Curve counting on abelian surfaces and threefolds

The wall-crossing behavior for Bridgeland’s stability conditions on the derived category of coherent sheaves on K3 or abelian surface is studied. We introduce two types of walls. One is called the wall for categories, where the t-structure encoded by stability condition is changed. The other is the wall for stabilities, where stable objects with prescribed Mukai vector may get destabilized. Some fundamental properties of walls and chambers are studied, including the behavior under Fourier–Mukai transforms. A wall-crossing formula of the counting of stable objects will also be derived. As an application, we will explain previous results on the birational maps induced by Fourier–Mukai transforms on abelian surfaces. These transformations turns out to coincide with crossing walls of certain property.

A K3 category is by definition a Calabi–Yau category of dimension two. Geometrically K3 categories occur as bounded derived categories of (twisted) coherent sheaves on K3 or abelian surfaces. A K3 category is generic if there are no spherical objects (or just one up to shift). We study stability conditions on K3 categories as introduced by Bridgeland and prove his conjecture about the topology of the stability manifold and the autoequivalences group for generic twisted projective K3, abelian surfaces, and K3 surfaces with trivial Picard group.

Given an abelian variety over a number field, its Sato–Tate group is a compact Lie group which conjecturally controls the distribution of Euler factors of the L L -function of the abelian variety. It was previously shown by Fité, Kedlaya, Rotger, and Sutherland that there are 52 groups (up to conjugation) that occur as Sato–Tate groups of abelian surfaces over number fields; we show here that for abelian threefolds, there are 410 possible Sato–Tate groups, of which 33 are maximal with respect to inclusions of finite index. We enumerate candidate groups using the Hodge-theoretic construction of Sato–Tate groups, the classification of degree-3 finite linear groups by Blichfeldt, Dickson, and Miller, and a careful analysis of Shimura’s theory of CM types that rules out 23 candidate groups; we cross-check this using extensive computations in GAP, SageMath, and Magma. To show that these 410 groups all occur, we exhibit explicit examples of abelian threefolds realizing each of the 33 maximal groups; we also compute moments of the corresponding distributions and numerically confirm that they are consistent with the statistics of the associated L L -functions.

The Picard group of S is generated by two effective divisors C and F with C = −2, F 2 = 0 and C · F = 1. It can be realized as an elliptic fibration over P with a unique section C, fibers F and λ = 2. Here λ = c1(π∗ω) is the first Chern class of the Hodge bundle π∗ω of the fibration π : S → P (see [H-M]). It is a standard result that the number of nodal fibers of an elliptic fibration are given by 12λ [H-M, p. 158]. So there are exactly 24 rational nodal curves in the linear series |F | for S general. This is the same special K3 surface used by Bryan and Leung in their counting of curves on K3 surfaces [B-L]. It is actually the attempt to understand their method that leads us to our proof. We will call a K3 surface with Picard lattice (1.1) a BL K3 surface. A BL K3 surface S lies on the boundary of the moduli space of K3 surfaces of genus g with C+gF as the corresponding primitive divisor. Every curve in the linear series |OS(C + gF )| is “totally reducible”, i.e., it consists of the −2 curve C and g elliptic “tails” attached to C. A curve D ∈ |OS(C + gF )| is the image of a stable rational map only if D = C ∪m1F1∪m2F2∪ ...∪m24F24, where F1, F2, ..., F24 are 24

We relate the Brauer group of a Kummer surface to the Brauer group of the corresponding abelian surface. For many pairs of elliptic curves over the rational numbers we prove that the Kummer surface attached to their product has trivial Brauer group.

Let S be a K3 surface and let E be an elliptic curve. We solve the reduced Gromov–Witten theory of the Calabi–Yau threefold $$S \times E$$ for all curve classes which are primitive in the K3 factor. In particular, we deduce the Igusa cusp form conjecture. The proof relies on new results in the Gromov–Witten theory of elliptic curves and K3 surfaces. We show the generating series of Gromov–Witten classes of an elliptic curve are cycle-valued quasimodular forms and satisfy a holomorphic anomaly equation. The quasimodularity generalizes a result by Okounkov and Pandharipande, and the holomorphic anomaly equation proves a conjecture of Milanov, Ruan and Shen. We further conjecture quasimodularity and holomorphic anomaly equations for the cycle-valued Gromov–Witten theory of every elliptic fibration with section. The conjecture generalizes the holomorphic anomaly equations for elliptic Calabi–Yau threefolds predicted by Bershadsky, Cecotti, Ooguri, and Vafa. We show a modified conjecture holds numerically for the reduced Gromov–Witten theory of K3 surfaces in primitive classes.

This study aimed to track the sources of water contamination in the Laboratorio de Camaroes Marinhos (LCM) pertaining to the Universidade Federal de Santa Catarina (UFSC); to quantify the bacterial load during the stages of shrimp production; to evaluate chlorination efficiency as desinfectant, as well as to evaluate the effect of oxitetracicline on larval quality by application of a multiple criteria method. Bacterial load was quantified by direct counting in plates with Marine Agar and TCBS Agar. A bacterial load of 1 x 103 CFU/mL (colony-forming units) was observed in the water used for larvae production. Differences in bacterial load were not detected between water with and without chlorine addition in the cisterns, indicating the low efficiency of conventional water disinfection. Similarly, the presence of filters not only did not reduce bacterial load in the water network, but was responsible for its increase. Water from hatchery presented a high counting of Vibrio genus, between 1 x 102 and 1 x 103 CFU/mL. When this load reached 1 x 104 CFU/mL, decrease in larval quality was detected, demonstrating that the application of multiple criteria may prevent enfermities. After antibiotic application (oxitetracicline) at 1ppm, the Vibrio load in hatchery water decreased as the larval quality improved. The bacteriological assessing has optimized production management and disinfection of water system in an effective and rational way, reducing the excessive use of chemicals and antimicrobians in marine shrimp hatchery.

