Note about holomorphic maps on a compact Riemann surface

有限维构造的自对偶性及相关模空间在闭黎曼曲面上的分层全纯辛空间

Finite-dimensional construction of self-duality and related moduli spaces over a closed Riemann surface as stratified holomorphic symplectic spaces

本文通过有限维全纯辛约简，在闭黎曼曲面上构建自对偶性及相关模空间的分层全纯辛结构。

A note on automorphisms and birational transformations of holomorphic symplectic manifolds

We give a necessary and sufficient condition for an automorphism of the Hilbert scheme of points on a K3 surface (non necessarily algebraic) to be induced by an automorphism of the surface. We prove furthermore that the group of birational transformations of a projective irreducible holomorphic symplectic manifold is finitely generated.

关于紧黎曼面之间全纯映射数量的改进界

Improved bounds on the number of holomorphic maps between compact Riemann surfaces

本文基于黎曼面的亏度给出非常数全纯映射数量的新上界，并改进了已知的界。方法结合了全纯微分的拉回研究、数论几何和雅可比品种理论的技术。

Riemann-Roch多项式、MBM类与贫乏IHS流形

Riemann-Roch Polynomials, MBM Classes and Poor IHS Manifolds

本文研究贫乏不可约全纯辛流形，证明其自然锥重合且无全纯叶理，并利用Riemann-Roch多项式根给出检测MBM类及有理曲线存在的统一准则，同时刻画了此类流形的形变空间。

Gromov Invariants for Holomorphic Maps from Riemann Surfaces to Grassmannians

Two compactifications of the space of holomorphic maps of fixed degree from a compact Riemann surface to a Grassmannian are studied. It is shown that the Uhlenbeck compactification has the structure of a projective scheme and is dominated by the algebraic compactification arising as a Grothendieck Quot scheme. The latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by Bradlow. This gives an effective way of computing certain intersection numbers (known as Gromov invariants) on the space of holomorphic maps into Grassmannians. We ca…

Note on holomorphic Morse inequalities tensoring with a coherent sheaf

In this note, we show various versions of holomorphic Morse inequalities tensoring with a coherent sheaf.

Proper holomorphic mappings onto symmetric products of a Riemann surface

We show that the structure of proper holomorphic maps between the $n$-fold symmetric products, $n\geq 2$, of a pair of non-compact Riemann surfaces $X$ and $Y$, provided these are reasonably nice, is very rigid. Specifically, any such map is determined by a proper holomorphic map of $X$ onto $Y$. This extends existing results concerning bounded planar domains, and is a non-compact analogue of a phenomenon observed in symmetric products of compact Riemann surfaces. Along the way, we also provide a condition for the complete hyperbolicity of all $n$-fold symmetric products of a non-compact Riema…

Notes on J-Holomorphic Maps

These notes present a systematic treatment of local properties of J-holomorphic maps and of Gromov's convergence for sequences of such maps, specifying the assumptions needed for all statements. In particular, only one auxiliary statement depends on the manifold being symplectic. The content of these notes roughly corresponds to Chapters 2 and 4 of McDuff-Salamon's book on the subject.

Projective structures on Riemann surface and natural differential operators

We investigate the holomorphic differential operators on a Riemann surface $M$. This is done by endowing $M$ with a projective structure. Let $\mathcal L$ be a theta characteristic on $M$. We explicitly describe the jet bundle $J^k(E\otimes {\mathcal L}^{\otimes n})$, where $E$ is a holomorphic vector bundle on $M$ equipped with a holomorphic connection, for all $k$ and $n$. This provides a description of holomorphic differential operators from $E\otimes {\mathcal L}^{\otimes n}$ to another holomorphic vector bundle $F$ using the natural isomorphism $\text{Diff}^k(E\otimes {\mathcal L}^{\otime…

紧黎曼面上全纯主丛的全纯李代数胚联络

Holomorphic Lie algebroid connections on holomorphic principal bundles on compact Riemann surfaces

研究紧黎曼面上Γ等变全纯李代数胚对全纯主G丛等变联络的影响，证明非分裂李代数胚时所有主丛都有等变联络，分裂时给出四个等价条件，并将结果转化到抛物G丛语境。

Moduli spaces of holomorphic triples over compact Riemann surfaces

A holomorphic triple over a compact Riemann surface consists of two holomorphic vector bundles and a holomorphic map between them. After fixing the topological types of the bundles and a real parameter, there exist moduli spaces of stable holomorphic triples. In this paper we study non-emptiness, irreducibility, smoothness, and birational descriptions of these moduli spaces for a certain range of the parameter. Our results have important applications to the study of the moduli space of representations of the fundamental group of the surface into unitary Lie groups of indefinite signature, whic…

Variation of the holomorphic determinant bundle

In this paper, we prove that the Grothendieck-Riemann-Roch formula in Deligne cohomology computing the determinant of the cohomology of a holomorphic vector bundle on the fibers of a proper submersion between abstract complex manifolds is invariant by deformation of the bundle.

