[Paper Review] Holomorphic rank-2 vector bundles on non-Kahler elliptic surfaces

Vasile Brînzănescu, Ruxandra Moraru|ArXiv.org|Jun 11, 2003

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes necessary and sufficient conditions for the existence of holomorphic rank-2 vector bundles on non-Kähler elliptic surfaces by extending the spectral construction to fibrations without a section. The key result provides a discriminant-based criterion involving the Néron-Severi group and base curve genus, resolving the existence problem for rank-2 bundles on these surfaces and identifying non-filtrable bundles via geometric invariants.

ABSTRACT

The existence problem for vector bundles on a smooth compact complex surface consists in determining which topological complex vector bundles admit holomorphic structures. For projective surfaces, Schwarzenberger proved that a topological complex vector bundle admits a holomorphic (algebraic) structure if and only if its first Chern class belongs to the Neron-Severi group of the surface. In contrast, for non-projective surfaces there is only a necessary condition for the existence problem (the discriminant of the vector bundles must be positive) and the difficulty of the problem resides in the lack of a general method for constructing non-filtrable vector bundles. In this paper, we close the existence problem in the rank-2 case, by giving necessary and sufficient conditions for the existence of holomorphic rank-2 vector bundles on non-K\" ahler elliptic surfaces.

Motivation & Objective

• To resolve the long-standing open problem of determining which topological rank-2 vector bundles on non-Kähler elliptic surfaces admit holomorphic structures.
• To extend the spectral construction method to non-Kähler elliptic fibrations without a section, where previous methods fail.
• To provide necessary and sufficient conditions for the existence of such bundles, particularly focusing on non-filtrable cases.
• To clarify how the geometry of the base curve (genus g) influences the discriminant gaps in the existence spectrum.
• To establish a correspondence between line bundles on the surface and sections of its relative Jacobian, enabling the construction of vector bundles.

Proposed method

• Adapts the Fourier-Mukai transform framework to non-Kähler elliptic fibrations by introducing a twisted version specific to these surfaces.
• Uses the relative Jacobian $ J(X) $ of the fibration to define a spectral curve that encodes the isomorphism class of the bundle over each fibre.
• Constructs holomorphic rank-2 bundles via elementary modifications of a base bundle $ E_0' $, starting from a bisection $ \overline{C}_0 $ of $ J(X) $ with minimal self-intersection.
• Introduces a key invariant $ \Delta_0 = m_{c_1} - d/2 $, where $ d $ is a non-negative integer determined by the geometry of the spectral curve, and uses it to define a threshold for discriminant values.
• Applies the condition $ \Delta(2,c_1,c_2) \geq \Delta_0 $, with equality allowing construction of bundles via the spectral correspondence.
• Employs the restriction of bundles to fibres, showing that any such bundle has degree zero on each smooth fibre, which is crucial for the spectral method.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for the existence of holomorphic rank-2 vector bundles on non-Kähler elliptic surfaces?
• RQ2 How does the absence of a section in the fibration affect the construction of vector bundles via spectral methods?
• RQ3 What role does the genus of the base curve $ B $ play in determining the possible values of the discriminant $ \Delta(2,c_1,c_2) $?
• RQ4 Can non-filtrable vector bundles be systematically constructed on non-Kähler elliptic surfaces, and under what conditions?
• RQ5 How does the relative Jacobian $ J(X) $ relate to the classification of line bundles and vector bundles on non-Kähler elliptic surfaces?

Key findings

• A holomorphic rank-2 vector bundle with Chern classes $ c_1 $ and $ c_2 $ exists on a non-Kähler elliptic surface $ X \to B $ if and only if $ \Delta(2,c_1,c_2) \geq m_{c_1} - d/2 $, where $ d $ is a non-negative integer determined by the spectral curve's geometry.
• When $ \Delta(2,c_1,c_2) \in [m_{c_1} - d/2, m_{c_1}) $, the corresponding vector bundles are non-filtrable, resolving a key difficulty in the non-projective case.
• For base curves of genus $ g < 2 $, the existence condition simplifies to $ \Delta(2,c_1,c_2) \geq 0 $, indicating no discriminant gaps.
• For genus $ g > 1 $, discriminant gaps appear when $ m_{c_1} > d/2 $, showing that the base curve's geometry directly constrains the existence spectrum.
• The construction yields infinitely many non-filtrable bundles whenever an irreducible bisection of $ J(X) $ exists, providing a systematic method for their generation.
• The spectral curve $ \overline{C}_0 $ of minimal self-intersection $ 8\Delta_0 $ serves as the starting point for building all such bundles via elementary modifications.