Vasile Brînzănescu

Holomorphic vector bundles over compact complex surfaces

Vector bundles over complex manifolds.- Facts on compact complex surfaces.- Line bundles over surfaces.- Existence of holomorphic vector bundles.- Classification of vector bundles.

A CLASS OF HARMONIC SUBMERSIONS AND MINIMAL SUBMANIFOLDS

We introduce the class of pseudo-horizontally homothetic maps from a Riemann manifold to a Kahler manifold, and we study some of their properties. For example, we prove that a pseudo-horizontally homothetic submersion is harmonic if and only if it has minimal fibres, and a pseudo-horizontally homothetic harmonic submersion pulls back complex submanifolds into minimal submanifolds.

Holomorphic vector bundles on primary Kodaira surfaces

It is in general unknown which topological complex vector bundles on a non-algebraic surface admit holomorphic structures. We solve this problem for primary Kodaira surfaces by using results of Kani on curves of genus two with elliptic differentials. Some of the corresponding moduli spaces will be smooth compact and holomorphically symplectic.

Moduli spaces of vector bundles over ruled surfaces

We study moduli spaces M(c1, c2, d, r) of isomorphism classes of algebraic 2-vector bundles with fixed numerical invariants c1, c2, d, r over a ruled surface. These moduli spaces are independent of any ample line bundle on the surface. The main result gives necessary and sufficient conditions for the nonemptiness of the space M(c1, c2, d, r) and we apply this result to the moduli spaces ML(c1, c2) of stable bundles, where L is an ample line bundle on the ruled surface. Introduction Let π : X → C be a ruled surface over a smooth algebraic curve C, defined over the complex number field C. Let f be a fibre of π. Let c1 ∈ Num(X) and c2 ∈ H 4(X,Z) ∼= Z be fixed. For any polarization L, denote the moduli space of rank-2 vector bundles stable with respect to L in the sense of Mumford-Takemoto by ML(c1, c2). Stable 2-vector bundles over a ruled surface have been investigated by many authors; see, for example [T1], [T2], [H-S], [Q1]. Let us mention that Takemoto [T1] showed that there is no rank-2 vector bundle (having c1.f even) stable with respect to every polarization L. In this paper we shall study algebraic 2-vector bundles over ruled surfaces, but we adopt another point of view: we shall study moduli spaces of (algebraic) 2-vector bundles over a ruled surface X, which are defined independent of any ample divisor (line bundle) on X, by taking into account the special geometry of a ruled surface (see [B], [B-St1], [B-St2] and also [Br1], [Br2], [W]). In Section 1 (put for the convenience of the reader) we present (see [B]) two numerical invariants d and r for a 2-vector bundle with fixed Chern classes c1 and c2 and we define the set M(c1, c2, d, r) of isomorphism classes of bundles with fixed invariants c1, c2, d, r. The integer d is given by the splitting of the bundle on the general fibre and the integer r is given by some normalization of the bundle. Recall that the set M(c1, c2, d, r) carries Received February 8, 1996. 111 https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000025332 Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 06 Dec 2018 at 05:10:24, subject to the Cambridge Core terms of use, available at 112 M. APRODU AND V. BRı̂NZǍNESCU a natural structure of an algebraic variety (see [B], [B-St1], [B-St2]). In Section 2 we study uniform vector bundles and we prove the existence of algebraic vector bundles given by extensions of line bundles and which are not uniform. In Section 3 the main result gives necessary and sufficient conditions for the non-emptiness of the space M(c1, c2, d, r) and we apply this result to the moduli space of stable bundles ML(c1, c2). §1. Moduli spaces of rank-2 vector bundles In this section we shall recall from ([B], [B-St1], [B-St2]) some basic notions and facts. The notations and the terminology are those of Hartshorne’s book [Ha]. Let C be a nonsingular curve of genus g over the complex number field and let π : X→C be a ruled surface over C. We shall write X ∼= P(E) where E is