Ample Divisor Research Articles

On the degeneracy of integral points and entire curves in the complement of nef effective divisors

As a consequence of the divisorial case of our recently established generalization of Schmidt's subspace theorem, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel aspect of our result is the attainment of a strong degeneracy conclusion (arithmetic quasi-hyperbolicity) under weak positivity assumptions on the divisors. The proof hinges on applying our recent theorem with a well-situated ample divisor realizing a certain lexicographical minimax. We also explore the connections with earlier work by other authors and make a conjecture regarding bounds for the numbers of divisors necessary, including consideration of the question of arithmetic hyperbolicity. Under the standard correspondence between statements in Diophantine approximation and Nevanlinna theory, one obtains analogous degeneration statements for entire curves.

On Fujita's freeness conjecture in dimension 5

Let X be a smooth projective variety of dimension 5 and A an integral ample divisor on X. We show that |KX+kA| is basepoint-free for any integer k≥6.

On sharp rates and analytic compactifications of asymptotically conical Kähler metrics

Let X be a complex manifold, and let S↪X be an embedding of a complex submanifold. Assuming that the embedding is (k−1)-linearizable or (k−1)-comfortably embedded, we construct via the deformation to the normal cone a diffeomorphism F from a small neighborhood of the zero section in the normal bundle NS to a small neighborhood of S in X such that F is in a precise sense holomorphic up to the (k−1)th order. Using this F, we obtain optimal estimates on asymptotic rates for asymptotically conical (AC) Calabi–Yau (CY) metrics constructed by Tian and Yau. Furthermore, when S is an ample divisor satisfying an appropriate cohomological condition, we relate the order of comfortable embedding to the weight of the deformation of the normal isolated cone singularity arising from the deformation to the normal cone. We also give an example showing that the condition of comfortable embedding depends on the splitting liftings. We then prove an analytic compactification result for the deformation of the complex structure on an affine cone that decays to any positive order at infinity. This can be seen as an analytic counterpart of Pinkham’s result on deformations of cone singularities with negative weights.

Symplectic cohomology rings of affine varieties in the topological limit

We construct a multiplicative spectral sequence converging to the symplectic cohomology ring of any affine variety X, with first page built out of topological invariants associated to strata of any fixed normal crossings compactification $$(M,{\mathbf {D}})$$ of X. We exhibit a broad class of pairs $$(M,{\mathbf {D}})$$ (characterized by the absence of relative holomorphic spheres or vanishing of certain relative GW invariants) for which the spectral sequence degenerates, and a broad subclass of pairs (similarly characterized) for which the ring structure on symplectic cohomology can also be described topologically. Sample applications include: (a) a complete topological description of the symplectic cohomology ring of the complement, in any projective M, of the union of sufficiently many generic ample divisors whose homology classes span a rank one subspace, (b) complete additive and partial multiplicative computations of degree zero symplectic cohomology rings of many log Calabi-Yau varieties, and (c) a proof in many cases that symplectic cohomology is finitely generated as a ring. A key technical ingredient in our results is a logarithmic version of the PSS morphism, introduced in our earlier work Ganatra and Pomerleano, arXiv:1611.06849.

Triviality Properties of Principal Bundles on Singular Curves. II

AbstractFor $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\rightarrow S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D\subset X$. We show, by constructing explicit examples, that the obstruction is nontrivial if $G$ is not simply connected, but it can be made to vanish by a faithfully flat base change, if $S$ is the spectrum of a dvr (and some other hypotheses). The vanishing of this obstruction is shown to be a sufficient condition for étale local triviality if $S$ is a smooth curve, and the singular locus of $X-D$ is finite over $S$.

On upper bounds of arithmetic degrees

Let $X$ be a smooth projective variety over $ \overline{\mathbb Q}$, and $f:X -rightarrow X$ be a dominant rational map. Let $\delta_{f}$ be the first dynamical degree of $f$ and $h_{X}:X( \overline{\mathbb Q})\to [1,\infty)$ be a Weil height function on $X$ associated with an ample divisor on $X$. We prove several inequalities which give upper bounds of the sequence $(h_X (f^n(P)))_{n\geq0}$ where $P$ is a point of $X( \overline{\mathbb Q})$ whose forward orbit by $f$ is well-defined. As a corollary, we prove that the upper arithmetic degree is less than or equal to the first dynamical degree; $ \overline{\alpha}_{f}(P) \leq \delta_{f}$. Furthermore, if the Picard number of $X$ is one, $f$ is algebraically stable and $\delta_{f}>1$, we prove that the limit defining canonical height $\lim_{n\to \infty} h_{X} (f^{n}(P)) \big/ \delta_f^n$ converges.

The Motive of a Smooth Theta Divisor

We prove a motivic version of the Lefschetz hyperplane theorem for a smooth ample divisor $\Theta$ on an Abelian variety. We use this to construct a motive $P$ that realizes the primitive cohomology of $\Theta$.

