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Bott vanishing for Fano threefolds

April 2024
Mathematische Zeitschrift 307(1)

Authors:

Bott proved a strong vanishing theorem for sheaf cohomology on projective space, namely that Hj(X,ΩXi⊗L)=0

H^j(X,\Omega ^i_X\otimes L)=0

for j>0

j>0

, i≥0

i\ge 0

, and L ample. This holds for toric varieties, but not for most other varieties. We classify the smooth Fano threefolds that satisfy Bott vanishing. There are many more than expected.

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Mathematische Zeitschrift (2024) 307:14

https://doi.org/10.1007/s00209-024-03468-x

Mathematische Zeitschrift

Bott vanishing for Fano threefolds

Burt Totaro1

Received: 25 February 2023 / Accepted: 4 February 2024 / Published online: 20 April 2024

Abstract

Bott proved a strong vanishing theorem for sheaf cohomology on projective space, namely

that Hj(X,

X⊗L)=0for j>0, i≥0, and Lample. This holds for toric varieties, but not

for most other varieties. We classify the smooth Fano threefolds that satisfy Bott vanishing.

There are many more than expected.

Keywords Bott vanishing ·Vanishing theorem ·Fano threefold ·Fano variety

Mathematics Subject Classiﬁcation 14F17 ·14J30 ·14J45

Contents

1 Introduction ............................................... 1

2 Vanishing theorems ........................................... 2

3 Cases where Bott vanishing fails ..................................... 4

4 Inductive approach to Bott vanishing .................................. 7

5 Higher direct images of differential forms ................................ 7

6 The Fano threefolds (2.30) and (3.19) .................................. 9

7 Higher direct images of differential forms, continued .......................... 12

8 First blow-up along a curve: (2.26) ...................................14

9 The Fano threefold (3.24) ........................................ 15

10 The Fano threefolds (3.15), (3.16), (3.18), (3.20), (3.21), (3.22), (3.23) ................ 16

11 The Fano threefolds (4.3), (4.4), (4.5), (4.6), (4.7), (4.8) ........................22

12 The Fano threefold (5.1) ......................................... 27

References .................................................. 30

1 Introduction

A smooth projective variety Xsatisﬁes Bott vanishing if

Hj(X,

X⊗L)=0

BBurt Totaro

totaro@math.ucla.edu

1Department of Mathematics, UCLA, Box 951555, Los Angeles 90095-1555, CA, USA

123

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