whose lc centers are at most 1-…
Varieties with pseudoeffective canonical divisor and their Kodaira dimension
本文研究复数域上光滑射影簇，围绕Shigefumi Mori提出的“簇为单直纹的当且仅当其小平维数为负”的猜想展开探讨，属于代数几何领域的理论研究。
Let $k$ be an $F$-finite field containing an infinite perfect field of positive characteristic. Let $(X, Δ)$ be a projective log canonical pair over $k$. In this note we show that, for a semi-ample divisor $D$ on $X$, there exists an effective $\mathbb{Q}$-divisor $Δ' \sim_{\mathbb Q} Δ+D$ such that $(X, Δ')$ is log canonical if there exists a log resolution of $(X, Δ)$.
From a block-diagonal $(n+1) \times (m+1) \times (m+1)$ tensor symmetric in the last two entries one obtains two varieties: an intersection of symmetric determinantal hypersurfaces $X$ in $n$-dimensional projective space, and an intersection of quadrics $\mathfrak{C}$ in $m$-dimensional projective space. Under mild technical assumptions, we characterize the accidental singularities of $X$ in terms of $\mathfrak{C}$. We apply our methods to algebraic curves and show how to construct theta characteristics of certain canonical curves of genera 3, 4, and 5, generalizing a classical construction of…
Stability of the canonical extension of tangent bundles on Picard-rank-1 Fano varieties
本文研究Picard数为1的Fano簇上切丛典范扩张的斜率稳定性，证明在指数整除维数或维数加一时切丛稳定性蕴含典范扩张的（半）稳定性，并给出其在曲线稳定向量丛模空间上的应用推论。
We prove a Fujita-type theorem for varieties with numerically trivial canonical bundle using properties of semihomogeneous bundles on abelian varieties. We combine our results with work of Riess on compact hyperkähler manifolds and work of Mukai, Pareschi and Yoshioka to obtain effective global generation statements for certain moduli spaces of sheaves on abelian surfaces. Among these is the statment that if $\cL$ is an ample line bundle on the Hilbert square $S^{[2]}$ of an abelian surface $S,$ then $\cL^{\otimes m}$ is globally generated for $m \geq 3.$
Semi-log canonical varieties are a higher-dimensional analogue of stable curves. They are the varieties appearing as the boundary $Δ$ of a log canonical pair $(X,Δ)$, and also appear as limits of canonically polarized varieties in moduli theory. For certain three-fold pairs $(X,Δ)$ we show how to compute the PL homeomorphism type of the dual complex of a dlt minimal model directly from the normalization data of $Δ$.
Generalising toric geometry we study compact varieties admitting lower dimensional torus actions. In particular we describe divisors on them in terms of convex geometry and give a criterion for their ampleness. These results may be used to study Fano varieties with small torus actions. As a first result we classify log del Pezzo C*-surfaces of Picard number 1 and Gorenstein index less than 4. In further examples we show how classification might work in higher dimensions and we give explicit descriptions of some equivariant smoothings of Fano threefolds.
We show that the family of semi log canonical pairs with ample log canonical class and with fixed volume is bounded.
Weakly special test configurations of log canonical Fano varieties
该研究针对严格对数典范Fano簇，证明补的所有lc位置均为dreamy型，并建立了(X,-K_X)的弱特殊测试构型与补的lc位置的对应关系，丰富了对数典范Fano簇的相关理论。
Given a natural number $l$ and a weak Fano $n$-fold $X$ with $\operatorname{dim}\overline{φ_{-lK_X}(X)}\geq n-1$, we study the lower bound of the anti-canonical volume and the upper bound of the anti-canonical stability index. The method can also be used to give similar bounds for polarised Calabi-Yau varieties.
