Let $M_{d}(¶^r)$ be the space of $(r+1)$-tuples $(f_0,...,f_r)$ modulo homothety, where $f_0,...,f_r$ are homogeneous polynomials of degree $d$ in two variables. Let $M_{d}^{\circ}(¶^r)$ be the open subset of $M_{d}(¶^r)$ such that $f_0,...,f_r$ have no common zeros. Then $M_{d}^{\circ}(¶^r)$ parametrizes the space of holomorphic maps of degree $d$ from $¶^1$ into $¶^r$. In general the boundary divisor $M_{d}(¶^r) \setminus M_{d}^{\circ}(¶^r)$ is not normal crossing. In this paper we will give a natural stratification of this boundary and show that we can process an iterated blow-ups along these strata (or its proper transformations) to obtain a compactification of $M_{d}^{\circ}(¶^n)$ with normal crossing divisors.
In this paper, we prove the moduli spaces of genus zero stable log maps to a large class of wonderful compactifications are irreducible and unirational.
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…
We construct quasi-projective moduli spaces of $K$-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily--Borel compactification and investigate a relation between one-dimensional boundary components and equivalence classes of rational Lagrangian fibrations defined on mirror manifolds.
We construct a new compactification of the moduli space of maps from pointed nonsingular projective stable curves to a nonsingular projective variety with prescribed ramification indices at the points. It is shown to be a proper Deligne-Mumford stack equipped with a natural virtual fundamental class.
In this paper we compute the multiple cover Gromov-Witten integrals (analog of the Aspinwall-Morrison formula) for the unramified compactification of the moduli space of stable maps to an embedded $\PO$ in a Calabi-Yau threefold $X$ with the normal bundle $\mathcal{O}_{\PO}(-1) \bigoplus \mathcal{O}_{\PO}(-1)$.
We provide a natural smooth projective compactification of the space of algebraic maps from the projective line to the projective space of dimension n by adding a divisor with simple normal crossings.
The paper studies the deformation theory of a holomorphic surjective map from a normal compact complex space to a compact Kaehler manifold and describes the component of the space of holomorphic maps, generalizing results in the projective case by Kebekus and the authors.
We present a new compactification $M(d,n)$ of the moduli space of self-maps of $\mathbb{CP}^1$ of degree $d$ with $n$ markings. It is constructed via GIT from the stable maps moduli space $\ ar M_{0,n}(\mathbb{CP}^1 \times \mathbb{CP}^1, (1,d))$. We show that it is the coarse moduli space of a smooth Deligne-Mumford stack and we compute its rational Picard group. Using the recursive boundary structure inherited from the stable maps space, we give an explicit algorithm for computing top-intersection numbers of divisors on $M(d,n)$. We also study the $m$-fold iteration map $M(d,n) \dashrightarro…
We compactify the classical moduli variety $A_g$ of principally polarized abelian varieties of complex dimension $g$ by attaching the moduli of flat tori of real dimensions at most $g$ in an explicit manner. Equivalently, we explicitly determine the Gromov-Hausdorff limits of principally polarized abelian varieties. This work is analogous to the first of our series (available at arXiv:1406.7772v2), which compactified the moduli of curves by attaching the moduli of metrized graphs. Then, we also explicitly specify the Gromov-Hausdorff limits along holomorphic family of abelian varieties and sho…
In this paper we show a general method to compactify certain open varieties by adding normal crossing divisors. This is done by proving that {\it blowing up along an arrangement of subvarieties} can be carried out. Important examples such as Ulyanov's configuration spaces, spaces of holomorphic maps, etc., are covered. Intersection ring and (non-recursive) Hodge polynomails are computed. Further general structures arising from the blowup process are described and studied.
We use the techniques of birational algebraic geometry and some combinatorial arguments related to weighted trees to study the structure of resolutions of compactifications of hypothetical counterexamples to the two-dimensional Jacobian Conjecture. We obtain especially detailed results for the Stein decomposition of such maps. We hope that our approach will inspire more birational geometers to seriously work on the Jacobian Conjecture.
