Workshop on Algebraic Geometry over complex
number ﬁeld or in positive characteristic-Around positivity of tangent sheaves and anti-canonical divisorsDate: 17th–20th September 2024. (2024 年 9 月 17–20 日)
Place: Osaka Metropolitan University (Sugimoto Campus), Faculty of Science, Bldg. E,
Room E408 (4F). (大阪公立大学 杉本キャンパス 理学部 E 棟大講究室 E408)

Program
9/17 (Tuesday)
13:00-14:00 Sho Tanimoto (Nagoya University)
Campana rational connectedness and weak approximation
14:30-15:30 Takuzo Okada (Kyushu University)
Birationally solid Fano 3-fold hypersurfaces
16:00-17:00 Taro Yoshino (The University of Tokyo)
Stable rationality of hypersurfaces in Grassmannian varieties
9/18 (Wednesday)
10:00-11:00 Akihiro Kanemitsu (Tokyo Metropolitan University)
Mukai pairs and associated K3 surfaces
11:00-11:30 Breaktime1
11:30-12:30 Chi-Kang Chang (National Center for Theoretical Sciences (NCTS))
Positivity of anticanonical divisors in algebraic ﬁbre spaces
14:30-15:30 Juanyong Wang (Academy of Mathematics and Systems Science,
Chinese Academy of Sciences (AMSS CAS))
An abundance-type result for tangent bundles of smooth Fano varieties
16:00-17:00 Guolei Zhong (Institute for Basic Science Center for Complex
Geometry (IBS-CCG))
Projective varieties with almost nef tangent sheaves and its dynamical application

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At this time, we will take a group photo of participants.

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9/19 (Thursday)
10:00-11:00 Hirotaka Onuki (The University of Tokyo)
On the eﬀective generation of direct images of pluricanonical bundles in mixed characteristic
11:30-12:30 Fuetaro Yobuko (Tokyo University of Science)
Quasi-F-splitting and positivity in positive characteristic
14:30-15:30 Hiromu Tanaka (The University of Tokyo)
Classiﬁcation of smooth Fano threefolds in positive characteristic
16:00-17:00 Yuta Takahashi (Chuo University)
Fano 4-folds with nef tangent bundle in positive characteristic
9/20 (Friday)
10:00-11:00 Wahei Hara (Kavli IPMU, The University of Tokyo)
Rank two weak Fano bundles over Fano threefolds of Picard rank one
11:30-12:30 Tatsuro Kawakami (Kyoto University)
Kodaira vanishing for smooth Fano threefolds in positive characteristic

9/17
10:00-11:00
11:30-12:30
14:30-15:30
16:00-17:00

Sho Tanimoto(13:00-14:00)
Takuzo Okada
Taro Yoshino

9/18
Akihiro Kanemitsu

9/19
Hirotaka Onuki

9/20Hara Wahei

Chi-Kang Chang

Fuetaro Yobuko

Tatsuro Kawakami

Juanyong Wang
Guolei Zhong

Hiromu Tanaka
Yuta Takahashi

2Organizers• Sho Ejiri (Osaka Metropolitan University)• Masataka Iwai (Osaka University)• Takayuki Koike (Osaka Metropolitan University)• Shin-ichi Matsumura (Tohoku University)• Yohsuke Matsuzawa (Osaka Metropolitan University)• Kenta Sato (Kyushu University)

Supports
This conference is supported by Osaka Central Advanced Mathematical Institute(MEXT Promotion of Distinctive Joint Research Center Program JPMXP0723833165),
Osaka Metropolitan University .

Homepage
We have posted various information on our website, including how to access to theconference room.
Homepage Link: https://masataka123.github.io/tangent_anticanonical/
You can also read the QR code below:

