The number of smooth varieties in an MMP on a 3-fold of Fano type

Technical Summary: The Number of Smooth Varieties in an MMP on a 3-Fold of Fano Type

Problem Statement
The Minimal Model Program (MMP) for threefolds frequently produces singular varieties, unlike the surface case where outcomes often remain smooth. This necessitates the study of singularities such as terminal, Kawamata log terminal (klt), and log canonical types. A natural question arises regarding the frequency of smooth varieties generated during the running of an MMP on a threefold of Fano type. Specifically, the paper investigates how many smooth varieties can appear in a $D$-MMP (where $D$ is a movable $\mathbb{Q}$-Cartier Weil divisor) on a threefold $X$ of Fano type. Previous work by [Ben85] established that if $D = -K_X$, no smooth varieties other than $X$ itself appear in a $-K_X$-MMP. This paper generalizes the problem to an arbitrary movable divisor $D$.

Methodology
The proof of the main theorem follows a four-step structure, closely adapting the argument found in [Kim24, Theorem 1.4]. The methodology relies on the interplay between cohomological invariants and the geometric properties of flips within the MMP.

1. Setup and Reduction to Flips: The author first establishes that for a movable divisor $D$ on a normal projective variety, any $D$-MMP consists exclusively of flips (Lemma 2.4). This is because divisorial contractions would contradict the movability of the strict transform of $D$. The proof proceeds by analyzing a sequence of flips $X_0 \dashrightarrow X_1 \dashrightarrow \dots \dashrightarrow X_n$. It is shown that if the initial variety $X$ is of Fano type, all subsequent varieties $X_i$ in the MMP sequence remain of Fano type.

2. Cohomological Monotonicity: The core of the argument involves defining a sequence of integers $d_i = h^1(X_i, \mathcal{O}_{X_i}(2mD_i))$, where $D_i$ is the strict transform of $D$ on $X_i$. Using Leray spectral sequences, resolutions of flips, and vanishing theorems (specifically Lemma 2.8 and the relative Kawamata-Viehweg vanishing theorem), the author proves that this sequence is non-increasing: $d_{i-1} \geq d_i$.

3. Strict Decrease Condition: The paper establishes a critical link between the smoothness (or Cartier property) of the divisor and the cohomological sequence. It is proven that if $d_{i-1} = d_i$, then $mD_i$ cannot be Cartier. This step utilizes the properties of the flipping contraction $\phi_i$, the vanishing of higher direct images, and a contradiction argument involving rational curves contracted by the flip. Specifically, if $mD_{i-1}$ were Cartier, one could construct a rational curve $C$ where the restriction of the line bundle leads to a vanishing cohomology group that is known to be non-zero for anti-ample bundles on $\mathbb{P}^1$.

4. Bounding the Count: By combining the monotonicity ($d_{i-1} \geq d_i$) with the strict decrease condition (if $mD_i$ is Cartier, then $d_{i-1} > d_i$), the author bounds the number of indices $i$ for which $mD_i$ is Cartier. Since the sequence $d_i$ consists of non-negative integers and starts at $h^1(X, 2mD)$, the number of strict decreases cannot exceed the initial value.

Key Contributions and Results

• Theorem 1.1: For a threefold $X$ of Fano type and a movable $\mathbb{Q}$-Cartier Weil divisor $D$, the number of varieties $X_i$ in a $D$-MMP where $mD_i$ is Cartier is bounded by $1 + h^1(X, \mathcal{O}_X(2mD))$.
• Corollary 1.2: A direct consequence of Theorem 1.1 is that the number of smooth varieties in such an MMP is bounded by $1 + h^1(X, \mathcal{O}_X(2D))$. This is because on a smooth variety, every Weil divisor is Cartier.
• Corollary 1.3 (Partial Converse to Kodaira Vanishing): The paper proves a partial converse to the Kodaira vanishing theorem for movable divisors on threefolds of Fano type. It states that if $X$ is of Fano type, $D$ is a movable $\mathbb{Q}$-Cartier Weil divisor, and $H^1(X, \mathcal{O}_X(mD)) = 0$ for any sufficiently divisible $m$, then $D$ is nef. The proof utilizes the fact that if $D$ were not nef, a $D$-MMP would terminate in a non-trivial way, contradicting the vanishing hypothesis via Theorem 1.1.

Significance and Claims
The paper claims to provide a quantitative bound on the occurrence of smooth varieties in the MMP for Fano type threefolds, extending known results for the anticanonical case to general movable divisors. The significance lies in connecting the geometric property of smoothness (or the Cartier property of divisors) to the cohomological invariant $h^1(X, 2D)$.

Furthermore, the paper presents Corollary 1.3 as a partial converse to the Kodaira vanishing theorem. While the standard Kodaira vanishing theorem asserts that nef divisors on Fano type varieties have vanishing higher cohomology, this result suggests that for movable divisors, the vanishing of $H^1$ is sufficient to guarantee nefness. The author notes that the movability assumption is essential, providing a counterexample (Remark 3.1) involving a blow-up of $\mathbb{P}^3$ where a non-nef divisor satisfies the cohomological vanishing condition but fails to be movable.

The work relies on established techniques from the MMP, including the Negativity Lemma, resolution of singularities, and vanishing theorems, synthesizing them to address the specific question of smoothness frequency in threefold MMPs.