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"Geometry, Differential"
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Gromov’s Theory of Multicomplexes with Applications to Bounded Cohomology and Simplicial Volume
The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper
The first aim of this paper is to lay the foundation of the theory of
multicomplexes. After setting the main definitions, we construct the singular multicomplex
In the second part of this work we apply the theory of multicomplexes to the study of the bounded
cohomology of topological spaces. Our constructions and arguments culminate in the complete proofs of Gromov’s Mapping Theorem (which
implies in particular that the bounded cohomology of a space only depends on its fundamental group) and of Gromov’s Vanishing Theorem,
which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity.
The
third and last part of the paper is devoted to the study of locally finite chains on non-compact spaces, hence to the simplicial volume
of open manifolds. We expand some ideas of Gromov to provide detailed proofs of a criterion for the vanishing and a criterion for the
finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the
eBook
Twistors, Quartics, and del Pezzo Fibrations
It has been known that twistor spaces associated to self-dual metrics on compact 4-manifolds are source of interesting examples of
non-projective Moishezon threefolds. In this paper we investigate the structure of a variety of new Moishezon twistor spaces. The
anti-canonical line bundle on any twistor space admits a canonical half, and we analyze the structure of twistor spaces by using the
pluri-half-anti-canonical map from the twistor spaces.
Specifically, each of the present twistor spaces is bimeromorphic to a
double covering of a scroll of planes over a rational normal curve, and the branch divisor of the double cover is a cut of the scroll by
a quartic hypersurface. In particular, the double covering has a pencil of Del Pezzo surfaces of degree two. Correspondingly, the
twistor spaces have a pencil of rational surfaces with big anti-canonical class. The base locus of the last pencil is a cycle of
rational curves, and it is an anti-canonical curve on smooth members of the pencil.
These twistor spaces are naturally classified
into four types according to the type of singularities of the branch divisor, or equivalently, those of the Del Pezzo surfaces in the
pencil. We also show that the quartic hypersurface satisfies a strong constraint and as a result the defining polynomial of the quartic
hypersurface has to be of a specific form.
Together with our previous result in Honda (“A new series of compact minitwistor
spaces and Moishezon twistor spaces over them”, 2010), the present result completes a classification of Moishezon twistor spaces whose
half-anti-canonical system is a pencil. Twistor spaces whose half-anti-canonical system is larger than pencil have been understood for a
long time before. In the opposite direction, no example is known of a Moishezon twistor space whose half-anti-canonical system is
smaller than a pencil.
Twistor spaces which have a similar structure were studied in Honda (“Double solid twistor spaces: the
case of arbitrary signature”, 2008 and “Double solid twistor spaces II: General case”, 2015) and they are very special examples among
the present twistor spaces.
eBook
Convexity of Singular Affine Structures and Toric-Focus Integrable Hamiltonian Systems
This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus
singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners),
analogous to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus
singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In
particular, near a focus singular point, they are locally convex and the local-global convexity principle still applies. They are also
globally convex under some natural additional conditions. However, when the monodromy is sufficiently large, the local-global convexity
principle breaks down and the base spaces can be globally non-convex, even for compact manifolds. As a surprising example, we construct
a 2-dimensional “integral affine black hole”, which is locally convex but for which a straight ray from the center can never escape.
eBook
The shape of a life : one mathematician's search for the universe's hidden geometry
\"Harvard geometer and Fields medalist Shing-Tung Yau has provided a mathematical foundation for string theory, offered new insights into black holes, and mathematically demonstrated the stability of our universe. In this autobiography, Yau reflects on his improbable journey to becoming one of the world's most distinguished mathematicians. Beginning with an impoverished childhood in China and Hong Kong, Yau takes readers through his doctoral studies at Berkeley during the height of the Vietnam War protests, his Fields Medal-winning proof of the Calabi conjecture, his return to China, and his pioneering work in geometric analysis. This new branch of geometry, which Yau built up with his friends and colleagues, has paved the way for solutions to several important and previously intransigent problems. With complicated ideas explained for a broad audience, this book offers readers not only insights into the life of an eminent mathematician, but also an accessible way to understand advanced and highly abstract concepts in mathematics and theoretical physics\"--Publisher's website.
Book
Partial Compactification of Monopoles and Metric Asymptotics
We construct a partial compactification of the moduli space,
eBook
New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in ℝⁿ
The aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable
conformal minimal surfaces in
All our new tools mentioned above apply to non-orientable minimal surfaces endowed with a fixed choice
of a conformal structure. This enables us to obtain significant new applications to the global theory of non-orientable minimal
surfaces. In particular, we construct proper non-orientable conformal minimal surfaces in
eBook