3Abstract
9/17 (Tuesday)
Sho Tanimoto (Nagoya University)
Campana rational connectedness and weak approximation
Campana and Abramovich introduced the notion of Campana points which interpolate
between rational points and integral points. Recently there are extensive activities onarithmetic geometry of Campana points and many conjectures have been proposed. In thistalk we discuss Campana curves/sections in the geometric setting. Campana introduced
the notion of Campana rational connectedness and conjectured that any klt Fano orbifold
is Campana rationally connected. We prove that weak approximation at good places
holds in the setting of Campana sections for any Campana rationally connected ﬁbration.
This is a generalization of theorems by Graber-Harris-Starr and Hassett-Tschinkel. A key
tool to this theorem is log geometry and the notion of moduli stack of stable log maps.
This is joint work with Qile Chen and Brian Lehmann.
Takuzo Okada (Kyushu University)
Birationally solid Fano 3-fold hypersurfaces
A Fano variety of Picard number 1 is birationally rigid if it is the unique Mori ﬁber
space in its birational equivalence class (up to isomorphism). Birational rigidity of a Fanovariety implies the irrationality of the variety. By relaxing the uniqueness requirement,
we introduce a notion of birational solidity which is weaker than birational rigidity but still
implies irrationality of the variety. We investigate Fano 3-folds that are embedded intoweighted projective 4-spaces as quasismooth hypersurfaces and explain the classiﬁcation
result of birationally solid ones which solves a conjecture posed by Abban-Cheltsov-Park.
Taro Yoshino (The University of Tokyo)
Stable rationality of hypersurfaces in Grassmannian varieties
In recent years, there has been a development in approaching rationality problems
through motivic methods. This approach requires the explicit construction of degeneration families over curves with favorable properties. However, the speciﬁc construction
is generally diﬃcult. Nicaise and Ottem combined combinatorial methods to constructdegeneration families of hypersurfaces in toric varieties and mentioned the stable rationality of a very general hypersurface in projective spaces. In this talk, we mention thefollowing two points: First, I introduce the notion of mock toric varieties, which are generalizations of toric varieties. Second, I combinatorially construct degeneration familiesof hypersurfaces in mock toric varieties, and I mention the irrationality of a very general
hypersurface in the complex Grassmannian variety Gr(2, n).

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9/18 (Wednesday)
Akihiro Kanemitsu (Tokyo Metropolitan University)
Mukai pairs and associated K3 surfaces
A Mukai pair (X, E) is a pair of a smooth projective variety X and an ample vector
bundle E with c1 (X) = c1 (E). Such pairs are completely classiﬁed when the rank of Eis at least dim X − 2. When the rank of E is dim X − 2, Mukai pairs are related to K3
surfaces; the zero locus of a section of E is a K3 surface. In this talk, we discuss thestructure of these K3 surfaces from the viewpoint of their moduli.
Chi-Kang Chang (National Center for Theoretical Sciences (NCTS))
Positivity of anticanonical divisors in algebraic ﬁbre spaces
The positivity of the anticanonical divisor is an important property closely related
to the geometric structure of varieties and algebraic ﬁbre spaces. For example, given
an algebraic ﬁbre space f : X → Y between two normal projective varieties with mild
singularities. If −KX is positive in some sense, then the positivity of −KX deeply aﬀects
the positivity of −KY . In this talk, we will introduce results concerning the positivity of−KX and −KY under certain conditions on the asymptotic base locus of −KX . In particular, our main result is: let F be a general ﬁbre of f , then the Iitaka-type inequality ofanticanonical divisors κ(X, −KX ) ≤ κ(F, −KF ) + κ(Y, −KY ) holds under the assumptionthat the stable base locus B(−KX ) does not dominant Y .
Juanyong Wang (Academy of Mathematics and Systems Science, Chinese
Academy of Sciences (AMSS CAS))
An abundance-type result for tangent bundles of smooth Fano varieties
Positive curvature condition imposed on a compact Kähler manifold usually has strong
restriction on its geometry. This principle has been manifested by the pioneering worksof Mori (1979) and Siu-Yau (1980) on the Hartshorne-Frankel conjecture, and by thesuccessive works of Howard-Smyth-Wu (1981), Mok-Zhong (1986) and Mok (1988) onthe generalized Frankel conjecture. In Campana-Peternell (1991), they propose to study
compact Kähler manifolds with nef tangent bundles, aiming at giving an algbro-geometric
generalization of Mori’s characterization of projective spaces by ampleness of the tangent
bundles. In view of Demailly-Peternell-Schneider (1994), these manifolds are shown to
be interpolations of complex tori and smooth Fano varieties with nef tangent bundles;
the latter are further conjectured to be rational homogeneous, i.e. they are quotients ofsemi-simple linear algebraic groups by parabolic subgroups (Campana-Peternell, 1991).
In this talk, I will explain our recent progress towards the Campana-Peternell conjecture:
we prove that the tangent bundle is big thus semiample.

