[Paper Review] A note on toric degeneration of a Bott-Samelson-Demazure-Hansen variety

B. Narasimha Chary|arXiv (Cornell University)|Oct 16, 2017

Algebraic Geometry and Number Theory23 references4 citations

TL;DR

This paper studies toric degenerations of Bott-Samelson-Demazure-Hansen (BSDH) varieties in the Kac-Moody setting using toric geometry. It classifies Fano, weak Fano, and log Fano BSDH varieties and their toric limits via combinatorial conditions on the Weyl group expression, proves vanishing theorems for cohomology of tangent and line bundles, and recovers and extends results from [PK16] using toric methods.

ABSTRACT

In this paper we study the geometry of toric degeneration of a Bott-Samelson-Demazure-Hansen (BSDH) variety. We give some applications to BSDH varieties. Precisely, we classify Fano, weak Fano and log Fano BSDH varieties and their toric limits in Kac-Moody setting. We prove some vanishing theorems for the cohomology of tangent bundle (and line bundles) on BSDH varieties. We also recover the results in arXiv:1604.01998, by toric methods.

Motivation & Objective

• To classify Fano, weak Fano, and log Fano BSDH varieties and their toric limits in the Kac-Moody setting.
• To prove vanishing theorems for the cohomology of tangent bundles and line bundles on BSDH varieties.
• To recover and extend the results of [PK16] using toric geometric techniques.
• To establish a connection between the combinatorics of Weyl group expressions and geometric properties of BSDH varieties via toric limits.
• To characterize extremal rays and Mori rays in the Mori cone of the toric limit using curve classes from the Bott tower structure.

Proposed method

• Utilizes toric geometry to analyze the limiting toric variety $Y_{\tilde{w}}$ of a BSDH variety $Z(\tilde{w})$.
• Identifies $Y_{\tilde{w}}$ as a Bott tower — an iterated $\mathbb{P}^1$-bundle over a point with decomposable rank-2 vector bundles.
• Applies the semi-continuity theorem to transfer properties from the toric limit $Y_{\tilde{w}}$ to the original BSDH variety $Z(\tilde{w})$.
• Introduces combinatorial conditions $N^1_i$ and $N^2_i$ on the Weyl group expression $\tilde{w} = s_{\beta_1} \cdots s_{\beta_r}$ to characterize Fano and weak Fano behavior.
• Uses the toric Kleiman criterion and the Toric Cone Theorem to relate ampleness of the anti-canonical bundle to the negativity of canonical divisor on extremal rays.
• Applies results from [Ch18] on Bott towers to characterize the Mori cone and extremal rays via curve classes $L_{I_i}$.

Experimental results

Research questions

• RQ1 Under what combinatorial conditions on the Weyl group expression $\tilde{w}$ is the BSDH variety $Z(\tilde{w})$ Fano or weak Fano in the Kac-Moody setting?
• RQ2 How do the cohomology groups of the tangent bundle and line bundles on $Z(\tilde{w})$ behave under toric degeneration?
• RQ3 Can the results of [PK16] on Fano and weak Fano toric limits be recovered and extended using toric geometry?
• RQ4 What is the structure of the Mori cone $\overline{NE}(Y_{\tilde{w}})$ and which extremal rays are Mori in the toric limit $Y_{\tilde{w}}$?
• RQ5 How do the curve classes $L_{I_i}$, associated to the Bott tower structure, generate the Mori cone and determine Fano properties?

Key findings

• If the Weyl group expression $\tilde{w}$ satisfies condition $I$, then both the toric limit $Y_{\tilde{w}}$ and the BSDH variety $Z(\tilde{w})$ are Fano.
• If $\tilde{w}$ satisfies condition $II$, then both $Y_{\tilde{w}}$ and $Z(\tilde{w})$ are weak Fano.
• The ample cone $Amp(Y_{\tilde{w}})$ of the toric limit is identified as a subcone of $Amp(Z(\tilde{w}))$.
• Vanishing theorems for $H^i(Z(\tilde{w}), T_{Z(\tilde{w})})$ hold when $\tilde{w}$ satisfies condition $I$, extending prior results to the Kac-Moody setting.
• The extremal rays of $\overline{NE}(Y_{\tilde{w}})$ are precisely the classes $L_{I_i}$, and they form a basis of $N_1(Y_{\tilde{w}})_{\mathbb{Z}}$.
• A smooth projective toric variety is Fano if and only if every extremal ray is Mori, and this criterion applies to $Y_{\tilde{w}}$ to characterize Fano behavior.