We show that a large class of maximally degenerating families of n-dimensional polarized varieties come with a canonical basis of sections of powers of the ample line bundle. The families considered are obtained by smoothing a reducible union of toric varieties governed by a wall structure on a real n-(pseudo-)manifold. Wall structures have previously been constructed inductively for cases with locally rigid singularities and by Gromov-Witten theory for mirrors of log Calabi-Yau surfaces and K3 surfaces by various combinations of the authors. For trivial wall structures on the n-torus we retri…
We study log canonical thresholds (also called global log canonical threshold or $α$-invariant) of $\mathbb{R}$-linear systems. We prove existence of positive lower bounds in different settings, in particular, proving a conjecture of Ambro. We then show that the Borisov-Alexeev-Borisov conjecture holds, that is, given a natural number $d$ and a positive real number $ε$, the set of Fano varieties of dimension $d$ with $ε$-log canonical singularities forms a bounded family. This implies that birational automorphism groups of rationally connected varieties are Jordan which in particular answers a…
We obtain new results on the geometry of Hilbert modular varieties in positive characteristic and morphisms between them. Using these results and methods of rigid geometry, we develop a theory of canonical subgroups for abelian varieties with real multiplication.
We show the existence of canonical heights of subvarieties for bounded sequences of morphisms and give some applications.
In 2002, Alexeev provided a modular interpretation for the toroidal compactification of $A_g$ for the 2nd Voronoi fan. In this paper we show that pairs $(X,Θ)$ in the boundary of this moduli space are semi-log canonical, the analog of log canonical in the non-normal setting. This extends the result of Koll${á}$r stating that principally polarized abelian pairs $(X,Θ)$ are log canonical.
Given a p-form defined on the smooth locus of a normal variety, and a resolution of singularities, we study the problem of extending the pull-back of the p-form over the exceptional set of the desingularization. For log canonical pairs and for certain values of p, we show that an extension always exists, possibly with logarithmic poles along the exceptional set. As a corollary, it is shown that sheaves of reflexive differentials enjoy good pull-back properties. A natural generalization of the well-known Bogomolov-Sommese vanishing theorem to log canonical threefold pairs follows.
Let $(X,B)$ be a log canonical pair over a normal variety $Z$ with maximal Albanese dimension. If $K_X+B$ is relatively abundant over $Z$ (for example, $K_X+B$ is relatively big over $Z$), then we prove that $K_X+B$ is abundant. In particular, the subadditvity of Kodaira dimensions $κ(K_X+B) \geq κ(K_F+B_F)+ κ(Z)$ holds, where $F$ is a general fiber, $K_F+B_F= (K_X+B)|_F$, and $κ(Z)$ means the Kodaira dimension of a smooth model of $Z$. We discuss several variants of this result in Section 4. We also give a remark on the log Iitaka conjecture for log canonical pairs in Section 5.
We show that the log canonical threshold of a generic determinantal variety and its generic link are the same.
We give a necessary and sufficient condition for the canonical divisor to vanish on a quasi-homogeneous affine algebraic variety.
Classical Serre-Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips its local moduli space with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus. In this paper, we construct canonical liftings modulo $p^2$ of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Fr…
We consider pairs (X,A), where X is a variety with klt singularities and A is a formal product of ideals on X with exponents in a fixed set that satisfies the Descending Chain Condition. We also assume that X has (formally) bounded singularities, in the sense that it is, formally locally, a subvariety in a fixed affine space defined by equations of bounded degree. We prove in this context a conjecture of Shokurov, predicting that the set of log canonical thresholds for such pairs satisfies the Ascending Chain Condition.
We prove the existence of integral canonical models of unitary Shimura varieties in arbitrary unramified mixed characteristic. Errata to [Va1] are also included.
We prove that a Fano variety (with arbitrary singularities) of dimension $n$ in positive characteristic is isomorphic to $\mathbb{P}^n$ if the Seshadri constant of the anti-canonical divisor at some smooth point is greater than $n$ and classify Fano varieties whose anti-canonical divisors have Seshadri constants $n$. In characteristic $p>5$ and dimension $3$, we also show that Fano varieties $X$ with Seshadri constants $ε(-K_X,x)>2+ε$ at some smooth point $x\in X$ (for some fixed $ε>0$) have bounded anti-canonical degrees.