5Guolei Zhong (Institute for Basic Science Center for Complex Geometry (IBSCCG))
Projective varieties with almost nef tangent sheaves and its dynamical application
It is expected that a certain positivity of tangent sheaves would give strong restrictionson the underlying varieties. In this talk, we ﬁrst study some structure properties of projective varieties with almost nef tangent sheaves. Then we give a dynamical application
to describe projective varieties admitting an int-ampliﬁed endomorphism with totally invariant ramiﬁcations. More precisely, if a projective klt variety X admits an int-ampliﬁed
endomorphism f such that outside an f −1 -invariant divisor D, f is quasi-étale, then weshow that there exists an f -equivariant quasi-étale Galois cover Y → X such that Yadmits a locally trivial ﬁbration to an abelian variety with all the ﬁbres being toric. Thisextends a theorem due to Hwang and Nakayama. The ﬁrst part of this talk is based ona joint work with Masataka Iwai and Shin-ichi Matsumura. The second part of this talkis based on a joint work with Sheng Meng.
9/19 (Thursday)
Hirotaka Onuki (The University of Tokyo)
On the eﬀective generation of direct images of pluricanonical bundles in mixed characteristic
In characteristic zero, Popa and Schnell proved an eﬀective global generation theorem
for direct images of pluricanonical bundles as a special case of their Fujita-type conjecture.
Ejiri showed an analogous result in positive characteristic. These results have been applied
to the study of direct images of pluricanonical bundles. Recently, substantial progress has
been made in birational geometry in mixed characteristic. In this talk, we present a mixed
characteristic analog of the theorem of Popa and Schnell and that of Ejiri.
Fuetaro Yobuko (Tokyo University of Science)
Quasi-F-splitting and positivity in positive characteristic
F-splitting and F-regularity have been well studied in birational geometry and commutative algebra in positive characteristic. Quasi-F-splitting is an extension of F-splitting.
In this talk, we discuss a relationship between quasi-F-splitting and positivity. The talkwill contain a joint work with Tatsuro Kawakami, Teppei Takamatsu, Hiromu Tanaka,
Jakub Witaszek and Shou Yoshikawa.
Hiromu Tanaka (The University of Tokyo)
Classiﬁcation of smooth Fano threefolds in positive characteristic
In the 1980s, Mori-Mukai completed the classiﬁcation of smooth Fano threefolds incharacteristic zero, based on work by Iskovskikh and Shokurov. In this talk, I will explain
an analogous result in positive characteristic.
6Yuta Takahashi (Chuo University)
Fano 4-folds with nef tangent bundle in positive characteristic
The positivity of the tangent bundle is expected to represent the geometric propertiesof algebraic varieties. An example of this is the Hartshorne conjecture, which states thata smooth projective variety deﬁned over an algebraically closed ﬁeld is the projective
space if the tangent bundle is ample. In this talk, we will consider the Campana-Peternell
conjecture which is a generalization of the Hartshorne conjecture. This states that asmooth Fano variety with nef tangent bundles are homogeneous varieties. Speciﬁcally,
we introduce the classiﬁcation problem of four-dimensional smooth Fano varieties with
nef tangent bundles in positive characteristic. Firstly, we introduce an overview of theCampana-Peternell conjecture and main theorem. Next, we discuss the outline of thisproof. This talk is based on the joint work with Kiwamu Watanabe.
9/20 (Friday)
Wahei Hara (Kavli IPMU, The University of Tokyo)
Rank two weak Fano bundles over Fano threefolds of Picard rank one
During this talk we discuss the classiﬁcation problem for rank two weak Fano bundles
over various Fano threefolds of Picard rank one. A vector bundle is said to be weak Fano
if its projectivization (in the sense of Grothendieck) has the nef and big anti-canonicalbundle. The classiﬁcation problem we discuss can be regarded as a part of the classiﬁcationof weak Fano 4-folds of Picard rank two, which is related to the study of Fano 4-folds ofPicard rank one by 2-ray game, for example. As usual, the classiﬁcation will be done bycase-by-case study of each Fano threefolds of Picard rank one. The aim of this talk is toshare a part of such studies with some general result we established to investigate weak
Fano bundles. This is all joint work with T.Fukuoka and D.Ishikawa.
Tatsuro Kawakami (Kyoto University)
Kodaira vanishing for smooth Fano threefolds in positive characteristic
We prove that smooth Fano threefolds in positive characteristic satisfy Kodaira vanishing. It is well-known that Frobenius split (F-split) varieties satisfy Kodaira vanishing.
However, many smooth Fano threefolds are not F-split. For this reason, we utilize quasiF-splitting, which was originally introduced by Yobuko, as a weaker condition than usual
F-splitting. In fact, we can prove that most Fano threefolds are quasi-F-split, and conclude Kodaira vanishing holds for all Fano threefolds. This talk is based on joint workwith Hiromu Tanaka.

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