Descendant and Fourier-Mukai equivalences for simple flops

This paper establishes a correspondence between the genus 0 descendant Gromov-Witten theories of two varieties related by a simple flop and demonstrates that this correspondence is compatible with the Fourier-Mukai equivalence induced by the flop.

The Gist

Imagine you have two different landscapes, let's call them Mountain A and Mountain B. To the naked eye, they look completely different. One has a valley that dips down, and the other has a hill that peaks up. In the world of mathematics (specifically algebraic geometry), these are two "smooth projective varieties."

Now, imagine there is a magical, invisible tunnel connecting a specific spot on Mountain A to a specific spot on Mountain B. This tunnel allows you to transform Mountain A into Mountain B without tearing or gluing anything in a messy way. Mathematicians call this a "Simple Flop." It's like taking a piece of a mountain, flipping it inside out, and reattaching it so the landscape changes shape but the underlying "soul" of the mountain remains the same.

This paper by Chen and Tseng is about proving that two very different ways of looking at these mountains are actually compatible.

The Two Languages of the Mountains

The authors are trying to translate between two different "languages" used to describe these shapes:

1. Language 1: The "Shape-Shifter" Dictionary (Fourier-Mukai Equivalence)
   Think of this as a dictionary that translates the structure of the mountain. It says, "If you have a specific type of rock formation (a sheaf) on Mountain A, here is exactly what it looks like on Mountain B." This is a deep, structural translation that mathematicians have known exists for a while. It's like saying, "The blueprint of the house on the left is mathematically identical to the blueprint of the house on the right, even if the paint colors are different."

2. Language 2: The "Tourist Guide" (Gromov-Witten Theory)
   This language is about movement and paths. Imagine you are a tourist walking on the mountain. You want to know: "If I walk in a straight line, how many ways can I loop around? How many paths go through the valley?" This is called "Gromov-Witten theory." It counts the possible paths (curves) you can take.

   • The "Descendant" part: This is a fancy way of saying, "Not just where you walk, but how you walk." It adds extra details, like "Did you stop to take a photo?" or "Did you walk fast or slow?" It's a more detailed, complex version of the tourist guide.

The Big Question

The authors wanted to know: If I use the "Shape-Shifter" dictionary to translate a rock formation from Mountain A to Mountain B, and then I use the "Tourist Guide" to count the paths on Mountain B, does it match the result if I count the paths on Mountain A first and then translate the answer?

In other words, do these two languages agree with each other?

The Solution: The "Deformation" Trick

To prove they agree, the authors used a clever trick called "Deformation to the Normal Cone."

Imagine you have a lump of clay (Mountain A). You want to see how it turns into Mountain B. Instead of just snapping it into place, you slowly stretch and squish the clay.

• You stretch the mountain until it splits into two pieces connected by a thin, rubbery bridge.
• One piece is the "old" mountain, and the other is a "new" piece that looks like a giant cone or a funnel.
• This bridge is the projective local model. It's a simplified, toy version of the complex transformation.

The authors realized that the complex, messy mountain is actually just a combination of:

1. A boring, unchanged part (the parts of the mountain far away from the flip).
2. This simple, toy "funnel" part where the magic happens.

The "Aha!" Moment

Because the "funnel" part is a simple, symmetrical shape (like a torus or a donut), mathematicians had already solved the puzzle for it in a previous paper. They knew exactly how the "Shape-Shifter" and "Tourist Guide" languages matched up for this simple funnel.

The authors' breakthrough was showing that:

1. The complex mountain is just the boring part + the simple funnel.
2. The "Shape-Shifter" dictionary works perfectly on the boring part (it's just the identity, nothing changes).
3. The "Tourist Guide" paths that matter for this specific flip only happen inside the simple funnel. The boring parts don't contribute to the special counts.

So, if the languages match for the simple funnel (which they do), and they match for the boring parts (which they do), then they must match for the whole mountain.

The Conclusion

The paper proves that the Fourier-Mukai equivalence (the structural dictionary) and the Descendant Gromov-Witten correspondence (the detailed path-counting guide) are perfectly synchronized.

In everyday terms:
Imagine you have two different maps of a city. One map shows the buildings (structure), and the other shows all the possible bus routes (paths). This paper proves that if you use a specific rule to translate the buildings from Map A to Map B, the bus routes will automatically translate correctly too. You don't have to re-calculate the bus routes from scratch; the structural translation guarantees the path translation is correct.

This is a big deal because it connects two major areas of modern mathematics, showing that the "shape" of a space and the "paths" you can take through it are deeply, beautifully linked, even when the space undergoes a dramatic transformation like a flop.

Technical Summary

1. Problem Statement

The paper addresses the compatibility between two fundamental concepts in algebraic geometry and mathematical physics regarding simple flops ($X \dashrightarrow X'$):

1. Derived Equivalence: The existence of a Fourier-Mukai equivalence between the derived categories of coherent sheaves $D^b(X)$ and $D^b(X')$, which induces an isomorphism on K-groups ($K(X) \cong K(X')$).
2. Gromov-Witten (GW) Equivalence: The conjecture that varieties related by crepant transformations (K-equivalences) possess equivalent Gromov-Witten theories. Specifically, the paper focuses on the correspondence between genus 0 descendant GW theories of $X$ and $X'$.

While the compatibility of these two structures has been proven for specific cases (e.g., toric wall-crossings, Hilbert-Chow morphisms), this paper aims to establish the commutativity of the diagram linking the K-theoretic Fourier-Mukai transform and the descendant GW correspondence for simple flops.

