Deformation limit and bimeromorphic embedding of Moishezon manifolds

Let [Formula: see text] be a holomorphic family of compact complex manifolds over an open disk in [Formula: see text]. If the fiber [Formula: see text] for each nonzero [Formula: see text] in an uncountable subset [Formula: see text] of [Formula: see text] is Moishezon and the reference fiber [Formula: see text] satisfies the local deformation invariance for Hodge number of type [Formula: see text] or admits a strongly Gauduchon metric introduced by D. Popovici, then [Formula: see text] is still Moishezon. We also obtain a bimeromorphic embedding [Formula: see text]. Our proof can be regarded as a new, algebraic proof of several results in this direction proposed and proved by Popovici in 2009, 2010 and 2013. However, our assumption with [Formula: see text] not necessarily being a limit point of [Formula: see text] and the bimeromorphic embedding are new. Our strategy of proof lies in constructing a global holomorphic line bundle over the total space of the holomorphic family and studying the bimeromorphic geometry of [Formula: see text]. S.-T. Yau’s solutions to certain degenerate Monge–Ampère equations are used.

Stable Higgs bundles over positive principal elliptic fibrations

Let M be a compact complex manifold of dimension at least three and Π : M → X a positive principal elliptic fibration, where X is a compact Kähler orbifold. Fix a preferred Hermitian metric on M. In [14], the third author proved that every stable vector bundle on M is of the form L⊕ Π ⃰ B0, where B0 is a stable vector bundle on X, and L is a holomorphic line bundle on M. Here we prove that every stable Higgs bundle on M is of the form (L ⊕ Π ⃰B0, Π ⃰ ɸX), where (B0, ɸX) is a stable Higgs bundle on X and L is a holomorphic line bundle on M.

The L2 ∂¯-Cauchy problem on pseudoconvex domains and applications

Let [Formula: see text] be a complex manifold of dimension [Formula: see text] and [Formula: see text] be a weakly pseudoconvex domain with smooth boundary in [Formula: see text]. Let [Formula: see text] be a holomorphic line bundle over [Formula: see text] which is positive on a neighborhood of [Formula: see text]. Let [Formula: see text] be the [Formula: see text]-times tensor product of [Formula: see text] for positive integer [Formula: see text]. The purpose of this paper is to study the [Formula: see text]-problem with support conditions in [Formula: see text] for forms of type [Formula: see text], [Formula: see text] with values in [Formula: see text]. Applications to the [Formula: see text]-problem for smooth forms on boundaries of [Formula: see text] are given.

A Stein Criterion Via Divisors for Domains Over Stein Manifolds

It is shown that a domain $X$ over a Stein manifold is Stein if the following two conditions are fulfilled: a) the cohomology group $H^i(X,\mathscr{O})$ vanishes for $i \geq 2$ and b) every topologically trivial holomorphic line bundle over $X$ admits a non-trivial meromorphic section. As a consequence we recover, with a different proof, a known result due to Siu stating that a domain $X$ over a Stein manifold $Y$ is Stein provided that $H^i(X,\mathscr{O})=0$ for $i \geq 1$.

Vanishing Theorems on Compact Hyper-kähler Manifolds

We prove that if B is a k-positive holomorphic line bundle on a compact hyper-kähler manifold M, then HpM,Ωq⊗B=0 for P>n+[k/2] with q a nonnegative integer. In a special case, k=0 and q=0, we recover a vanishing theorem of Verbitsky’s with a little stronger assumption.

RANK ONE CONNECTIONS ON ABELIAN VARIETIES, II

Given a holomorphic line bundle L on a compact complex torus A, there are two naturally associated holomorphic ΩA-torsors over A: one is constructed from the Atiyah exact sequence for L, and the other is constructed using the line bundle [Formula: see text], where α is the addition map on A × A, and p1 is the projection of A × A to the first factor. In [I. Biswas, J. Hurtvbise and A. K. Raina, Rank one connections on abelian varieties, Internat. J. Math.22 (2011) 1529–1543], it was shown that these two torsors are isomorphic. The aim here is to produce a canonical isomorphism between them through an explicit construction.

On moduli spaces of Hitchin pairs

LetXbe a compact Riemann surfaceXof genus at–least two. Fix a holomorphic line bundleLoverX. Letbe the moduli space of Hitchin pairs (E, φ ∈H0(End0(E) ⊗L)) overXof rankrand fixed determinant of degreed. The following conditions are imposed:(i)deg(L) ≥ 2g−2,r≥ 2, andL⊗rKX⊗r;(ii)(r, d) = 1; and(iii)ifg= 2 thenr≥ 6, and ifg= 3 thenr≥ 4.We prove that that the isomorphism class of the varietyuniquely determines the isomorphism class of the Riemann surfaceX. Moreover, our analysis shows thatis irreducible (this result holds without the additional hypothesis on the rank for low genus).

CHEN–RUAN COHOMOLOGY OF SOME MODULI SPACES, II

Let X be a compact connected Riemann surface of genus at least two. Let r be a prime number and ξ → X a holomorphic line bundle such that r is not a divisor of degree ξ. Let [Formula: see text] denote the moduli space of stable vector bundles over X of rank r and determinant ξ. By Γ we will denote the group of line bundles L over X such that L⊗r is trivial. This group Γ acts on [Formula: see text] by the rule (E, L) ↦ E ⊗ L. We compute the Chen–Ruan cohomology of the corresponding orbifold.