Three new examples of 4-dimensional irreducible symplectic V-manifolds are constructed. Two of them are relative compactified Prymians of a family of genus-3 curves with involution, and the third one is obtained from a Prymian by Mukai's flop. They have the same singularities as two of Fujiki's examples, namely, 28 isolated singular points analytically equivalent to the Veronese cone of degree 8, but a different Euler number. The family of curves used in this construction forms a linear system on a K3 surface with involution. The structure morphism of both Prymians to the base of the family is a Lagrangian fibration in abelian surfaces with polarization of type (1,2). No example of such fibration is known on nonsingular irreducible symplectic varieties.

Let S be a smooth surface over the complex number field r An a u t o m o r p h i s m q~ of S is cohomologically trivial (resp. numerically trivial) if the induced a u t o m o r p h i s m q~* of the cohomology ring H * (S, 7/) (resp. H * (S, I1~)) is trivial. We denote by A o (S) (resp. A (S)) the quot ient o f the group o fcohomolog ica l ly (resp. numerically) trivial au tomorph i sms of S by its connected component . I t is known tha t A (S) is a finite g roup i f S is Kiihler, [4, 3]. A (S) is even trivial, for example, in the case of ra t ional surfaces, abelian surfaces and K3 surfaces, [9, 5]. In this article, we shall s tudy A (S) for an Enriques surface S. In this case, it is no more true that Ao (S) is always trivial. So far two examples are known.

A remarkable result of Peter O’Sullivan asserts that the algebra epimorphism from the rational Chow ring of an abelian variety to its rational Chow ring modulo numerical equivalence admits a (canonical) section. Motivated by Beauville’s splitting principle, we formulate a conjectural Section Property which predicts that for smooth projective holomorphic symplectic varieties there exists such a section of algebra whose image contains all the Chern classes of the variety. In this paper, we investigate this property for (not necessarily symplectic) varieties with a Chow motive of abelian type. We introduce the notion of a symmetrically distinguished abelian motive and use it to provide a sufficient condition for a smooth projective variety to admit such a section. We then give a series of examples of varieties for which our theory works. For instance, we prove the existence of such a section for arbitrary products of varieties with Chow groups of finite rank, abelian varieties, hyperelliptic curves, Fermat cubic hypersurfaces, Hilbert schemes of points on an abelian surface or a Kummer surface or a K3 surface with Picard number at least 19, and generalized Kummer varieties. The latter cases provide evidence for the conjectural Section Property and exemplify the mantra that the motives of holomorphic symplectic varieties should behave as the motives of abelian varieties, as algebra objects.

Logarithmic space bounded complexity classes such as L and NL play a central role in space bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space bounded models developed by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). They defined the operators para_W and para_β for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators para_W and para_β by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, para_{W[1]} and para_{βtail}. Then, we consider counting versions of all four operators applied to logspace and obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0,1)-matrices is #para_{βtail} L-hard and can be written as the difference of two functions in #para_{βtail} L. These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of #para_{βtail} L under parameterised logspace parsimonious reductions coincides with #para_β L, that is, modulo parameterised reductions, tail-nondeterminism with read-once access is the same as read-once nondeterminism. Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions. Also, we show that the counting classes defined can naturally be characterised by parameterised variants of classes based on branching programs in analogy to the classical counting classes. Finally, we underline the significance of this topic by providing a promising outlook showing several open problems and options for further directions of research.

These notes are based on a series of talks given by the authors at the CIMPA Summer School on Algebraic Geometry and Hypergeometric Functions held in Istanbul in Summer of 2005. They provide an introduction to recent work on the complex ball uniformization of the moduli spaces of del Pezzo surfaces, K3 surfaces and algebraic curves of lower genus. We discuss the relationship of these constructions with the Deligne-Mostow theory of periods of hypergeometric differential forms. For convenience to a non-expert reader we include an introduction to the theory of periods of integrals on algebraic varieties with emphasis on abelian varieties and K3 surfaces.

We give the counterexamples for a conjecture of Fiedler concerning the ribbon genus of a link.

We give a one-parameter family of Bridgeland stability conditions on the derived category of a smooth projective complex surface S and describe "wall-crossing behavior'' for objects with the same invariants as \mathcal O_C(H) when H generates Pic (S) and C \in |H| . If, in addition, S is a K3 or Abelian surface, we use this description to construct a sequence of fine moduli spaces of Bridgeland-stable objects via Mukai flops and generalized elementary modifications of the universal coherent sheaf. We also discover a natural generalization of Thaddeus' stable pairs for curves embedded in the moduli spaces.

O objetivo foi analisar a customização (Contagem Ágil) de uma métrica funcional de tamanho com foco na redução do esforço do processo de contagem de projetos de manutenção, experimentar e validar. Realizou-se pesquisa bibliográfica. Foram analisadas 13.320 contagens de manutenção detalhadas (APF) de uma organização brasileira. Em 68%, a contagem estimada era menor que a detalhada. As funções transacionais foram as que mais impactaram as diferenças entre estimadas e detalhadas. Foi aplicada a Contagem Ágil (customização da NESMA) nas contagens com tamanho acima de 51 PF (2915 contagens) e o resultado foi promissor (variação inferior a 1,21% com relação a contagem detalhada).