Holomorphic sections of line bundles on the jet spaces of the Riemann sphere

Fix a point $t_0$ in the circle $S^1$. The space $J^k(t_0, \mathbb{P}^1)$ of $k$-jets at $t_0$ of $C^{\infty}$ maps from $S^1$ to the Riemann sphere $\mathbb{P}^1$ is a $k+1$ dimensional complex algebraic manifold. We identify a class of holomorphic sections of line bundles on $J^k(t_0, \mathbb{P}^1)$.

L^2-Riemann-Roch Inequalities for Covering Manifolds

We study the existence of $L^2$ holomorphic sections of invariant line bundles over Galois coverings of Zariski open sets in Moishezon manilolds. We show that the von Neuman dimension of the space of $L^2$ holomorphic sections is bounded below under reasonable curvature conditions. We also give criteria for a a compact complex space with isolated singularities and some related strongly pseudoconcave manifolds to be Moishezon. Their coverings are then studied with the same methods. As applications we give weak Lefschetz theorems using the Napier--Ramachandran proof of the Nori theorem.

The Geometry of the Space of Holomorphic Maps from a Riemann Surface into Complex Projective Space

In this paper we study the topology of the spaces Hol(M,P{n},k) of (basepoint preserving) holomorphic maps of a given degree k from a Riemann surface M of genus g>0 into the n-th complex projective space P{n}, n>0. Using symmetric products of the surface as well as the geometry of the associated Abel-Jacobi maps, we construct an explicit topological compactification of Hol(M,P{n},k) which allow us to construct spectral sequences with known E_1-terms that can be used to determine the homology groups of Hol(M,P{n},k). Complete calculations are given for the elliptic case (g=1). Complete rational…

A canonical connection on bundles on Riemann surfaces and Quillen connection on the theta bundle

We investigate the symplectic geometric and differential geometric aspects of the moduli space of connections on a compact Riemann surface $X$. Fix a theta characteristic $K^{1/2}_X$ on $X$; it defines a theta divisor on the moduli space ${\mathcal M}$ of stable vector bundles on $X$ of rank $r$ degree zero. Given a vector bundle $E \in {\mathcal M}$ lying outside the theta divisor, we construct a natural holomorphic connection on $E$ that depends holomorphically on $E$. Using this holomorphic connection, we construct a canonical holomorphic isomorphism between the following two: \begin{enumer…

On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces

We study holomorphic $(n+1)$-chains $E_n\to E_{n-1} \to >... \to E_0$ consisting of holomorphic vector bundles over a compact Riemann surface and homomorphisms between them. A notion of stability depending on $n$ real parameters was introduced in the work of the first two authors and moduli spaces were constructed by the third one. In this paper we study the variation of the moduli spaces with respect to the stability parameters. In particular we characterize a parameter region where the moduli spaces are birationally equivalent. A detailed study is given for the case of 3-chains, generalizing…

紧致黎曼面上随机全纯截面的洞事件

Hole event for random holomorphic sections on compact Riemann surfaces

本文研究了紧致黎曼面上正线丛全纯截面在固定开子集外不为零的条件零点期望，证明了当幂趋于无穷时零点等分布，并由此产生了一个禁止集。

Quaternionic holomorphic geometry: Pluecker formula, Dirac eigenvalue estimates and energy estimates of harmonic 2-tori

The paper develops the fundamentals of quaternionic holomorphic curve theory. The holomorphic functions in this theory are conformal maps from a Riemann surface into the 4-sphere, i.e., the quaternionic projective line. Basic results such as the Riemann-Roch Theorem for quaternionic holomorphic vector bundles, the Kodaira embedding and the Pluecker relations for linear systems are proven. Interpretations of these results in terms of the differential geometry of surfaces in 3- and 4-space are hinted at throughout the paper. Applications to estimates of the Willmore functional on constant mean c…

紧 Riemann 曲面上 Lie 代数胚联络的一个判据

A criterion for Lie algebroid connections on a compact Riemann surface

针对稳定全纯向量丛所定义的紧 Riemann 曲面上全纯 Lie 代数胚，本文给出向量丛容许 Lie 代数胚联络的充要判据。

The Pontryagin rings of moduli spaces of arbitrary rank holomorphic bundles over a Riemann surface

The cohomology of the moduli spaces of stable bundles M(n,d), of coprime rank n and degree d, over a Riemann surface (of genus g > 1) have been intensely studied over the past three decades. We prove in this paper that the Pontryagin ring of M(n,d) vanishes in degrees above 2n(n-1)(g-1) and that this bound is strict (i.e. there exists a non-zero element of degree 2n(n-1)(g-1) in Pont(M(n,d)).) This result is a generalisation of a 1967 Newstead conjecture that Pont(M(2,1)) vanished above 4(g-1) (or equivalently that β^g =0.) These results have been independently proved by Lisa Jeffrey and Jonat…

Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface

We prove formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) on a Riemann surface of genus g greater than or equal to 2. We also use these formulas for intersection numbers to obtain a proof of the Verlinde formula for the dimension of the space of holomorphic sections of a line bundle over M(n,d).