Divisors on matroids and their volumes

The classical volume polynomial in algebraic geometry measures the degrees of ample (and nef) divisors on a smooth projective variety. We introduce an analogous volume polynomial for matroids, and give a complete combinatorial formula. For a realizable matroid, we thus obtain an explicit formula for the classical volume polynomial of the associated wonderful compactification. We then introduce a new invariant called the shifted rank volume of a matroid as a particular specialization of its volume polynomial, and discuss its algebro-geometric and combinatorial properties in connection to linear series on blow-ups of projective spaces.

Logarithmic vanishing theorems for effective q-ample divisors

Let $X$ be a compact K\"ahler manifold and $D$ be a simple normal crossing divisor. If $D$ is the support of some effective $k$-ample divisor, we show $$ H^q(X,\Omega^p_X(\log D))=0,\quad \text{for}\quad p+q>n+k.$$

Positivity functions for curves on algebraic varieties

This is the second part of our work on Zariski decomposition structures, where we compare two different volume type functions for curve classes. The first function is the polar transform of the volume for ample divisor classes. The second function captures the asymptotic geometry of curves analogously to the volume function for divisors. We prove that the two functions coincide, generalizing Zariski's classical result for surfaces to all varieties. Our result confirms the log concavity conjecture of the first named author for weighted mobility of curve classes in an unexpected way, via Legendre-Fenchel type transforms. We also give a number of applications to birational geometry, including a refined structure theorem for the movable cone of curves.

Polarized endomorphisms of normal projective threefolds in arbitrary characteristic

Let $X$ be a projective variety over an algebraically closed field $k$ of arbitrary characteristic $p \ge 0$. A surjective endomorphism $f$ of $X$ is $q$-polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ and integer $q > 1$. Suppose $f$ is separable and $X$ is $\mathbb{Q}$-Gorenstein and normal. We show that the anti-canonical divisor $-K_X$ is numerically equivalent to an effective $\mathbb{Q}$-Cartier divisor, strengthening slightly the conclusion of Boucksom, de Fernex and Favre (Theorem C) and also covering singular varieties over an algebraically closed field of arbitrary characteristic. Suppose $f$ is separable and $X$ is normal. We show that the Albanese morphism of $X$ is an algebraic fibre space and $f$ induces polarized endomorphisms on the Albanese and also the Picard variety of $X$, and $K_X$ being pseudo-effective and $\mathbb{Q}$-Cartier means being a torsion $\mathbb{Q}$-divisor. Let $f^{Gal}:\overline{X}\to X$ be the Galois closure of $f$. We show that if $p>5$ and co-prime to $deg\, f^{Gal}$ then one can run the minimal model program (MMP) $f$-equivariantly, after replacing $f$ by a positive power, for a mildly singular threefold $X$ and reach a variety $Y$ with torsion canonical divisor (and also with $Y$ being a quasi-\'etale quotient of an abelian variety when $\dim(Y)\le 2$). Along the way, we show that a power of $f$ acts as a scalar multiplication on the Neron-Severi group of $X$ (modulo torsion) when $X$ is a smooth and rationally chain connected projective variety of dimension at most three.

On the existence of Ulrich bundles on blown-up varieties at a point

The objective is to show the construction of an Ulrich vector bundle on the blowing-up $\widetilde X$ of a nonsingular projective variety $X$ at a closed point, where the original variety is embedded by a very ample divisor $H$ and carries an Ulrich vector bundle. In order to achieve this result, we aim to find a suitable very ample divisor on $\widetilde X$, which is dependent on $H$. At the end, we take into consideration some applications to surfaces with regards to minimal models and their Kodaira dimension.

On Fano complete intersections in rational homogeneous varieties

Complete intersections inside rational homogeneous varieties provide interesting examples of Fano manifolds. For example, if $X = \cap_{i=1}^r D_i \subset G/P$ is a general complete intersection of $r$ ample divisors such that $K_{G/P}^* \otimes \mathcal{O}_{G/P}(-\sum_i D_i)$ is ample, then $X$ is Fano. We first classify these Fano complete intersections which are locally rigid. It turns out that most of them are hyperplane sections. We then classify general hyperplane sections which are quasi-homogeneous.

Building blocks of amplified endomorphisms of normal projective varieties

Let X be a normal projective variety. A surjective endomorphism $$f{:}X\rightarrow X$$ is int-amplified if $$f^*L - L =H$$ for some ample Cartier divisors L and H. This is a generalization of the so-called polarized endomorphism which requires that $$f^*H\sim qH$$ for some ample Cartier divisor H and $$q>1$$. We show that this generalization keeps all nice properties of the polarized case in terms of the singularity, canonical divisor, and equivariant minimal model program.