normalized. Let us denote by e the divisor on C corresponding to ∧2 E and by e = − deg(e). We fix a point p0 ∈ C and a fibre f0 = π (p0) of X. Let C0 be a section of π such that OX(C0) ∼= OP(E)(1). Any element of Num(X) ∼= H2(X,Z) can be written aC0 + bf0 with a, b ∈ Z. We shall denote by OC(1) the invertible sheaf associated to the divisor p0 on C. If L is an element of Pic(C) we shall write L = OC(k)⊗L0, where L0 ∈ Pic0(C) and k = deg(L). We also denote by F (aC0 + bf0) = F ⊗OX(a) ⊗ π OC(b) for any sheaf F on X and any a, b ∈ Z. Let E be an algebraic rank-2 vector bundle on X with fixed numerical Chern classes c1 = (α, β) ∈ H 2(X,Z) ∼= Z × Z, c2 = γ ∈ H 4(X,Z) ∼= Z, where α, β, γ ∈ Z. Since the fibres of π are isomorphic to P1 we can speak about the generic splitting type of E and we have E|f ∼= Of (d) ⊕ Of (d ′ ) for a general fibre f , where d ′ ≤ d, d + d ′ = α. The integer d is the first numerical invariant of E. The second numerical invariant is obtained by the following normalization: −r = inf{l| ∃L ∈ Pic(C),deg(L) = l, s.t. H(X,E(−dC0) ⊗ π L) 6= 0}. We shall denote by M(α, β, γ, d, r) or M(c1, c2, d, r) or M the set of isomorphism classes of algebraic rank-2 vector bundles on X with fixed Chern classes c1, c2 and invariants d and r. With these notations we have the following result (see [B]): https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000025332 Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 06 Dec 2018 at 05:10:24, subject to the Cambridge Core terms of use, available at MODULI VECTOR BUNDLES OVER RULED SURFACES 113 Theorem 1. For every vector bundle E ∈ M(c1, c2, d, r) there exist L1, L2 ∈ Pic0(C) and Y ⊂ X a locally complete intersection of codimension 2 in X, or the empty set, such that E is given by an extension 0→OX(dC0 + rf0)⊗π ∗L2→E→OX(d ′ C0 + sf0)⊗π ∗L1⊗IY →0, (1) where c1 = (α, β) ∈ Z × Z, c2 = γ ∈ Z, d + d ′ = α, d ≥ d ′ , r + s = β, l(c1, c2, d, r) := γ + α(de− r) − βd+ 2dr − d 2e = deg(Y ) ≥ 0. Remark. By applying Theorem 1 we can obtain the canonical extensions used in [Br1], [Br2]. Indeed, let us suppose first that d > d ′ . From the exact sequence (1) it follows that OC(r) ⊗ L2 ∼= π∗E(−dC0) so OX(rf0) ⊗ π L2 ∼= π π∗E(−dC0) and OX(dC0 + rf0) ⊗ π L2 ∼= (π π∗E(−dC0))(dC0). If d = d ′ then, by applying π∗ to the short exact sequence 0→OX(rf0) ⊗ π L2→E(−dC0)→OX(sf0) ⊗ π L1 ⊗ IY →0 it follows the exact sequence 0→OC(r) ⊗ L2→π∗E(−dC0)→OC(s) ⊗ L1 ⊗OC(−Z1)→0, where Z1 is an effective divisor on C with the support π(Y ). With the notation Z = π(Z1), by applying π ∗ (π is a flat morphism) we obtain the following commutative diagram with exact rows 0 -OX(rf0) ⊗ πL2 E(−dC0) -OX(sf0) ⊗ πL1 ⊗ IY 0 0 -OX(rf0) ⊗ πL2 ππ∗E(−dC0) OX(sf0) ⊗ πL1 ⊗ IZ 0 6 o id 6φ 6ψ https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000025332 Downloaded from https://www.cambridge.org/core. IP address: 54.70.40.11, on 06 Dec 2018 at 05:10:24, subject to the Cambridge Core terms of use, available at 114 M. APRODU AND V. BRı̂NZǍNESCU From the injectivity of ψ we obtain the injectivity of φ. Because of OX(sf0) ⊗ π ∗L1 ⊗ IY ⊂Z ∼= Coker ψ ∼= Coker φ

Rank-two vector bundles on Hirzebruch surfaces

We survey some parts of the vast literature on vector bundles on Hirzebruch surfaces, focusing on the rank-two case.

Hodge Invariants of Higher-Dimensional Analogues of Kodaira Surfaces

In the paper [17], Sankaran gives a construction of some complex analytic manifolds, which are higher-dimensional analogues of Inoue parabolic surfaces, by using methods of toric geometry (see also [9, 16]). Some higher-dimensional analogues of Kodaira surfaces are obtained as hypersurfaces in these Inoue manifolds. In this paper we construct another higher-dimensional analogues of primary Kodaira surfaces and we compute their invariants as the Hodge numbers.

From string theory to algebraic geometry and back

We describe some facts in physics which go up to the modern string theory and the related concepts in algebraic geometry. Then we present some recent results on moduli‐spaces of vector bundles on non‐Kahler Calabi‐Yau 3‐folds and their consequences for heterotic string theory.