Answering a problem raised by Lazarsfeld, Hwang and Mok proved that a surjective holomorphic map from a rational homogeneous space of Picard number 1 onto projective manifold different from projective space must be a biholomorphism. THe aim of this paper is to generalized this result to irreducible rational homogeneous space of higher Picard number.
A 1-parameter variation of Hodge structures corresponds to a holomorphic, horizontal, locally liftable map into a classifying space of Hodge structures. In this paper it is shown that such a map has a limit in the reductive Borel-Serre compactification of the classifying space. The boundary component in which the limit lies is a union over possible polarizations of classifying spaces of Hodge structures on the primitive parts. It is discussed which boundary components can contain such limit points.
We use the theory of twisted stable maps to Deligne-Mumford stacks to construct compactifications of the moduli space of pairs $(X \to C, S + F)$ where $X \to C$ is a fibered surface, $S$ is a sum of sections, $F$ is a sum of marked fibers, and $(X, S + F)$ is a stable pair in the sense of the minimal model program. This generalizes the work of Abramovich-Vistoli, who compactified the moduli space of fibered surfaces without marked fibers. Furthermore, we compare our compactification to Alexeev's space of stable maps and the KSBA compactification of stable pairs. As an application, we describe…
In this paper, we study deformations of holomorphic Poisson maps which extend Horikawa's series of papers on deformations of holomorphic maps in the context of holomorphic Poisson deformations. In appendices, we present deformations of Poisson morphisms in the language of functors of Artin rings which is the algebraic version of deformations of holomorphic Poisson maps. We identity first-order deformations and obstructions.
We give a new proof of the classification of normal singular surface germs admitting non-invertible holomorphic self-maps and due to J. Wahl. We then draw an analogy between the birational classification of singular holomorphic foliations on surfaces, and the dynamics of holomorphic maps. Following this analogy, we introduce the notion of minimal holomorphic model for holomorphic maps. We give sufficient conditions which ensure the uniqueness of such a model.
Drinfeld's relative compactification plays a basic role in the theory of automorphic sheaves, and its singularities encode representation-theoretic information in the form of intersection cohomology. We introduce a resolution of singularities consisting of stable maps from nodal deformations of the curve into twisted flag varieties. As an application, we prove that the twisted intersection cohomology sheaf on Drinfeld's compactification is universally locally acyclic over the moduli stack of $G$-bundles at points sufficiently antidominant relative to their defect.
We show that any separated essentially finite-type map $f$ of noetherian schemes globally factors as $f = hi$ where $i$ is an injective localization map and $h$ a separated finite-type map. In particular, via Nagata's compactification theorem, $h$ can be chosen to be proper. We apply these results to Grothendieck duality. We also obtain other factorization results and provide essentialized versions of many general results such as Zariski's Main Theorem, Chow's Lemma, and blow-up descriptions of birational maps.
We prove that the space of surjective holomorphic maps from a compact complex manifold to a compact Kähler manifold with trivial canonical class and finite fundamental group is discrete.
We study deformations of holomorphic maps of compact, complex, Kähler manifolds. In particular, we describe a generalization of Bloch's semiregularity map that annihilates obstructions to deform holomorphic maps with fixed codomain.
We use orbifold structures to deduce degeneracy statements for holomorphic maps into logarithmic surfaces. We improve former results in the smooth case and generalize them to singular pairs. In particular, we give applications on nodal surfaces and complements of singular plane curves.
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.
We obtain results on approximation of holomorphic maps by algebraic maps, jet transversality theorems for holomorphic and algebraic maps, and the homotopy principle for holomorphic submersions of Stein manifolds to certain algebraic manifolds.
