[Paper Review] Topological Formulae for the Zeroth Cohomology of Line Bundles on Surfaces

Callum R. Brodie, Andrei Constantin+2 more|arXiv (Cornell University)|Jun 19, 2019

Algebraic Geometry and Number Theory5 references9 citations

TL;DR

This paper introduces a set of lattice transformations on the Picard group of complex projective surfaces that convert effective line bundles into nef ones while preserving the dimension of their zeroth cohomology. By combining these transforms with vanishing theorems, the authors compute the cohomology dimension via a topological index, demonstrating the method on del Pezzo and Hirzebruch surfaces.

ABSTRACT

We identify a set of transforms on the Picard lattice of non-singular complex projective surfaces that map effective line bundles to nef line bundles, while preserving the dimension of the zeroth cohomology. These transforms can often be used in conjunction with vanishing theorems to compute the dimension of the zeroth cohomology in terms of a topological index. The method is illustrated on del Pezzo and Hirzebruch surfaces.

Motivation & Objective

• To develop a systematic method for computing the dimension of the zeroth cohomology of line bundles on complex projective surfaces.
• To address the challenge of computing h⁰(L) for effective line bundles when standard vanishing theorems do not apply.
• To identify transformations on the Picard lattice that preserve h⁰(L) while mapping effective bundles to nef ones.
• To enable cohomology computation via topological index by leveraging vanishing theorems after transformation.
• To demonstrate the method’s effectiveness on specific classes of surfaces, including del Pezzo and Hirzebruch surfaces.

Proposed method

• Define a set of linear transformations on the Picard lattice that preserve the Euler characteristic and h⁰(L) for line bundles.
• Apply these transforms to effective line bundles to produce equivalent nef line bundles with identical h⁰(L).
• Utilize known vanishing theorems (e.g., Kodaira vanishing) for nef line bundles to simplify cohomology computation.
• Express the dimension h⁰(L) as a topological index after transformation, relying on the Riemann-Roch formula.
• Verify the method’s consistency by comparing results on known surfaces such as del Pezzo and Hirzebruch surfaces.
• Use the invariance of h⁰(L) under the lattice transformations to reduce general cohomology problems to computable topological invariants.

Experimental results

Research questions

• RQ1 Can effective line bundles on complex projective surfaces be transformed into nef line bundles while preserving the dimension of their zeroth cohomology?
• RQ2 To what extent can vanishing theorems be applied to compute h⁰(L) after such transformations?
• RQ3 Can the dimension of the zeroth cohomology be expressed purely in terms of topological invariants after transformation?
• RQ4 How do these transforms behave on specific surfaces like del Pezzo and Hirzebruch surfaces?
• RQ5 What is the relationship between the original line bundle’s cohomology and the topological index of its transformed counterpart?

Key findings

• The proposed lattice transformations preserve the dimension of the zeroth cohomology of line bundles while mapping effective bundles to nef ones.
• The method enables the computation of h⁰(L) via the topological index after applying transforms that allow the use of vanishing theorems.
• On del Pezzo surfaces, the method successfully computes h⁰(L) for all effective line bundles using topological data.
• For Hirzebruch surfaces, the transforms yield consistent cohomology dimensions that match known results.
• The approach establishes a general framework for cohomology computation that relies on lattice symmetries and topological invariants.
• The invariance of h⁰(L) under the transforms confirms that cohomological data can be extracted from topological formulae after transformation.