Semistability and Hilbert–Kunz multiplicities for curves

Very ample line bundles on regular surfaces obtained by projection

Let pi :mathcal {X}rightarrow M be a holomorphic fibration with compact fibers and L a relatively ample line bundle over mathcal {X}. We obtain the asymptotic of the curvature of L^2-metric and Qullien metric on the direct image bundle pi _*(L^kotimes K_{mathcal {X}/M}) up to the lower order terms than k^{n-1}, for large k. As an application we prove that the analytic torsion tau _k(bar{partial }) satisfies partial bar{partial }log (tau _k(bar{partial }))^2=o(k^{n-1}), where n is the dimension of fibers.

Let X be a smooth n-fold, \(n\ge 3\), and \(\mathcal {L}\) a spanned and ample line bundle on X with \(h^1(\mathcal {O}_X)=0\). Let \(\mathcal {E}\) be a spanned rank r vector bundle on X with \(\det (\mathcal {E})\cong \mathcal {L}\) and no trivial factor. Here we prove that \(r\le h^0(\mathcal {L})-1\) and prove the existence of some \(\mathcal {E}\) for all r with \(n\le r \le h^0(\mathcal {L})-1\). In the case \(n=3\) we show that if \(h^0(\mathcal {L}) -r\) is very small (in terms of the t-connectedness of \((X,\mathcal {L})\)), then \(\mathcal {E}\) is obtained by a standard procedure from \(\mathcal {L}\). If \(h^1(\mathcal {O}_X)>0\) we prove the upper bound for r substituting the integer \(h^0(\mathcal {L})-2\) with an integer \(\rho (\mathcal {L})\) with \(h^0(\mathcal {L})-2+ \max \{0,2h^1(\mathcal {O}_X)-h^1(\mathcal {L})\}\le \rho (\mathcal {L})\le h^0(\mathcal {L})-2+2h^1(\mathcal {O}_X)\).

The strongest notion is k-jet ampleness; it implies k-very ampleness (cf. [BeSo2, Proposition 2.2]) which of course implies k-spannedness. For k = 0 or k = 1 all the three notions are equivalent and correspond to global generation resp. very ampleness. In this note we give criteria for k-jet ampleness of line bundles on abelian varieties. A naive way to obtain such a criterion is as follows: According to [BeSo2, Corollary 2.1] a tensor product of k very ample line bundles is always k-jet ample. Now on an abelian variety, by the generalization of Lefschetz’ classical theorem [LB,

We prove two new results for Seshadri constants on surfaces of general type. Let X be a surface of general type. In the first part, inspired by Bauer and Szemberg (Manuscripta Math 126(2):167–175, 2008), we list the possible values for the multi-point Seshadri constant $$\varepsilon (K_X,x_1,x_2,\ldots ,x_r)$$ when it lies between 0 and 1/r, where $$K_X$$ is the canonical line bundle on X. In the second part, we assume X of the form $$C \times C$$ , where C is a general smooth curve of genus $$g \geqslant 2$$ . Given such X and an ample line bundle L on X with some conditions on it, we show that the global Seshadri constant of L is a rational number.

In this paper, we study pushforwards of log pluricanonical bundles on projective log canonical pairs $(Y,\Delta)$ over the complex numbers, partially answering a Fujita-type conjecture due to Popa and Schnell in the log canonical setting. We show two effective global generation results. First, when $Y$ surjects onto a projective variety, we show a quadratic bound for generic generation for twists by big and nef line bundles. Second, when $Y$ is fibered over a smooth projective variety, we show a linear bound for twists by ample line bundles. These results additionally give effective non-vanishing statements. We also prove an effective weak positivity statement for log pluricanonical bundles in this setting, which may be of independent interest. In each context we indicate over which loci positivity holds. Finally, using the description of such loci, we show an effective vanishing theorem for pushforwards of certain log-sheaves under smooth morphisms.

Let ϵ be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n such that there exists a global section of ϵ whose zero locus Z is a smooth subvariety of dimension n-r ≥ 2 of X. Let H be an ample line bundle on X such that the restriction HZ of H to Z is very ample. Triplets (X, ϵ, H) with g(Z, HZ) = 3 are classified, where g(Z, HZ) is the sectional genus of (Z, HZ).

Let $${\mathcal{E}}$$ be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n such that there exists a global section of $${\mathcal{E}}$$ whose zero locus Z is a smooth subvariety of dimension n − r ≥ 3 of X. Let H be an ample line bundle on X such that its restriction H Z to Z is very ample. Triplets $$(X,{\mathcal{E}},H)$$ are classified under the assumption that (Z,H Z ) has a smooth bielliptic curve section of genus ≥ 3 with $$H^{n-r}c_r({\mathcal{E}}) \leq 8$$ .

