On intersection cohomology with torus action of complexity one, II

This paper establishes that the decomposition theorem components for contraction maps of torus actions of complexity one are intersection cohomology complexes of even codimensional subvarieties, leading to the vanishing of odd-dimensional intersection cohomology for rational complete varieties of this type and providing explicit formulas for the Betti numbers of affine trinomial hypersurfaces based on their defining equations.

The Gist

Imagine you are an architect trying to understand the shape of a very strange, twisted building. This building isn't made of bricks, but of mathematical equations. It has a special property: it can be stretched, shrunk, and rotated by a group of "magic hands" (called a torus action).

The paper you're asking about is like a master guidebook for understanding the hidden skeleton of these buildings, specifically a class of them called "complexity one."

Here is the story of the paper, broken down into simple concepts:

1. The Building and the "Magic Hands"

In math, a Torus is like a donut shape (or a collection of them). When we say a shape has a "torus action," imagine the shape is being spun, stretched, or twisted by these donut-shaped hands.

• Complexity 0: These are the "easy" buildings (Toric varieties). They are so regular that you can describe their entire shape just by looking at a simple drawing of cones and polygons (like a blueprint made of triangles).
• Complexity 1: These are the "tricky" buildings. They are almost as regular as the easy ones, but they have one extra wrinkle. They are like a bundle of easy buildings tied together along a single curved line (like a necklace of beads).

The authors are studying these "Complexity 1" buildings to figure out their Intersection Cohomology.

2. What is "Intersection Cohomology"? (The X-Ray Vision)

Imagine taking a photo of a building. If the building is smooth, the photo is clear. But if the building has cracks, sharp corners, or self-intersections (singularities), the photo gets blurry or distorted.

Intersection Cohomology is like a super-powered X-ray. It ignores the messy surface cracks and tells you about the true underlying shape of the building. It asks: "If we smoothed out all the sharp edges, what would the core structure look like?"

The big question the authors wanted to answer was: "Can we predict the shape of this X-ray just by looking at the blueprint (the equations)?"

3. The "Contraction" Trick (Flattening the Building)

To solve this, the authors use a clever trick called Contraction.
Imagine your twisted building is a crumpled piece of paper. The "Contraction Map" is a process where you gently press the paper flat.

• Some parts of the paper flatten out perfectly.
• Other parts get squished into a smaller, simpler shape.

The authors proved that when you flatten this specific type of building (Complexity 1), the resulting flat shape is actually a Toric Variety (the "easy" building from step 1). This is huge because we already know how to calculate the X-ray of easy buildings!

The Discovery: The paper shows that the "messy" parts of the original building that get squished during this flattening process are actually just simple, flat slices (subvarieties) of even dimensions. This means the "noise" in the X-ray is very predictable.

4. The "Odd-Number" Mystery

One of the coolest results is about Odd Dimensions.
In the world of these twisted buildings, the authors proved a surprising rule: If the building is "rational" (meaning it can be built from simple pieces like a standard cube), then its X-ray has no "odd-numbered" frequencies.

Think of it like a musical chord. If the building is rational, it only plays even notes (2, 4, 6...). If you hear an odd note (1, 3, 5...), you know the building is "complicated" and not rational. This gives mathematicians a quick way to check if a shape is simple or complex just by listening to its "mathematical music."

5. The "Weight Matrix" (The Recipe Book)

The paper also provides a cookbook. If you give them the Weight Matrix (a grid of numbers that describes how the magic hands twist the building), they can tell you exactly what the X-ray looks like.

They tested this on Trinomial Hypersurfaces. These are buildings defined by an equation with three terms, like $A + B + C = 0$.

• The Analogy: Imagine a recipe that says "Mix 3 cups of flour, 2 cups of sugar, and 1 cup of salt."
• The authors showed that if you know the numbers in your recipe (the exponents in the equation), you can calculate the exact "X-ray" of the resulting shape without ever having to build it.

