代数几何/微分几何/辛几何/公制几何学术速递[1.10]
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math.AG代数几何，共计5篇
math.DG微分几何，共计4篇
math.SG辛几何，共计1篇
math.MG公制几何，共计2篇
1.math.AG代数几何:
【1】 Fano manifolds with Lefschetz defect 3
标题：Lefschetz亏数为3的Fano流形
链接：https://arxiv.org/abs/2201.02413
备注：30 pages, 2 figures
摘要：Let X be a smooth, complex Fano variety, and delta(X) its Lefschetz defect.
It is known that if delta(X) is at least 4, then X is isomorphic to a product
SxT, where dim T=dim X-2. In this paper we prove a structure theorem for the
case where delta(X)=3. We show that there exists a smooth Fano variety T with
dim T=dim X-2 such that X is obtained from T with two possible explicit
constructions; in both cases there is a P^2-bundle Z over T such that X is the
blow-up of Z along three pairwise disjoint smooth, irreducible, codimension 2
subvarieties. Then we apply the structure theorem to Fano 4-folds, to the case
where X has Picard number 5, and to Fano varieties having an elementary
divisorial contraction sending a divisor to a curve. In particular we complete
the classification of Fano 4-folds with delta(X)=3.
【2】 Deligne-Beilinson cohomology of the universal K3 surface
标题：泛K3曲面的Deligne-Beilinson上同调
链接：https://arxiv.org/abs/2201.02367
备注：26 pages, any comments are welcome!
摘要：O'Grady's generalized Franchetta conjecture (GFC) is concerned with
codimension 2 algebraic cycles on universal polarized K3 surfaces. In
\cite{BL17}, this conjecture has been studied in the Betti cohomology groups.
Following a suggestion of Voisin, we investigate this problem in the
Deligne-Beilinson (DB) cohomology groups. In this paper, we develop the theory
of Deligne-Beilinson cohomology groups on separated (smooth) Deligne-Mumford
stacks. Using the automorphic cohomology group and Noether-Lefschetz theory, we
compute the 4-th DB-cohomology group of universal oriented polarized K3
surfaces with at worst an $A_1$-singularity and show that GFC for such family
holds in DB-cohomology. In particular, this confirms O'Grady's original
conjecture in DB cohomology.
【3】 On M-O.Ore determinants
标题：关于M-O矿的决定因素
链接：https://arxiv.org/abs/2201.02361
摘要：The existence of certain Fq-spaces of differential forms of the projective
line over a field K containing Fq leads us to prove an identity linking the
determinant of the Moore matrix of n indeterminates with the determinant of the
Moore matrix of the cofactors of its first row. These same spaces give an
interpretation of Elkies pairing in terms of residues of differential forms.
This pairing puts in duality the Fq-vector space of the roots of a Fq-linear
polynomial and that of the roots of its reversed polynomial.
【4】 Asymptotic slopes and strong semistability on surfaces
标题：曲面上的渐近斜率与强半稳定性
链接：https://arxiv.org/abs/2201.02329
备注：Comments welcome
摘要：In this article we study asymptotic slopes of strongly semistable vector
bundles on a smooth projective surface. A connection between asymptotic slopes
and strong restriction theorem of a strongly semistable vector bundle is shown.
We also give an equivalent criterion of strong semistability of a vector bundle
in terms of its asymptotic slopes under some assumptions on the surface and on
the bundle.
【5】 Polynomial Dynamical Systems and Differentiation of Genus 4 Hyperelliptic Functions
标题：多项式动力系统与亏格4的超椭圆函数的微分
链接：https://arxiv.org/abs/2201.02462
摘要：We give an explicit solution to the problem of differentiation of
hyperelliptic functions in genus 4 case. We describe explicitly the polynomial
Lie algebras and polynomial dynamical systems connected to this problem.
2.math.DG微分几何:
【1】 Semilinear Li & Yau inequalities
标题：半线性Li&Yau不等式
链接：https://arxiv.org/abs/2201.02530
摘要：We derive an adaptation of Li & Yau estimates for positive solutions of
semilinear heat equations on Riemannian manifolds with nonnegative Ricci
tensor. We then apply these estimates to obtain a Harnack inequality and to
discuss monotonicity, convexity, decay estimates and triviality of ancient and
eternal solutions.
