Peter Beelen, Mrinmoy Datta, Masaaki Homma
In this article we prove a conjecture formulated by A.B. Soerensen in 1991 on the maximal number of $\mathbb{F}_{q^2}$-rational points on the intersection of a non-degenerate Hermitian surface and a surface of degree $d \le q.$
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In 1991 Sørensen proposed a conjecture for the maximum number of points on the intersection of a surface of degree $d$ and a non-degenerate Hermitian surface in $\PP^3(\Fqt)$. The conjecture was proven to be true by Edoukou in the case when $d=2$. In this paper, we prove that the conjecture is true for $d=3$ and $q \ge 8$. We further determine the second highest number of rational points on the intersection of a cubic surface and a non-degenerate Hermitian surface. Finally, we classify all the cubic surfaces that admit the highest and second highest number of points in common with a non-degene…
Discriminants of Hermitian forms, maximal degenerations and Kontsevich's tropical approach to the Hodge conjecture
本文受 Markman 与 Kontsevich 启发，阐明 Weil 型阿贝尔 2n 维族仅在判别式为 $(-1)^n$ 时具极大幂幺退化，并给出判别式定义及相关事实。
We study the functional codes $C_h(X)$ defined by G. Lachaud in $\lbrack 10 \rbrack$ where $X \subset {\mathbb{P}}^N$ is an algebraic projective variety of degree $d$ and dimension $m$. When $X$ is a hermitian surface in $PG(3,q)$, Sørensen in \lbrack 15\rbrack, has conjectured for $h\le t$ (where $q=t^2$) the following result : $$# X_{Z(f)}(\mathbb{F}_{q}) \le h(t^{3}+ t^{2}-t)+t+1$$ which should give the exact value of the minimum distance of the functional code $C_h(X)$. In this paper we resolve the conjecture of Sørensen in the case of quadrics (i.e. $h=2$), we show the geometrical structu…
Maximum number of points on an intersection of a cubic threefold and a non-degenerate Hermitian threefold
本文证明了Edoukou关于非退化Hermitian三重超曲面与三次三重超曲面公共点最大数的猜想在d=3且q≥7时成立。
We advance the understanding of K-theory of quadratic forms by computing the slices of the motivic spectra representing hermitian K-groups and Witt-groups. By an explicit computation of the slice spectral sequence for higher Witt-theory, we prove Milnor's conjecture relating Galois cohomology to quadratic forms via the filtration of the Witt ring by its fundamental ideal. In a related computation we express hermitian K-groups in terms of motivic cohomology.
We give a bound on the number of isolated, essential singularities of determinantal quartic surfaces in 3-space. We also provide examples of different configurations of real singularities on quartic surfaces with a definite Hermitian determinantal representation, and conjecture an extension of a theorem by Degtyarev and Itenberg.
An explicit form of the standard conjecture of Hodge type for Hermitian varieties, with $p$-adic applications
本文给出 Hermitian 簇上 Hodge 型标准猜想的显式初等证明，利用对偶极图控制相交形式，并证明 $p$-adic 初等因子整除 $q^m$，从而得到代数环的数值等价性及猜想对有限覆盖簇的推广。
Let $G$ be a simple algebraic group and $P$ a parabolic subgroup of $G$ with abelian unipotent radical $P^u$, and let $B$ be a Borel subgroup of $G$ contained in P. Let $\mathfrak{p}^u$ be the Lie algebra of $P^u$ and let $L$ be a Levi factor of $P$, then $L$ is a Hermitian symmetric subgroup of $G$ and $B$ acts with finitely many orbits both on $\mathfrak{p}^u$ and on $G/L$. In this paper we study the Bruhat order of the $B$-orbits in $\mathfrak{p}^u$ and in $G/L$, proving respectively a conjecture of Panyushev and a conjecture of Richardson and Ryan.
Let X be a noetherian scheme of finite Krull dimension, having 2 invertible in its ring of regular functions, an ample family of line bundles, and a global bound on the virtual mod-2 cohomological dimensions of its residue fields. We prove that the comparison map from the hermitian K-theory of X to the homotopy fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence in general, and an integral equivalence when X has no formally real residue field. We also show that the comparison map between the higher Grothendieck-Witt (hermitian K-) theory of X and its étale version is …
The cohomological Kudla conjecture for unitary Shimura varieties
通过构造紧化酉Shimura簇上特殊环的上同调类的Kudla-Millson生成级数的自然延拓，证明了它们是全纯Hermitian模形式，从而证明了Kudla和Bruinier-Rosu-Zemel猜想在所有余维数（至中维）的上同调版本，并发展了Hermitian拟模形式理论，证明了特殊环的Zariski闭包的生成级数是Hermitian拟模形式。
Lattices with skew-Hermitian forms over division algebras and unlikely intersections
本文研究带正对合可除代数上的斜埃尔米特型格，证明Albert I、II型下这类格子格存在有效有界指数的正交基，并将结果用于非典型交领域，在大Galois轨道猜想下证明单PEL I、II型特殊子簇与曲线交的Zilber-Pink猜想，且证得部分II型的该Galois轨道猜想。
We study the functional codes $C_2(X)$ defined on a projective variety $X$, in the case where $X \subset {\mathbb{P}}^3$ is a non-degenerate hermitian surface. We first give some bounds for $# X_{Z(\mathcal{Q})}(\mathbb{F}_{q})$, which are better than the ones known. We compute the number of codewords reaching the second weight. We also estimate the third weight, show the geometrical structure of the codewords reaching this third weight and compute their number. The paper ends with a conjecture on the fourth weight and the fifth weight of the code $C_2(X)$.