Summary: What did they actually do?

1. They found a shortcut: Instead of struggling with the complex, twisted shape, they showed you can flatten it into a simpler shape and add back a few predictable "patches."
2. They gave a formula: They wrote down a mathematical recipe that takes the numbers from the building's definition and spits out its "X-ray" (Betti numbers).
3. They solved a mystery: They proved that for these specific shapes, if the shape is "rational," it has no "odd" hidden features.

In a nutshell: This paper is a translation guide. It takes the confusing, twisted language of complex algebraic shapes and translates it into a simple, predictable language of cones, recipes, and even-numbered notes. It allows mathematicians to look at a messy equation and instantly know the hidden geometry inside.

Technical Summary

Here is a detailed technical summary of the paper "On Intersection Cohomology with Torus Action of Complexity One, II" by Marta Agustín Vicente, Narasimha Chary Bonala, and Kevin Langlois.

1. Problem Statement and Context

The paper addresses the computation and structural understanding of the intersection cohomology (specifically the Betti numbers) of complete complex algebraic varieties $X$ endowed with an action of an algebraic torus $T = (\mathbb{C}^*)^n$.

• Complexity: The central invariant is the complexity of the action, defined as $c(X) = \dim X - \dim T + \dim T_0$ (where $T_0$ is the kernel).
  • Complexity 0: These are toric varieties, whose geometry is fully described by combinatorial data (fans, polytopes). Their intersection cohomology is well-understood via the $f$-vector of the fan.
  • Complexity 1: This class includes varieties like $\mathbb{C}^*$-surfaces and generalizations of toric varieties. While a combinatorial description exists (via divisorial fans over a smooth projective curve), computing intersection cohomology directly from defining equations or understanding the topological constraints (e.g., vanishing of odd cohomology) remained an open challenge.
• Goal: The authors aim to complete a program initiated in previous works [4, 5] to provide explicit formulas for intersection cohomology Betti numbers for complexity-one varieties, specifically linking these invariants to the defining equations of the variety.

2. Methodology

The authors employ a multi-step approach combining algebraic geometry, combinatorics, and sheaf theory:

1. Decomposition Theorem Analysis:

   • They utilize the Decomposition Theorem (Beilinson-Bernstein-Deligne-Gabber) applied to the contraction map $\pi: \tilde{X} \to X$. Here, $\tilde{X}$ is the normalization of the graph of the rational quotient $X \dashrightarrow C$ (where $C$ is the curve parameterizing general orbits).
   • They analyze the direct image $\pi_* IC_{\tilde{X}}$ to relate the intersection cohomology of $X$ to that of the "contraction space" $\tilde{X}$ (which is a toroidal variety).

2. Weight Package Theory:

   • To handle linear torus actions (subvarieties of $\mathbb{P}^\ell$ or $\mathbb{A}^\ell$), they introduce the concept of a weight package $\theta = (N, N, F, S, \Sigma)$.
   • This package encodes the torus action via a weight matrix $F$ (defining the inclusion $T \hookrightarrow G$) and a section $S$.
   • They establish a correspondence between the weight package and divisorial fans (the combinatorial objects describing complexity-one varieties). Specifically, they show how to construct the divisorial fan of the normalization of a variety directly from the weight matrix and the defining equations.

3. Finite Group Actions and Local Systems:

   • A crucial technical step involves proving that the local systems appearing in the decomposition theorem for contraction maps are trivial.
   • They utilize results on finite group actions on toric varieties and Seifert torus bundles to show that the decomposition of $\pi_* IC_{\tilde{X}}$ involves only intersection cohomology complexes of even codimensional subvarieties with trivial coefficients.

4. Combinatorial Formulas:

   • They derive explicit formulas for the Poincaré polynomials using $g$-polynomials (for local intersection cohomology of cones) and $h$-polynomials (for complete fans/divisorial fans).