We describe a new approach to the definition of the moduli functor of stable varieties. While there is wide agreement as to what classes of varieties should appear, the notion of a family of stable surfaces is quite subtle, as key numerical invariants may fail to be constant in flat families. Our approach is to add natural stack structure to stable varieties. For example, given a canonical model we take the global-quotient stack structure arising from its realization as Proj of the canonical ring. Deformations of the stack structure preserve key numerical invariants of the stable variety, incl…
In this paper, we formulate the geometric Bogomolov conjecture for abelian varieties, and give some partial answers to it. In fact, we insist in a main theorem that under some degeneracy condition, a closed subvariety of an abelian variety does not have a dense subset of small points if it is a non-special subvariety. The key of the proof is the study of the minimal dimension of the components of a canonical measure on the tropicalization of the closed subvariety. Then we can apply the tropical version of equidistribution theory due to Gubler. This article includes an appendix by Walter Gubler…
Principal bundles on toroidal compactifications of integral canonical models of abelian-type Shimura varieties
本文研究了在积分典范模型的托里奥利紧致化上，主$\mathcal{G}^c$-和$M^c$-丛的典范扩展构造方法。
Optimal upper bounds for anti-canonical volumes of singular toric Fano varieties
本文给出了d维1/q-lc环面Fano簇反典范体积的最优上界，并构造例子证明最优性，同时给出恰含一个内部格点的d维格单纯形体积的最优上界。
In this paper we prove that for any smooth projective variety of Albanese fiber dimension two and of general type, the 6-canonical map is birational. And we also show that the 5-canonical map is birational for any such variety with some geometric restrictions.
We prove the existence of integral canonical models of Shimura varieties of preabelian type with respect to primes of characteristic at least 5.
We give the first examples of smooth Fano and Calabi-Yau varieties violating the (narrow) canonical strip hypothesis, which concerns the location of the roots of Hilbert polynomials of polarised varieties. They are given by moduli spaces of rank 2 bundles with fixed odd-degree determinant on curves of sufficiently high genus, hence our Fano examples have Picard rank 1, index 2, are rational, and have moduli. The hypotheses also fail for several other closely related varieties.
In this article we develop a new way of systematically constructing infinitely many families of smooth subvarieties $X$ of any given dimension $m$, $m \geq 3$, and any given codimension in $\mathbb P^N$, embedded by complete subcanonical linear series, and, in particular, in the range of Hartshorne's conjecture. We accomplish this by showing the existence of everywhere non--reduced schemes called ropes, embedded in $\mathbb P^N$, and by smoothing them. In the range $3 \leq m < N/2$, we construct smooth subvarieties, embedded by complete subcanonical linear series, that are not complete interse…
This is a survey of the three main methods developed in the last 15 years to prove the existence of integral canonical models of Shimura varieties of Hodge type. The only new part is formed by corrections to results of Kisin.
A widely believed conjecture predicts that curves of bounded geometric genus lying on a variety of general type form a bounded family. One may even ask whether the canonical degree of a curve $C$ in a variety of general type is bounded from above by some expression $aχ(C)+b$, where $a$ and $b$ are positive constants, with the possible exceptions corresponding to curves lying in a strict closed subset (depending on $a$ and $b$). A theorem of Miyaoka proves this for smooth curves in minimal surfaces, with $a>3/2$. A conjecture of Vojta claims in essence that any constant $a>1$ is possible provid…
Relations among fundamental invariants play an important role in algebraic geometry. It is known that an $n$-dimensional variety of general type with nef canonical divisor and canonical singularities, whose image $Y$ under the canonical map is of maximal dimension, satisfies $K_X^n \ge 2 (p_g-n)$. We investigate the very interesting extremal situation $K_X^n=2(p_g-n)$, which appears in a number of geometric situations. Since these extremal varieties are natural higher dimensional analogues of Horikawa surfaces, we name them Horikawa varieties. These varieties have been previously dealt with in…
In this note we show that the apolar cubic forms associated to codimension two linear sections of canonical curves of genus at least eleven are special with respect to their presentation as sums of cubes.
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