The spectral theory of generalized Laplacians associated to integrable metrics on compact Riemann surfaces

We extend the spectral theory of generalized Laplacians to integrable metrics on compact Riemann surfaces. As a consequence, we attach in a direct way, a holomorphic analytic torsion to any integrable metrics. We also provide a different approach to define the holomorphic analytic torsion. We prove that both approaches agree.

Bordered Riemann surfaces in C^2

One of the oldest open problems in the classical function theory is whether every open Riemann surface admits a proper holomorphic embedding into C^2. In this paper we prove the following Theorem: If D is a bordered Riemann surface whose closure admits an injective immersion in C^2 that is holomorphic in D, then D admits a proper holomorphic embedding in C^2. The most general earlier results are due to J. Globevnik and B. Stensones (Math. Ann. 303 (1995), 579-597) and E. F. Wold (Internat. J. Math. 17 (2006), 963-974). We give an explicit and elementary construction that does not require the T…

On Kaehler structures over symmetric products of a Riemann surface

Given a positive integer $n$ and a compact connected Riemann surface $X$, we prove that the symmetric product $S^n(X)$ admits a Kaehler form of nonnegative holomorphic bisectional curvature if and only if $\text{genus}(X) \leq 1$. If $n$ is greater than or equal to the gonality of $X$, we prove that $S^n(X)$ does not admit any Kaehler form of nonpositive holomorphic sectional curvature.

Computation of the topological type of a real Riemann surface

We present an algorithm for the computation of the topological type of a real compact Riemann surface associated to an algebraic curve, i.e., its genus and the properties of the set of fixed points of the anti-holomorphic involution $τ$, namely, the number of its connected components, and whether this set divides the surface into one or two connected components. This is achieved by transforming an arbitrary canonical homology basis to a homology basis where the $\mathcal{A}$-cycles are invariant under the anti-holomorphic involution $τ$.

Holomorphic $\mathfrak{sl}(2,\mathbb C)$-systems with Fuchsian monodromy (with an appendix by Takuro Mochizuki)

For every integer $g \,\geq\, 2$ we show the existence of a compact Riemann surface $Σ$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${\mathcal O}^{\oplus 2}_Σ$ admits holomorphic connections with $\text{SL}(2,{\mathbb R})$ monodromy and maximal Euler class. Such a monodromy representation is known to coincide with the Fuchsian uniformizing representation for some Riemann surface of genus $g$. The construction carries over to all very stable and compatible real holomorphic structures for the topologically trivial rank two bundle over $Σ$ and gives the existence of holo…

A Riemann-Roch-Hirzebruch formula for traces of differential operators

Let D be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an n-dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, for the Lefschetz number of D as the integral over the manifold of a differential form. The class of this differential form is obtained via formal differential geometry from the canonical generator of the Hochschild cohomology of the algebra of differential operators in a formal neighbourhood of a point. If D is the identity, the formula reduces to the Riemann--Roch--Hirzebruch formula.

紧黎曼曲面上随机零点空洞概率的研究

Hole probabilities of random zeros on compact Riemann surfaces

本文研究紧黎曼曲面上随机全纯截面零点的空洞概率，建立了其收敛速度估计。

Dessins d'enfants and some holomorphic structures on the Loch Ness Monster

The classical theory of dessin d'enfants, which are bipartite maps on compact orientable surfaces, are combinatorial objects used to study branched covers between compact Riemann surfaces and the absolute Galois group of the field of rational numbers. In this paper, we show how this theory is naturally extended to non-compact orientable surfaces and, in particular, we observe that the Loch Ness monster (the surface of infinite genus with exactly one end) admits infinitely many regular dessins d'enfants (either chiral or reflexive). In addition, we study different holomorphic structures on the …

K-theory of the moduli of bundles over a Riemann surface and deformations of the Verlinde algebra

I conjecture that index formulas for $K$-theory classes on the moduli of holomorphic $G$-bundles over a compact Riemann surface $Σ$ are controlled, in a precise way, by Frobenius algebra deformations of the Verlinde algebra of $G$. The Frobenius algebras in question are twisted $K$-theories of $G$, equivariant under the conjugation action, and the controlling device is the equivariant Gysin map along the "product of commutators" from $G^{2g}$ to $G$. The conjecture is compatible with naive virtual localization of holomorphic bundles, from $G$ to its maximal torus; this follows by localization …