aCM vector bundles on projective surfaces of nonnegative Kodaira dimension

In this paper, we contribute to the construction of families of arithmetically Cohen–Macaulay (aCM) indecomposable vector bundles on a wide range of polarized surfaces [Formula: see text] for [Formula: see text] an ample line bundle. In many cases, we show that for every positive integer [Formula: see text] there exists a family of indecomposable aCM vector bundles of rank [Formula: see text], depending roughly on [Formula: see text] parameters, and in particular they are of wild representation type. We also introduce a general setting to study the complexity of a polarized variety [Formula: see text] with respect to its category of aCM vector bundles. In many cases we construct indecomposable vector bundles on [Formula: see text] which are aCM for all ample line bundles on [Formula: see text].

On non-proper intersections and local intersection numbers

Given equidimensional (generalized) cycles $$\mu _1$$ μ 1 and $$\mu _2$$ μ 2 on a complex manifold Y we introduce a product $$\mu _1\diamond _{Y} \mu _2$$ μ 1 ⋄ Y μ 2 that is a generalized cycle whose multiplicities at each point are the local intersection numbers at the point. If Y is projective, then given a very ample line bundle $$L\rightarrow Y$$ L → Y we define a product $$\mu _1{\bullet _L}\mu _2$$ μ 1 ∙ L μ 2 whose multiplicities at each point also coincide with the local intersection numbers. In addition, provided that $$\mu _1$$ μ 1 and $$\mu _2$$ μ 2 are effective, this product satisfies a Bézout inequality. If $$i:Y\rightarrow {\mathbb P}^N$$ i : Y → P N is an embedding such that $$i^*\mathcal O(1)=L$$ i ∗ O ( 1 ) = L , then $$\mu _1{\bullet _L}\mu _2$$ μ 1 ∙ L μ 2 can be expressed as a mean value of Stückrad–Vogel cycles on $${\mathbb P}^N$$ P N . There are quite explicit relations between $${\diamond }_Y$$ ⋄ Y and $${\bullet _L}$$ ∙ L .

A lower bound for $$\chi ({\mathcal {O}}_S)$$

Let $$(S,{\mathcal {L}})$$ ( S , L ) be a smooth, irreducible, projective, complex surface, polarized by a very ample line bundle $${\mathcal {L}}$$ L of degree $$d > 25$$ d > 25 . In this paper we prove that $$\chi (\mathcal O_S)\ge -\frac{1}{8}d(d-6)$$ χ ( O S ) ≥ - 1 8 d ( d - 6 ) . The bound is sharp, and $$\chi ({\mathcal {O}}_S)=-\frac{1}{8}d(d-6)$$ χ ( O S ) = - 1 8 d ( d - 6 ) if and only if d is even, the linear system $$|H^0(S,{\mathcal {L}})|$$ | H 0 ( S , L ) | embeds S in a smooth rational normal scroll $$T\subset {\mathbb {P}}^5$$ T ⊂ P 5 of dimension 3, and here, as a divisor, S is linearly equivalent to $$\frac{d}{2}Q$$ d 2 Q , where Q is a quadric on T. Moreover, this is equivalent to the fact that a general hyperplane section $$H\in |H^0(S,{\mathcal {L}})|$$ H ∈ | H 0 ( S , L ) | of S is the projection of a curve C contained in the Veronese surface $$V\subseteq {\mathbb {P}}^5$$ V ⊆ P 5 , from a point $$x\in V\backslash C$$ x ∈ V \ C .

The asymptotic of curvature of direct image bundle associated with higher powers of a relatively ample line bundle

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

On Monge–Ampère Volumes of Direct Images

This paper is devoted to the study of the asymptotics of Monge–Ampère volumes of direct images associated with high tensor powers of an ample line bundle. We study the leading term of this asymptotics and provide a classification of bundles saturating the topological bound of Demailly. In the special case of high symmetric powers of ample vector bundles, this provides a characterization of those admitting projectively flat Hermitian structures.

Special Ulrich bundles on regular surfaces with non–negative Kodaira dimension

Let S be a regular surface endowed with a very ample line bundle $$\mathcal O_S(h_S)$$ O S ( h S ) . Taking inspiration from a very recent result by D. Faenzi on K3 surfaces, we prove that if $${\mathcal O}_S(h_S)$$ O S ( h S ) satisfies a short list of technical conditions, then such a polarized surface supports special Ulrich bundles of rank 2. As applications, we deal with general embeddings of regular surfaces, pluricanonically embedded regular surfaces and some properly elliptic surfaces of low degree in $$\mathbb {P}^{N}$$ P N . Finally, we also discuss about the size of the families of Ulrich bundles on S and we inspect the existence of special Ulrich bundles on surfaces of low degree.

Differentiability of Relative Volumes Over an Arbitrary Non-Archimedean Field

Given an ample line bundle $L$ on a geometrically reduced projective scheme defined over an arbitrary non-Archimedean field, we establish a differentiability property for the relative volume of two continuous metrics on the Berkovich analytification of $L$, extending previously known results in the discretely valued case. As applications, we provide fundamental solutions to certain non-Archimedean Monge–Ampère equations and generalize an equidistribution result for Fekete points. Our main technical input comes from determinant of cohomology and Deligne pairings.

ADJUNCTION FOR VARIETIES WITH A ℂ* ACTION

Let X be a complex projective manifold, L an ample line bundle on X, and assume that we have a ℂ* action on (X;L). We classify such triples (X; L;ℂ*) for which the closure of a general orbit of the ℂ* action is of degree ≤ 3 with respect to L and, in addition, the source and the sink of the action are isolated fixed points, and the ℂ* action on the normal bundle of every fixed point component has weights ±1. We treat this situation by relating it to the classical adjunction theory. As an application, we prove that contact Fano manifolds of dimension 11 and 13 are homogeneous if their group of automorphisms is reductive of rank ≥ 2.

On the set of divisors with zero geometric defect

Let {f:\mathbb{C}\to X} be a transcendental holomorphic curve into a complex projective manifold X. Let L be a very ample line bundle on {X.} Let s be a very generic holomorphic section of L and D the zero divisor given by {s.} We prove that the geometric defect of D (defect of truncation 1) with respect to f is zero. We also prove that f almost misses general enough analytic subsets on X of codimension 2.