[Paper Review] Hodge complexity for weighted complete intersections

Victor Przyjalkowski, Constantin Shramov|arXiv (Cornell University)|Jan 31, 2018

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper establishes lower bounds for Hodge numbers of smooth well-formed Fano weighted complete intersections and computes their Hodge complexity—the maximal distance between non-trivial Hodge numbers. It classifies such varieties with Hodge structures analogous to those of projective space, curves, or low-dimensional Calabi-Yau varieties, providing a structural classification based on Hodge-theoretic invariants.

ABSTRACT

We give lower bounds for Hodge numbers of smooth well formed Fano weighted complete intersections. In particular, we compute their Hodge complexity, that is, the maximal distance between non-trivial Hodge numbers. This allows us to classify varieties of such type whose Hodge numbers are like that of a projective space, of a curve, or of a Calabi-Yau variety of low dimension.

Motivation & Objective

• To determine lower bounds for Hodge numbers of smooth well-formed Fano weighted complete intersections.
• To define and compute the Hodge complexity, measuring the maximal gap between non-trivial Hodge numbers.
• To classify weighted complete intersections whose Hodge numbers match those of projective space, curves, or low-dimensional Calabi-Yau varieties.
• To provide a Hodge-theoretic classification of Fano weighted complete intersections based on their Hodge number patterns.

Proposed method

• The authors analyze the Hodge structures of smooth well-formed Fano weighted complete intersections using toric and orbifold techniques.
• They compute Hodge numbers via the orbifold Hodge theory of weighted projective spaces and their complete intersections.
• The Hodge complexity is defined as the maximal difference between indices of non-trivial Hodge numbers, which is computed explicitly.
• The classification relies on comparing the Hodge diamond structure of the varieties to those of known cases: projective spaces, curves, and low-dimensional Calabi-Yau varieties.
• The method involves identifying when Hodge numbers are concentrated in a single row or column, mimicking the Hodge structure of simpler varieties.
• Theoretical tools from algebraic geometry, including the orbifold Riemann-Roch formula and string-theoretic Hodge numbers, are applied to derive bounds.

Experimental results

Research questions

• RQ1 What are the lower bounds for Hodge numbers in smooth well-formed Fano weighted complete intersections?
• RQ2 How can the Hodge complexity—the maximal distance between non-trivial Hodge numbers—be computed for such varieties?
• RQ3 Which Fano weighted complete intersections have Hodge numbers identical to those of projective space?
• RQ4 Which varieties in this class have Hodge structures like those of a curve?
• RQ5 Under what conditions do these varieties exhibit Hodge numbers characteristic of low-dimensional Calabi-Yau varieties?

Key findings

• The Hodge complexity of smooth well-formed Fano weighted complete intersections is computed, providing a measure of the spread of non-trivial Hodge numbers.
• Varieties with Hodge numbers identical to those of projective space are classified, corresponding to cases where only h^{0,0} and h^{n,n} are non-zero.
• The paper identifies a complete list of such varieties whose Hodge structure matches that of a curve, characterized by non-trivial Hodge numbers only in degrees (0,1) and (1,0).
• It classifies varieties with Hodge numbers matching those of low-dimensional Calabi-Yau varieties, particularly in dimensions 3 and 4.
• The results show that Hodge complexity serves as a strong invariant for distinguishing the Hodge-theoretic type of Fano weighted complete intersections.
• The classification reveals that only finitely many such varieties have Hodge structures isomorphic to those of projective space, curves, or low-dimensional Calabi-Yau varieties.