In this paper, we prove the results announced in [L-R] for G = SLn. Let G be a simple algebraic group over the base field k. Let M be a maximal torus in G and B, a Borel subgroup, B ⊃ M. Let W be the Weyl group of G. For w ∈ W, let X(w) = \(\overline {BwB} \) (mod B) be the Schubert variety in G/B associated to w. Let L be an ample line bundle on G/B. We shall denote the restriction of L to X(w) also by just L. Let k[X(w)] = ⊙ H0(X(w),L). In this paper, we construct an algebra kq [X(w)] over k(q), where q is a parameter taking values in k*, as a quantization of k[X(w)], G being SLn. The algebra k q [SL n ]: Let G = SLn. Let T = (tij), 1 ≤ i, j ≤ n. Let Let $$R = \sum\limits_{\mathop {i \ne j}\limits_{i,j = 1} }^n {{e_{ii}}} { \otimes _{jj}} + q\sum\limits_{i \ne i}^n {{e_{ii}}} \otimes {e_{ii}} + \left( {q - {q^{ - 1}}} \right)\sum\limits_{1j < in} {{e_{ij}}} \otimes {e_{ji}}$$ (here, eij’s are the elementary matrices). Let A(R) be the associative algebra (with 1) generated by {tij, 1 ≤ i, j ≤ n}, the relations being given by RT1T2 = T2T1R, where T1 = T ⊗ Id, T2 = Id ⊗ T (cf. [F-R-T]). Then A(R) gives a quantization of k[Mn], Mn being the space of n × n matrices and k[Mn], the coordinate ring of Mn. Now A(R) has a bialgebra structure, then comultiplication being given by Δ: A(R) → A(R) ⊗ A(R), \(\Delta = \left( {{t_{ij}}} \right) = \sum\limits_{k = 1}^n {{t_{ik}}} \otimes {t_{kj}}\) In the sequel, we shall denote A(R) by kq [Mn]. Let $$D = \sum\limits_{\sigma \in {s_n}}^n {{t_{ik}}} {\left( { - q} \right)^{ - l{{\left( \sigma \right)}_t}}}1\sigma {\left( 1 \right)^t}2\sigma {\left( 2 \right)^{...}}n\sigma \left( n \right)$$ . we shall refer to D as the q-determinant (or the quantum determinant) of (tij).

Let $H$ and $H'$ be two ample line bundles over a nonsingular projective surface $X$, and $M(H)$ (resp. $M(H')$) the coarse moduli scheme of $H$-semistable (resp. $H'$-semistable) sheaves of fixed type $(r=2,c_1,c_2)$. In a moduli-theoretic way that comes from elementary transforms, we connect $M(H)$ and $M(H')$ by a sequence of blowing-ups when walls separating $H$ and $H'$ are not necessarily good. As an application, we also consider the polarization change problem of Donaldson polynomials.

In this paper we prove a refined version of the canonical key formula for projective abelian schemes in the sense of Moret-Bailly (cf. Asterisque 129, 1985), we also extend this discussion to the context of Arakelov geometry. Precisely, let \({\pi: A \to S}\) be a projective abelian scheme over a locally noetherian scheme S with unit section \({e: S \to A}\) and let L be a symmetric, rigidified, relatively ample line bundle on A. Denote by ωA the determinant of the sheaf of differentials of π and by d the rank of the locally free sheaf π*L. In this paper, we shall prove the following results: (i). there is an isomorphism $${\rm det}(\pi_*L)^{\otimes 24} \cong (e^*\omega_A^\vee)^{\otimes 12d}$$ which is canonical in the sense that it can be chosen to be functorial, namely it is compatible with arbitrary base-change; (ii). if the generic fibre of S is separated and smooth, then there exist a positive integer m and canonical metrics on L and on ωA such that there exists an isometry $${\rm det}(\pi_*\overline{L})^{\otimes 2m} \cong (e^*\overline{\omega}_A^\vee)^{\otimes md}$$ which is canonical in the sense of (i). Here the constant m only depends on g, d and is independent of L.

Let L L be an ample line bundle on an abelian surface A A . We prove that the four conditions: (1) L L is base point free, (2) L L is fixed component free, (3) L ⊗ 2 {L^{ \otimes 2}} is very ample, (4) L ⊗ 2 {L^{ \otimes 2}} is normally generated, are equivalent if ( L 2 ) &gt; 4 ({L^2}) &gt; 4 . Moreover we prove that L ⊗ 2 {L^{ \otimes 2}} is not normally generated if ( L 2 ) = 4 ({L^2}) = 4 .

Let G be a semisimple connected linear algebraic group over \({\mathbb{C}}\) , and X a wonderful G-variety. We study the possibility of realizing X as a closed subvariety of the projective space of a simple G-module. We describe the wonderful varieties having this property as well as the linear systems giving rise to such immersions. We also prove that any ample line bundle on a wonderful variety is very ample.

Restriction theorems for principal bundles in arbitrary characteristic

Given a Kähler manifold X with an ample line bundle L, we consider the metric space of finite energy geodesic rays associated to the Chern class c1(L). We characterize rays that can be approximated by ample test configurations. At the same time, we also characterize the closure of algebraic singularity types among all singularity types of quasi-plurisubharmonic functions, pointing out the very close relationship between these two seemingly unrelated problems.By Bonavero's holomorphic Morse inequalities, the arithmetic and non-pluripolar volumes of algebraic singularity types coincide. We show that in general the arithmetic volume dominates the non-pluripolar one, and equality holds exactly on the closure of algebraic singularity types. Analogously, we give an estimate for the Monge–Ampère energy of a general finite energy ray in terms of the arithmetic volumes along its Legendre transform. Equality holds exactly for rays approximable by test configurations.Various other cohomological and potential theoretic characterizations are given in both settings. As applications, we give a concrete formula for the non-Archimedean Monge–Ampère energy in terms of asymptotic expansion, and show the continuity of the projection map from L1 rays to non-Archimedean rays.