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Shifting numbers of abelian varieties via bounded t-structures 通过有界t结构的阿贝尔变异数的移位
IF 0.6 4区 数学 Q2 Mathematics
Manuscripta Mathematica
Pub Date : 2023-11-27 DOI: 10.1007/s00229-023-01525-z
Yu-Wei Fan

The shifting numbers measure the asymptotic amount by which an endofunctor of a triangulated category translates inside the category, and are analogous to Poincaré translation numbers that are widely used in dynamical systems. Motivated by this analogy, Fan–Filip raised the following question: “Do the shifting numbers define a quasimorphism on the group of autoequivalences of a triangulated category?” An affirmative answer was given by Fan–Filip for the bounded derived category of coherent sheaves on an elliptic curve or an abelian surface, via properties of the spaces of Bridgeland stability conditions on these categories. We prove in this article that the question has an affirmative answer for abelian varieties of arbitrary dimensions, generalizing the result of Fan–Filip. One of the key steps is to establish an alternative definition of the shifting numbers via bounded t-structures on triangulated categories. In particular, the full package of a Bridgeland stability condition (a bounded t-structure, and a central charge on a charge lattice) is not necessary for the purpose of computing the shifting numbers.

移位数测量了一个三角化范畴的内函子在范畴内平移的渐近量，类似于在动力系统中广泛使用的庞加莱平移数。受到这个类比的启发，Fan-Filip提出了以下问题:“移位的数是否定义了三角化范畴的自等价群上的拟同构?”利用椭圆曲线或阿贝曲面上相干束的布里奇兰稳定性条件的空间性质，给出了该类上相干束的有界派生范畴的肯定答案。推广了Fan-Filip的结果，证明了该问题对于任意维的阿贝尔变有一个肯定的答案。其中一个关键步骤是通过三角分类上的有界t结构建立移动数的另一种定义。特别是，布里奇兰稳定性条件的完整包(有界t结构和电荷格上的中心电荷)对于计算移位数的目的是不必要的。
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引用次数: 0
Linear hyperelliptic Hodge integrals 线性超椭圆Hodge积分
IF 0.6 4区 数学 Q2 Mathematics
Manuscripta Mathematica
Pub Date : 2023-11-27 DOI: 10.1007/s00229-023-01519-x
Adam Afandi

We provide a closed form expression for linear Hodge integrals on the hyperelliptic locus. Specifically, we find a succinct combinatorial formula for all intersection numbers on the hyperelliptic locus with one (lambda )-class, and powers of a (psi )-class pulled back along the branch map. This is achieved by using Atiyah–Bott localization on a stack of stable maps into the orbifold (left[ {mathbb {P}}^1/{mathbb {Z}}_2right] ).

给出了超椭圆轨迹上的线性Hodge积分的一个封闭表达式。具体地说，我们找到了一个简洁的组合公式，用于在一个(lambda ) -类的超椭圆轨迹上的所有交点数，以及一个(psi ) -类的幂沿着分支映射向后拉。这是通过在一堆稳定的轨道图(left[ {mathbb {P}}^1/{mathbb {Z}}_2right] )上使用阿蒂亚-博特定位来实现的。
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引用次数: 3
The Dirichlet problem for prescribed curvature equations of p-convex hypersurfaces p-凸超曲面规定曲率方程的Dirichlet问题
IF 0.6 4区 数学 Q2 Mathematics
Manuscripta Mathematica
Pub Date : 2023-11-17 DOI: 10.1007/s00229-023-01522-2
Weisong Dong

In this paper, we study the Dirichlet problem for p-convex hypersurfaces with prescribed curvature. We prove that there exists a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition. An interior curvature estimate is also obtained.