【2】 Singularity models in the three-dimensional Ricci flow
标题：三维Ricci流中的奇点模型
链接：https://arxiv.org/abs/2201.02522
备注：This is survey paper which will appear in the KIAS Expositions
摘要：The Ricci flow is a natural evolution equation for Riemannian metrics on a
given manifold. The main goal is to understand singularity formation. In his
spectacular 2002 breakthrough, Perelman achieved a qualitative understanding of
singularity formation in dimension $3$. More precisely, Perelman showed that
every finite-time singularity to the Ricci flow in dimension $3$ is modeled on
an ancient $\kappa$-solution. Moreover, Perelman proved a structure theorem for
ancient $\kappa$-solutions in dimension $3$.
In this survey, we will discuss recent developments which have led to a
complete classification of all the singularity models in dimension $3$.
Moreover, we give an alternative proof of the classification of noncollapsed
steady gradient Ricci solitons in dimension $3$ (originally proved by the
author in 2012).
【3】 The Stability of Generalized Ricci Solitons
标题：广义Ricci孤子的稳定性
链接：https://arxiv.org/abs/2201.02264
摘要：In this paper, I compute the second variation formula of the generalized
Einstein-Hilbert functional and prove that a Bismut-flat, Einstein manifold is
linearly stable under some curvature assumption. In the last part of the paper,
I prove that dynamical stability and the linear stability are equivalent on a
steady gradient generalized Ricci soliton $(g,H,f)$ which generalizes the
result done by Kr\"oncke, Haslhofer, Sesum, Raffero and Vezzoni.
【4】 Magnetic trajectories on 2-step nilmanifolds
标题：二阶零流形上的磁迹
链接：https://arxiv.org/abs/2201.02258
摘要：The aim of this work is the study of magnetic trajectories on nilmanifolds
but also of the associated magnetic fields. The magnetic equation is written
and the corresponding solutions are found for a family of Lorentz forces. The
existence question of magnetic fields is analyzed, obtaining non-existence
results. This is extended to prove the non-existence of uniform magnetic fields
under certain hypotheses. Finally some examples are computed in the Heisenberg
Lie groups $H_n$ for $n=3,5$, showing differences with the case of exact forms.
Interesting magnetic trajectories related to elliptic integrals appear in
$H_3$. The question of existence of closed or periodic magnetic trajectories
for every energy level on Lie groups or on compact quotients is treated.
3.math.SG辛几何:
【1】 Completeness of derived interleaving distances and sheaf quantization of non-smooth objects
标题：导出交织距离的完备性与非光滑物体的束量化
链接：https://arxiv.org/abs/2201.02598
备注：33 pages, 1 figure, comments are welcome
摘要：We investigate sheaf-theoretic methods to deal with non-smooth objects in
symplectic geometry. We show the completeness of a derived category of sheaves
with respect to the interleaving-like distance and construct a sheaf
quantization of a hameomorphism. We also develop Lusternik-Schnirelmann theory
in the microlocal theory of sheaves. With these new tools, we prove an
Arnold-type theorem for the image of the zero-section under a hameomorphism by
a purely sheaf-theoretic method.
4.math.MG公制几何:
【1】 Exploring the Steiner-Soddy Porism
标题：斯泰纳-索迪流浪主义探微
链接：https://arxiv.org/abs/2201.02222
备注：12 pages, 6 figures, 1 table
摘要：We explore properties and loci of a Poncelet family of polygons -- called
here Steiner-Soddy -- whose vertices are centers of circles in the Steiner
porism, including conserved quantities, loci, and its relationship to other
Poncelet families.
【2】 Isoperimetric 3- and 4-bubble results on $\mathbb{R}$ with density $|x|$
链接：https://arxiv.org/abs/2201.02197
备注：1 figure. arXiv admin note: text overlap with arXiv:2201.01808
摘要：We study the isoperimetric problem on $\mathbb{R}^1$ with a prescribed
density function $f(x) = |x|$. Under these conditions, we find that
isoperimetric $3$-bubble and $4$-bubble results satisfy a regular structure. As
our regions increase in size, the intervals that form them alternate
back-and-forth across the origin, with the smaller regions closer to the
origin. This expands on previously known observations about the single- and
double-bubble results on $\mathbb{R}$ with density $|x|^p$.
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