We construct a compactification of the moduli space of Drinfeld modules of rank $r$ and level $N$ as a moduli space of $A$-reciprocal maps. This is closely related to the Satake compactification, but not exactly the same. The construction involves some technical assumptions on $N$ that are satisfied for a cofinal set of ideals $N$. In the special case $A={\mathbb F}_q[t]$ and $N=(t^n)$ we obtain a presentation for the graded ideal of Drinfeld cusp forms of level $N$ and all weights and can deduce a dimension formula for the space of cusp forms of any weight. We expect the same results in gener…
We give a brief survey on the entropy of holomorphic self maps $f$ of compact Kähler manifolds, and rational dominating self maps $f$ of smooth projective varieties. We emphasize the connection between the entropy and the spectral radii of the induced action of $f$ on the homology of the compact manifold. The main conjecture for the rational maps states that modulo birational isomorphism all various notions of entropy and the spectral radii are equal.
Let X be a complex nonsingular affine algebraic variety, K a holomorphically convex subset of X, and Y a homogeneous variety for some complex linear algebraic group. We prove that a holomorphic map f:K-->Y can be uniformly approximated on K by regular maps K-->Y if and only if f is homotopic to a regular map K-->Y. However, it can happen that a null homotopic holomorphic map K-->Y does not admit uniform approximation on K by regular maps X-->Y. Here, a map g:K-->Y is called holomorphic (resp. regular) if there exist an open (resp. a Zariski open) neighborhood U of K in X and a holomorphic (res…
We construct the deformation functor associated to a couple of morphisms of differential graded Lie algebras, and use it to study the infinitesimal deformations of a holomorphic map of compact complex manifolds. In particular, in the case of Kähler manifolds we describe a generalizzation of the semiregularity map.
We construct the deformation functor associated with a pair of morphisms of differential graded Lie algebras, and use it to study infinitesimal deformations of holomorphic maps of compact complex manifolds. In particular, using L-infinity structures, we give an explicit description of the differential graded Lie algebra that controls this problem.
We consider the maximal number of arbitrary points in a special fibre that can be simultaneously approached by points in one sequence of general fibres. Several results about this topological invariant and their applications describe the structure of holomorphic maps. In particular, we get a lower bound on the number of points in the general fibre of a generically finite map.
In this paper we provide sufficient conditions for maps of vector bundles on smooth projective varieties to be uniquely determined by their degeneracy schemes. We then specialize to holomorphic distributions and foliations. In particular, we provide sufficient conditions for distributions of arbitrary rank on projective spaces to be uniquely determined by their singular schemes.
We develop a new approach to the study of the multipoint loci of holomorphic maps between complex manifolds. We relate the $k$-fold locus to the curvilinear component of the Hilbert scheme of $k$ points on the source space of the map, and using equivariant localisation, we derive a closed iterated multipoint residue formula. Our work is motivated by ideas and conjectures of M. Kazarian and R. Rimányi on residual polynomials.
We construct $d$-closed and $dd^c$-closed positive currents associated to a holomorphic map $ϕ$ via cluster points of normalized weighted truncated image currents. They are constructed using analogues of the Ahlfors length-area inequality in higher dimensions. Such classes of currents are referred to as Ahlfors currents. We give some applications to equidistribution problems in value distribution theory.
We show that the periods of the holomorphic Eisenstein series of level 1, parallel weight 2, on a Hilbert modular surface are not rational, even up to scaling. This is deduced from a study of the mixed Hodge structure on the cohomology of the Hilbert modular surface, where we find extension classes related to the units of the real quadratic field. We prove similar results for Hilbert modular varieties of all dimensions, and produce extensions of Galois representations in étale cohomology. The key point is to study the restriction of the canonical extension of the Hodge bundle to the boundary o…
We find good dynamical compactifications for arbitrary polynomial mappings of C^2 and use them to show that the degree growth sequence satisfies a linear integral recursion formula. For maps of low topological degree we prove that the Green function is well behaved. For maps of maximum topological degree, we give normal forms.
Let $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle $\mathscr{O}_{T_M^*}(1)$ is not pseudo-effective and $\mathscr{O}_{T_N^*}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$. In particular, we prove that any holomorphic map from a compact complex manifold $M$ with RC-positive tangent bundle to a compact complex manifold $N$ with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature …
