Complete scalar-flat Kähler metrics on affine algebraic manifolds

Let $$(X,L_{X})$$ ( X , L X ) be an n-dimensional polarized manifold. Let D be a smooth hypersurface defined by a holomorphic section of $$L_{X}$$ L X . We prove that if D has a constant positive scalar curvature Kähler metric, $$X {\setminus } D$$ X \ D admits a complete scalar-flat Kähler metric, under the following three conditions: (i) $$n \ge 6$$ n ≥ 6 and there is no nonzero holomorphic vector field on X vanishing on D, (ii) the average of a scalar curvature on D denoted by $${\hat{S}}_{D}$$ S ^ D satisfies the inequality $$0< 3 {\hat{S}}_{D} < n(n-1)$$ 0 < 3 S ^ D < n ( n - 1 ) , (iii) there are positive integers $$l(>n),m$$ l ( > n ) , m such that the line bundle $$K_{X}^{-l} \otimes L_{X}^{m}$$ K X - l ⊗ L X m is very ample and the ratio m/l is sufficiently small.

On the set of divisors with zero geometric defect

Let {f:\mathbb{C}\to X} be a transcendental holomorphic curve into a complex projective manifold X. Let L be a very ample line bundle on {X.} Let s be a very generic holomorphic section of L and D the zero divisor given by {s.} We prove that the geometric defect of D (defect of truncation 1) with respect to f is zero. We also prove that f almost misses general enough analytic subsets on X of codimension 2.

Convergence of the Yang–Mills–Higgs flow on Gauged Holomorphic maps and applications

The symplectic vortex equations admit a variational description as global minimum of the Yang–Mills–Higgs functional. We study its negative gradient flow on holomorphic pairs [Formula: see text] where [Formula: see text] is a connection on a principal [Formula: see text]-bundle [Formula: see text] over a closed Riemann surface [Formula: see text] and [Formula: see text] is an equivariant map into a Kähler Hamiltonian [Formula: see text]-manifold. The connection [Formula: see text] induces a holomorphic structure on the Kähler fibration [Formula: see text] and we require that [Formula: see text] descends to a holomorphic section of this fibration. We prove a Łojasiewicz type gradient inequality and show uniform convergence of the negative gradient flow in the [Formula: see text]-topology when [Formula: see text] is equivariantly convex at infinity with proper moment map, [Formula: see text] is holomorphically aspherical and its Kähler metric is analytic. As applications we establish several results inspired by finite dimensional GIT: First, we prove a certain uniqueness property for the critical points of the Yang–Mills–Higgs functional which is the analogue of the Ness uniqueness theorem. Second, we extend Mundet’s Kobayashi–Hitchin correspondence to the polystable and semistable case. The arguments for the polystable case lead to a new proof in the stable case. Third, in proving the semistable correspondence, we establish the moment–weight inequality for the vortex equation and prove the analogue of the Kempf existence and uniqueness theorem.

Positivity of metrics on conic neighborhoods of 1-convex submanifolds

Let [Formula: see text] be a holomorphic submersion from a complex manifold [Formula: see text] onto a 1-convex manifold [Formula: see text] with exceptional set [Formula: see text] and [Formula: see text] a holomorphic section. Let [Formula: see text] be a plurisubharmonic exhaustion function which is strictly plurisubharmonic on [Formula: see text] with [Formula: see text] For every holomorphic vector bundle [Formula: see text] there exists a neighborhood [Formula: see text] of [Formula: see text] for [Formula: see text] conic along [Formula: see text] such that [Formula: see text] can be endowed with Nakano strictly positive Hermitian metric. Let [Formula: see text] [Formula: see text] be a given holomorphic function. There exist finitely many bounded holomorphic vector fields defined on a Stein neighborhood [Formula: see text] of [Formula: see text] conic along [Formula: see text] with zeroes of arbitrary high order on [Formula: see text] and such that they generate [Formula: see text] Moreover, there exists a smaller neighborhood [Formula: see text] such that their flows remain in [Formula: see text] for sufficiently small times thus generating a local dominating spray.

A generalized Poincaré-Lelong formula

We prove a generalization of the classical Poincaré-Lelong formula. Given a holomorphic section $f$, with zero set $Z$, of a Hermitian vector bundle $E\to X$, let $S$ be the line bundle over $X\setminus Z$ spanned by $f$ and let $Q=E/S$. Then the Chern form $c(D_Q)$ is locally integrable and closed in $X$ and there is a current $W$ such that ${dd}^cW=c(D_E)-c(D_Q)-M,$ where $M$ is a current with support on $Z$. In particular, the top Bott-Chern class is represented by a current with support on $Z$. We discuss positivity of these currents, and we also reveal a close relation with principal value and residue currents of Cauchy-Fantappiè-Leray type.

HODGE POLYNOMIALS OF THE MODULI SPACES OF PAIRS

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic pair on X is a couple (E, ϕ), where E is a holomorphic bundle over X of rank n and degree d, and ϕ ∈ H0(E) is a holomorphic section. In this paper, we determine the Hodge polynomials of the moduli spaces of rank 2 pairs, using the theory of mixed Hodge structures. We also deal with the case in which E has fixed determinant.

BIRATIONAL EQUIVALENCE OF HIGGS MODULI

In this paper, we study triples of the form (E, θ, ϕ) over a compact Riemann surface, where (E, θ) is a Higgs bundle and ϕ is a global holomorphic section of the Higgs bundle. Our main result is an description of a birational equivalence which relates geometrically the moduli space of Higgs bundles of rank r and degree d to the moduli space of Higgs bundles of rank r-1 and degree d.

L2 Extension for jets of holomorphic sections of a Hermitian line Bundle

Let (X, ω) be a weakly pseudoconvex Kähler manifold, Y ⊂ X a closed submanifold defined by some holomorphic section of a vector bundle over X, and L a Hermitian line bundle satisfying certain positivity conditions. We prove that for any integer k > 0, any section of the jet sheaf which satisfies a certain L2 condition, can be extended into a global holomorphic section of L over X whose L2 growth on an arbitrary compact subset of X is under control. In particular, if Y is merely a point, this gives the existence of a global holomorphic function with an L2 norm under control and with prescribed values for all its derivatives up to order k at that point. This result generalizes the L2 extension theorems of Ohsawa-Takegoshi and of Manivel to the case of jets of sections of a line bundle. A technical difficulty is to achieve uniformity in the constant appearing in the final estimate. To this end, we make use of the exponential map and of a Rauch-type comparison theorem for complete Riemannian manifolds.

Generalized transversely projective structure on a transversely holomorphic foliation

The results of Biswas (2000) are extended to the situation of transversely projective foliations. In particular, it is shown that a transversely holomorphic foliation defined using everywhere locally nondegenerate maps to a projective spaceℂℙn, and whose transition functions are given by automorphisms of the projective space, has a canonical transversely projective structure. Such a foliation is also associated with a transversely holomorphic section ofN⊗−kfor eachk∈[3,n+1], whereNis the normal bundle to the foliation. These transversely holomorphic sections are also flat with respect to the Bott partial connection.

MODULI OF STABLE PAIRS FOR HOLOMORPHIC BUNDLES OVER RIEMANN SURFACES

It this paper we study the space of gauge equivalence classes of pairs [Formula: see text] where [Formula: see text] represents a holomorphic structure on a complex bundle, E, over a closed Riemann Surface, and ϕ is a holomorphic section. We define a space of stable pairs and consider the moduli space problem for this space. The space of stable pairs, [Formula: see text], is related to the space of solution to the Vortex (Hermitian-Yang-Mills-Higgs) equation. Using the parameter, τ, which appears in this equation we can define subspaces [Formula: see text] within [Formula: see text]. We show that under suitable restrictions on τ and the degree of E, the space [Formula: see text] is naturally a finite dimensional, Hausdorff, compact Kähler manifold. We show further that there is a natural holomorphic map from this space onto the Seshadri compactification of the moduli space of stable bundles and that this map is generically a fibration.