3. Key Contributions and Results

A. Structural Decomposition (Theorem A)

The paper proves a refined version of the decomposition theorem for the contraction map $\pi: \tilde{X} \to X$:
$$\pi_* IC_{\tilde{X}} \simeq IC_X \oplus \bigoplus_{O \in \text{Orb}^{\text{even}}(E)} (\iota_O)_* IC_{\bar{O}}^{\oplus s_O}$$

• Significance: The sum runs only over even codimensional torus orbits contained in the exceptional locus $E$. The local systems are trivial. This implies that the "correction terms" in the cohomology are purely combinatorial and depend on the geometry of the orbit closures.

B. Rationality Criterion (Theorem B)

A major topological consequence is a cohomological criterion for rationality:

• Result: A complete variety $X$ with a complexity-one torus action is rational if and only if its odd-dimensional intersection cohomology vanishes ($IH^{2j+1}(X; \mathbb{Q}) = 0$).
• Mechanism: This is derived from the fact that the contraction space $\tilde{X}$ has the same odd cohomology as the base curve $C$. If $X$ is rational, $C$ must be $\mathbb{P}^1$ (genus 0), leading to the vanishing of odd terms.

C. Explicit Formulas for Betti Numbers (Theorem C)

The authors provide a concrete formula for the Poincaré polynomial $P_X(t)$ of a complete normal $T$-variety $X$ with divisorial fan $\mathcal{E}$:
$$P_X(t) = h_{\tilde{\mathcal{E}}}(t) - \sum_{\tau \in HF(\mathcal{E})} g_{n(\tau)}(\tau) t^{c(\tau)} h(\text{Star}(\mathcal{E}, \tau); t^2)$$

• Here, $h_{\tilde{\mathcal{E}}}$ is the $h$-polynomial of the contraction space, $HF(\mathcal{E})$ represents cones corresponding to the exceptional locus, and $g, h$ are combinatorial polynomials associated with the fan structure.

D. From Equations to Cohomology (Theorem D)

The paper solves the inverse problem: How to compute intersection cohomology from the defining equations?

• They introduce an algorithm to construct the divisorial fan of the normalization of a projective subvariety $X \subset \mathbb{P}^\ell$ with linear torus action directly from its weight matrix $F$.
• This allows the application of Theorem C to varieties defined by explicit polynomial equations.

E. Affine Trinomial Hypersurfaces (Theorem E)

As a primary application, the authors compute the intersection cohomology of affine trinomial hypersurfaces defined by $T_1^{n_1} + T_2^{n_2} + T_3^{n_3} = 0$.

• They derive a closed-form expression for the Poincaré polynomial of the contraction space and the variety itself in terms of the exponents $n_i$ and the greatest common divisors of the coefficients.
• This generalizes previous results on specific surfaces to higher-dimensional hypersurfaces.

4. Significance

• Bridging Combinatorics and Geometry: The paper successfully extends the combinatorial toolkit of toric geometry (fans, $f$-vectors) to the broader class of complexity-one varieties, providing a "dictionary" between defining equations and topological invariants.
• Computational Power: By reducing the computation of intersection cohomology to matrix operations and polynomial calculations on fans, the results make these invariants accessible for explicit examples (like trinomial hypersurfaces) that were previously difficult to analyze.
• Topological Insights: The proof of the vanishing of odd intersection cohomology for rational varieties provides a strong structural constraint on the topology of these spaces, confirming that their "singular" topology behaves similarly to smooth rational varieties in this specific context.
• Foundation for Further Research: The methods developed (weight packages, lifting fans, contraction divisorial fans) provide a framework for studying more general torus actions and potentially extending these results to higher complexities or different fields.

In summary, this paper completes a systematic program to understand the topology of complexity-one torus varieties, offering both deep theoretical insights (decomposition structure, rationality criteria) and practical computational tools (formulas derived from weight matrices).