Maximum number of points of intersection of a non-degenerate Hermitian variety and a cubic hypersurface
本文证明了 Edoukou-Ling-Xing 猜想在三次超曲面情形（d=3，q≥7）下成立，即非退化 Hermitian 簇与三次超曲面的最大交点数仅由 d 个特殊超平面并集达到。
On a construction of hermitian metrics on holomorphic vector bundles
本文针对Griffiths关于全纯向量丛正曲率埃尔米特度量的猜想，证明无法通过函子性纤维构造从关联线丛O_{PE}(1)的正曲率度量得到该向量丛的正曲率度量，为该猜想的相关研究提供了关键结论。
The Griffiths conjecture asserts that every ample vector bundle $E$ over a compact complex manifold $S$ admits a hermitian metric with positive curvature in the sense of Griffiths. In this article we give a sufficient condition for a positive hermitian metric on $\mathcal{O}_{\mathbb{P}(E^*)}(1)$ to induce a Griffiths positive $L^2$-metric on the vector bundle $E$. This result suggests to study the relative Kähler-Ricci flow on $\mathcal{O}_{\mathbb{P}(E^*)}(1)$ for the fibration $\mathbb{P}(E^*)\to S$. We define a flow and give arguments for the convergence.
In this paper, we prove that there are certain relations among representation densities and provide an efficient way to compute representation densities by using these relations. As an application, we compute some arithmetic intersection numbers of special cycles on unitary Shimura varieties and propose a conjecture on these.
In [GM1], we defined a G_R-K_C invariant subset C(S) of G_C for each K_C-orbit S on every flag manifold G_C/P and conjectured that the connected component C(S)_0 of the identity would be equal to the Akhiezer-Gindikin domain D if S is of nonholomorphic type by computing many examples. In this paper, we first prove (in Theorem 1.3 and Corollary 1.4) this conjecture for the open K_C-orbit S on an ``arbitrary'' flag manifold generalizing the result of Barchini. This conjecture for closed S was solved in [WZ1], [WZ2] (Hermitian cases) and [FH] (non-Hermitian cases). We also deduce an alternative p…
Notes on the deformed Hermitian-Yang-Mills equations and the large scaling limits of stability conditions
本文基于Arcara和Miles的猜想，证明了复平面上的线束在大尺度极限下稳定性与变形Hermite-Yang-Mills度量的存在等价，扩展了Stoppa对紧致托里表面的结果。
We find a new obstruction to the existence of solutions of the Hull-Strominger system, which goes beyond the balanced property of the Calabi-Yau manifold $(X,Ω)$ and the Mumford-Takemoto slope stability of the bundle over it. The basic principle is the construction of a (possibly indefinite) Hermitian-Einstein metric on the holomorphic string algebroid associated to a solution of the system, provided that the connection $\nabla$ on the tangent bundle is Hermitian-Yang Mills. Using this, we define a family of Futaki invariants obstructing the existence of solutions in a given balanced class. Ou…
In [GM1], we defined a G_R-K_C invariant subset C(S) of G_C for each K_C-orbit S on every flag manifold G_C/P and conjectured that the connected component C(S)_0 of the identity would be equal to the Akhiezer-Gindikin domain D if S is of nonholomorphic type. This conjecture was proved for closed S in [WZ2,WZ3,FH,M4] and for open S in [M4]. It was proved for the other orbits in [M5] when G_R is of non-Hermitian type. In this paper, we prove the conjecture for an arbitrary non-closed K_C-orbit when G_R is of Hermitian type. Thus the conjecture is completely solved affirmatively.
A proof of the Kudla-Rapoport conjecture for Krämer models
本文证明了分歧处酉 Rapoport-Zink 空间 Krämer 模型的 Kudla-Rapoport 猜想，确立了特殊闭链算术相交数与修正导出局部密度的精确等式，并据此放宽了酉 Shimura 簇算术 Siegel-Weil 公式的局部假设。
Modularity of Higher Theta Series III: Proof of the Modularity Conjecture
该研究证明了埃尔米特 shtuka 模堆栈上高阶θ级数的模性猜想，确立一般线性 shtuka 的超模性，还证明低余秩希钦堆栈的迹猜想，实现特殊周期的虚基本类为范畴迹。
It contains the proof of a very general $\partial\bar\partial$-lemma, together with a decomposition theorem for currents with values in a (singular) Hermitian line bundle. As a corollary, we establish the Kähler version on an injectivity theorem due to O. Fujino in the projective case.
Towards the Colmez Conjecture
本文通过定义部分CM型的高度并推导算术周期公式，提出将特殊点高度与L函数特殊值联系起来的猜想，从而推进科尔梅兹猜想的几何表述。
We present a complex frame of eleven vectors in 4-space and prove that it defines injective measurements. That is, any rank-one $4\times 4$ Hermitian matrix is uniquely determined by its values as a Hermitian form on this collection of eleven vectors. This disproves a recent conjecture of Bandeira, Cahill, Mixon, and Nelson. We use algebraic computations and certificates in order to prove injectivity.
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