2. Methodology

The authors employ a combination of deformation theory, degeneration formulas, and reduction to local models. The core strategy involves the following steps:

A. The Commutative Diagram

The goal is to prove the commutativity of the following diagram (Equation 0.2 in the paper):
$$\begin{array}{ccc} K(X) & \xrightarrow{FM} & K(X') \\ \Psi_X \downarrow & & \downarrow \Psi_{X'} \\ \tilde{H}_X & \xrightarrow{U} & \tilde{H}_{X'} \end{array}$$

• $FM$: The isomorphism induced by the Fourier-Mukai transform $R(p_{X'})_* L p_X^*$.
• $\Psi_X, \Psi_{X'}$: Maps from K-groups to the multi-valued Givental space $\tilde{H}$, constructed using Iritani's integral structure (involving Chern characters, Gamma classes, and operators $\mu, \rho, \deg_0$).
• $U$: A linear symplectic transformation equating the genus 0 descendant GW theories of $X$ and $X'$, derived from the analytic continuation of ancestor theories.

B. Degeneration to the Normal Cone

To handle the global geometry, the authors utilize deformation to the normal cone.

• They construct a family $\mathcal{X} \to \mathbb{P}^1$ where the general fiber is $X$ and the special fiber $X_0$ is a union of the blow-up $Bl_Z X$ and a projective bundle $P_Z(N_{Z/X} \oplus \mathcal{O}_Z)$.
• Similarly, they construct a family for $X'$.
• Crucially, they show that the birational map $X \dashrightarrow X'$ deforms to a family where the map is the identity on the blow-up components and a projective local model on the projective bundle component.

C. Reduction to Projective Local Models

The proof reduces the global problem to a local one:

• The simple flop is locally modeled as a toric crepant wall-crossing between projective bundles:
  $$\mathbb{P}(\mathcal{O}_{\mathbb{P}^r}^{\oplus r+1} \oplus \mathcal{O}_{\mathbb{P}^r}) \dashrightarrow \mathbb{P}(\mathcal{O}_{\mathbb{P}^r} \oplus \mathcal{O}_{\mathbb{P}^r}^{\oplus r+1})$$
• Since this local model is toric, the authors can apply existing results (specifically [5]) regarding the compatibility of Fourier-Mukai and GW theories for toric wall-crossings.

D. Specialization and Injectivity

The authors use the specialization map for cohomology and Chow groups. By embedding the cohomology of $X$ and $X'$ into the cohomology of their degenerate fibers (via injectivity of the specialization map), they reduce the verification of the global equality to verifying it on the components of the degenerate fiber.

• They prove that the operators defining $\Psi$ (Chern character, Gamma class, etc.) are compatible with the specialization map.
• They show that the transformation $U$ acts as the identity on the "blow-up" components and acts as the known toric transformation $U_P$ on the "projective local model" components.

3. Key Contributions

1. Proof of Commutativity for Simple Flops: The paper provides a rigorous proof that the Fourier-Mukai equivalence and the descendant Gromov-Witten correspondence commute for simple flops. This confirms that the derived equivalence is compatible with the enumerative geometry of the varieties.
2. Degeneration Technique for K-Theory: The authors develop a method to degenerate elements of K-groups and their Fourier-Mukai images to the normal cone. This allows the reduction of global geometric problems to local toric models.
3. Explicit Construction of the Correspondence: The paper explicitly constructs the map $U$ in the context of descendant theories (involving $\psi$-classes) by relating it to the fundamental solutions of quantum differential equations ($S^{-1}$) and their analytic continuations.
4. Generalization of Compatibility: It adds "simple flops" to the list of known cases where derived equivalence implies GW equivalence, joining toric wall-crossings and Hilbert-Chow morphisms.

4. Main Results

• Theorem 0.2: The diagram (0.2) is commutative. Specifically, for any class $\alpha \in K(X)$:
  $$\Psi_{X'}(FM(\alpha)) = U \Psi_X(\alpha)$$
  where $U$ is the symplectic transformation relating the descendant GW theories.
• Reduction Lemma: The global commutativity is reduced to the commutativity of the diagram for the projective local model (a toric crepant transformation), which is already known to hold.
• Compatibility of Operators: The paper demonstrates that the operators $\mu_X$, $\rho_X$, and $\deg_0^X$ (defining the Iritani map $\Psi$) are compatible with the degeneration maps $\Phi$ and $\Phi'$.

5. Significance

• Verification of the Crepant Transformation Conjecture: This work provides strong evidence for the broader conjecture that K-equivalent varieties have equivalent Gromov-Witten theories. It bridges the gap between the categorical (derived) and enumerative (GW) aspects of mirror symmetry and birational geometry.
• Methodological Advancement: The technique of using deformation to the normal cone to reduce global K-theoretic problems to local toric models is a powerful tool. It suggests a pathway for tackling more complex K-equivalences, such as ordinary flops (mentioned in Remark 3.1), where the exceptional locus is a projective bundle over a general base.
• Foundation for Higher Genus: While the paper focuses on genus 0, the construction of the map $U$ and the compatibility of the integral structure lay the groundwork for extending these results to all genera, as hinted by the discussion of ancestor potentials in Section 1.2.

In summary, Chen and Tseng successfully demonstrate that the deep structural equivalence provided by Fourier-Mukai transforms is perfectly aligned with the enumerative invariants of simple flops, unifying derived category theory with Gromov-Witten theory in this specific but significant geometric context.