LINEAR SYSTEMS ON IRREGULAR VARIETIES

Let$X$be a normal complex projective variety,$T\subseteq X$a subvariety of dimension$m$(possibly$T=X$) and$a:X\rightarrow A$a morphism to an abelian variety such that$\text{Pic}^{0}(A)$injects into$\text{Pic}^{0}(T)$; let$L$be a line bundle on$X$and$\unicode[STIX]{x1D6FC}\in \text{Pic}^{0}(A)$a general element.We introduce two new ingredients for the study of linear systems on$X$. First of all, we show the existence of a factorization of the map$a$, called theeventual map of $L$ on $T$, which controls the behavior of the linear systems$|L\otimes \unicode[STIX]{x1D6FC}|_{|T}$, asymptotically with respect to the pullbacks to the connected étale covers$X^{(d)}\rightarrow X$induced by the$d$-th multiplication map of$A$.Second, we define the so-calledcontinuous rank function$x\mapsto h_{a}^{0}(X_{|T},L+xM)$, where$M$is the pullback of an ample divisor of$A$. This function extends to a continuous function of$x\in \mathbb{R}$, which is differentiable except possibly at countably many points; when$X=T$we compute the left derivative explicitly.As an application, we give quick short proofs of a wide range of newClifford–Severi inequalities, i.e., geographical bounds of the form$$\begin{eqnarray}\displaystyle \text{vol}_{X|T}(L)\geqslant C(m)h_{a}^{0}(X_{|T},L), & & \displaystyle \nonumber\end{eqnarray}$$where$C(m)={\mathcal{O}}(m!)$depends on several geometrical properties of$X$,$L$or$a$.

Triviality properties of principal bundles on singular curves

For $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\to S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D \subset X$. We show, by constructing explicit examples, that the obstruction is nontrivial if $G$ is not simply connected but it can be made to vanish, if $S$ is the spectrum of a dvr (and some other hypotheses), by a faithfully flat base change. The vanishing of this obstruction is shown to be a sufficient condition for etale local triviality if $S$ is a smooth curve, and the singular locus of $X-D$ is finite over $S$.

Lifting weighted blow-ups

Let f\colon X \to Z be a local, projective, divisorial contraction between normal varieties of dimension n with \mathbb Q -factorial singularities. Let Y \subset X be a f -ample Cartier divisor and assume that f_{|Y}\colon Y \to W has a structure of a weighted blow-up. We prove that f\colon X \to Z , as well, has a structure of weighted blow-up. As an application we consider a local projective contraction f\colon X \to Z from a variety X with terminal \mathbb Q -factorial singularities, which contracts a prime divisor E to an isolated \mathbb Q -factorial singularity P\in Z , such that -(K_X + (n-3)L) is f -ample, for a f -ample Cartier divisor L on X . We prove that (Z,P) is a hyperquotient singularity and f is a weighted blow-up.

Hilbert–Kunz density function and asymptotic Hilbert–Kunz multiplicity for projective toric varieties

For a toric pair (X,D), where X is a projective toric variety of dimension d−1≥1 and D is a very ample T-Cartier divisor, we show that the Hilbert–Kunz density function HKd(X,D)(λ) is the d−1 dimensional volume of P‾D∩{z=λ}, where P‾D⊂Rd is a compact d-dimensional set (which is a finite union of convex polytopes).We also show that, for k≥1, the function HKd(X,kD) can be replaced by another compactly supported continuous function φkD which is ‘linear in k’. This gives the formula for the associated coordinate ring (R,m):limk→∞⁡eHK(R,mk)−e0(R,mk)/d!kd−1=e0(R,m)(d−1)!∫0∞φD(λ)dλ, where φD (see Proposition 1.2) is solely determined by the shape of the polytope PD, associated to the toric pair (X,D). Moreover φD is a multiplicative function for Segre products.This yields explicit computation of φD (and hence the limit), for smooth Fano toric surfaces with respect to anticanonical divisor. In general, due to this formulation in terms of the polytope PD, one can explicitly compute the limit for two dimensional toric pairs and their Segre products.We further show that (Theorem 6.3) the renormalized limit takes the minimum value if and only if the polytope PDtiles the spaceMR=Rd−1 (with the lattice M=Zd−1). As a consequence, one gets an algebraic formulation of the tiling property of any rational convex polytope.

On a motivic interpretation of primitive, variable and fixed cohomology

AbstractIf M is a smooth projective variety whose motive is Kimura finite‐dimensional and for which the standard Lefschetz Conjecture B holds, then the motive of M splits off a primitive motive whose cohomology is the primitive cohomology. Under the same hypotheses on M, let X be a smooth complete intersection of ample divisors within M. Then the motive of X is the sum of a variable and a fixed motive inducing the corresponding splitting in cohomology. I also give variants with group actions.

Semistability of Frobenius Direct Image of Representations of Cotangent Bundles

Let k be an algebraically closed field of characteristic p > 0, X a smooth projective variety over k with a fixed ample divisor H, FX : X → X the absolute Frobenius morphism on X. Let E be a rational GLn(k)-bundle on X, and ρ : GLn(k) → GLm(k) a rational GLn(k)-representation of degree at most d such that ρ maps the radical RGLn(k)) of GLn(k) into the radical R(GLm(k)) of GLm(k). We show that if \(F_X^{N*}(E)\) is semistable for some integer \(N \ge {\max {_{0 < r < m}}}(_r^m) \cdot {\log _p}(dr)\), then the induced rational GLm(k)-bundle E(GLm(k)) is semistable. As an application, if dimX = n, we get a sufficient condition for the semistability of Frobenius direct image \(F_{X*}(\rho*(\Omega_X^1))\), where \(\rho*(\Omega_X^1)\) is the vector bundle obtained from \(\Omega_X^1\) via the rational representation ρ.