本文研究了具有规定曲率的p-凸超曲面的Dirichlet问题。证明了在齐次边界条件下存在满足规定曲率方程的图形超曲面。得到了内部曲率的估计。
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引用次数: 0
Polyharmonic surfaces in 3-dimensional homogeneous spaces 三维齐次空间中的多谐曲面
4区 数学 Q2 Mathematics
Manuscripta Mathematica
Pub Date : 2023-11-13 DOI: 10.1007/s00229-023-01520-4
S.  Montaldo, C.  Oniciuc, A.  Ratto
Abstract In the first part of this paper we shall classify proper triharmonic isoparametric surfaces in 3-dimensional homogeneous spaces (Bianchi-Cartan-Vranceanu spaces, shortly BCV-spaces). We shall also prove that triharmonic Hopf cylinders are necessarily CMC. In the last section we shall determine a complete classification of CMC r -harmonic Hopf cylinders in BCV-spaces, $$r ge 3$$ r ≥ 3 . This result ensures the existence, for suitable values of r , of an ample family of new examples of r -harmonic surfaces in BCV-spaces.
摘要本文第一部分对三维齐次空间(bianchi - cartan - vrancanu空间，简称bcv空间)中的固有三谐等参曲面进行了分类。我们还将证明三谐波霍普夫圆柱必然是CMC。在最后一节中，我们将确定bcv空间中CMC r -谐波Hopf圆柱的完整分类，$$r ge 3$$ r≥3。这个结果保证了在合适的r值下，在bcv空间中存在大量的r -调和曲面的新实例。
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引用次数: 1
Cuspidal components of Siegel modular forms for large discrete series representations of $$textrm{Sp}_4({mathbb {R}})$$ 的倒轴分量的西格尔模形式的大离散级数表示 $$textrm{Sp}_4({mathbb {R}})$$
4区 数学 Q2 Mathematics
Manuscripta Mathematica
Pub Date : 2023-11-13 DOI: 10.1007/s00229-023-01513-3
Shuji Horinaga, Hiro-aki Narita
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引用次数: 0
Existence of self-similar Dirichlet forms on post-critically finite fractals in terms of their resistances 后临界有限分形上自相似狄利克雷形式的存在性及其阻力
4区 数学 Q2 Mathematics
Manuscripta Mathematica
Pub Date : 2023-11-09 DOI: 10.1007/s00229-023-01521-3
Guanhua Liu
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引用次数: 1
On algebraic Chern classes of flat vector bundles 平面向量束的代数Chern类
4区 数学 Q2 Mathematics
Manuscripta Mathematica
Pub Date : 2023-11-09 DOI: 10.1007/s00229-023-01518-y
Adrian Langer
Abstract We show that under some assumptions on the monodromy group some combinations of higher Chern classes of a flat vector bundle are torsion in the Chow group. Similar results hold for flat vector bundles that deform to such flat vector bundles (also in case of quasi-projective varieties). The results are motivated by Bloch’s conjecture on Chern classes of flat vector bundles on smooth complex projective varieties but in some cases they give a more precise information. We also study Higgs version of Bloch’s conjecture and analogous problems in the positive characteristic case.
摘要在单群的某些假设下，证明了平面向量束的高Chern类的某些组合在Chow群中是挠性的。类似的结果也适用于变形为这种平面向量束的平面向量束(也适用于拟射影变体)。这些结果是由布洛赫关于光滑复射影变体上的平坦向量束的陈类的猜想所激发的，但在某些情况下它们给出了更精确的信息。我们还研究了希格斯版布洛赫猜想和正特征情况下的类似问题。
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引用次数: 1
On the rationality of certain Fano threefolds 论法诺的某些合理性
4区 数学 Q2 Mathematics
Manuscripta Mathematica
Pub Date : 2023-11-06 DOI: 10.1007/s00229-023-01514-2
Ciro Ciliberto
Abstract In this paper we study the rationality problem for Fano threefolds $$Xsubset {mathbb P}^{p+1}$$ X ⊂ P p + 1 of genus p , that are Gorenstein, with at most canonical singularities. The main results are: (1) a trigonal Fano threefold of genus p is rational as soon as $$pgeqslant 8$$ p ⩾ 8 (this result has already been obtained in Przyjalkowski et al. (Izv Math 69(2):365–421, 2005), but we give here an independent proof); (2) a non-trigonal Fano threefold of genus $$pgeqslant 7$$ p ⩾ 7 containing a plane is rational; (3) any Fano threefold of genus $$pgeqslant 17$$ p ⩾ 17 is rational; (4) a Fano threefold of genus $$pgeqslant 12$$ p ⩾ 12 containing an ordinary line $$ell $$ ℓ in its smooth locus is rational.
摘要本文研究了具有至多正则奇点的格伦斯坦(Gorenstein)属P的Fano三倍$$Xsubset {mathbb P}^{p+1}$$ X∧P P + 1的合理性问题。主要结果是:(1)当$$pgeqslant 8$$ p小于8时，p属的三角形Fano三倍是有理的(这个结果已经在Przyjalkowski等人中获得(Izv Math 69(2): 365-421, 2005)，但我们在这里给出了一个独立的证明);(2)含有平面的$$pgeqslant 7$$ p小于7属的非三角形Fano三倍是有理的;(3)属$$pgeqslant 17$$ p大于或等于17的任何Fano三倍是合理的;(4)属$$pgeqslant 12$$ p小于12的Fano三倍在其平滑位点中包含一条普通线$$ell $$ r是合理的。
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引用次数: 0
Boundedness of regular del Pezzo surfaces over imperfect fields 不完全域上正则del Pezzo曲面的有界性
4区 数学 Q2 Mathematics
Manuscripta Mathematica
Pub Date : 2023-10-30 DOI: 10.1007/s00229-023-01517-z
Hiromu Tanaka
Abstract For a regular del Pezzo surface X , we prove that $$|-12K_X|$$ | - 12 K X | is very ample. Furthermore, we also give an explicit upper bound for the volume $$K_X^2$$ K X 2 which depends only on $$[k: k^p]$$ [ k : k p ] for the base field k . As a consequence, we obtain the boundedness of geometrically integral regular del Pezzo surfaces.
摘要对于正则del Pezzo曲面X，证明了$$|-12K_X|$$ | - 12 K X |是非常充足的。此外，我们还给出了体积$$K_X^2$$ K x2的显式上界，该上界仅取决于基础场K的$$[k: k^p]$$ [K: kp]。由此得到几何积分正则del Pezzo曲面的有界性。
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引用次数: 5
Formal ternary laws and Buchstaber’s 2-groups 正式的三元定律和Buchstaber的两个基团
4区 数学 Q2 Mathematics
Manuscripta Mathematica
Pub Date : 2023-10-30 DOI: 10.1007/s00229-023-01512-4
Jean Fasel, David Coulette, Frédéric Déglise, Jens Hornbostel
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引用次数: 1
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