This volume is an expansion of lectures given by the author at the Park City Mathematics Institute (Utah) in 2008, and on other occasions. The purpose of this volume is to describe analytic techniques useful in the study of questions pertaining to linear series, multiplier ideals, and vanishing theorems for algebraic vector bundles. The author aims to be concise in his exposition, assuming that the reader is already somewhat acquainted with the basic concepts of sheaf theory, homological algebra, and complex differential geometry. In the final chapters, some very recent questions and open problems are addressed--such as results related to the finiteness of the canonical ring and the abundance conjecture, and results describing the geometric structure of Kahler varieties and their positive cones.

/pdf/analytic-methods-in-algebraic-geometry-3gcf7ignmq.pdf

Analytic Methods in Algebraic Geometry

LetX be a finite connectedCW-complex. Suppose that its fundamental group π is residually finite, i.e. there is a nested sequence ... ⊂ Г
 m
 + 1 ⊂ Г
 m
 ⊂ ... ⊂ π of in π normal subgroups of finite index whose intersection is trivial. Then we show that thep-thL
2-Betti number ofX is the limit of the sequenceb
p(Xm)/[π:Г
 m
 ] whereb
p(Xm) is the (ordinary)p-th Betti number of the finite covering ofX associated with Г
 m
 .

ApproximatingL 2-invariants by their finite-dimensional analogues

Oxford University Press in Africa, 1927â80

These notes are an expanded version of lectures delivered at the AMS Summer School on Algebraic Geometry, held at Santa Cruz in July 1995 The main goal of the notes is to study complex varieties (mostly compact or projective algebraic ones), through a few geometric questions related to hyperbolicity in the sense of Kobayashi A convenient framework for this is the category of “directed manifolds”, that is, the category of pairs (X, V ) whereX is a complex manifold and V a holomorphic subbundle of TX  If X is compact, the pair (X, V ) is hyperbolic if and only if there are no nonconstant entire holomorphic curves f : C → X tangent to V (Brody’s criterion) We describe a construction of projectivized kjet bundles PkV , which generalizes a construction made by Semple in 1954 and allows to analyze hyperbolicity in terms of negativity properties of the curvature More precisely, πk : PkV → X is a tower of projective bundles over X and carries a canonical line bundle OPkV (1) ; the hyperbolicity of X is then conjecturally equivalent to the existence of suitable singular hermitian metrics of negative curvature on OPkV (−1) for k large enough The direct images (πk)⋆OPkV (m) can be viewed as bundles of algebraic differential operators of order k and degree m, acting on germs of curves and invariant under reparametrization Following an approach initiated by Green and Griffiths, we establish a basic Ahlfors-Schwarz lemma in the situation when OPkV (−1) has a (possibly singular) metric of negative curvature, and we infer that every nonconstant entire curve f : C → V tangent to V must be contained in the base locus of the metric This basic result is then used to obtain a proof of the Bloch theorem, according to which the Zariski closure of an entire curve in a complex torus is a translate of a subtorus Our hope, supported by explicit Riemann-Roch calculations and other geometric considerations, is that the Semple bundle construction should be an efficient tool to calculate the base locus Necessary or sufficient algebraic criteria for hyperbolicity are then obtained in terms of inequalities relating genera of algebraic curves drawn on the variety, and singularities of such curves We finally describe some techniques introduced by Siu in value distribution theory, based on a use of meromorphic connections These techniques have been developped later by Nadel to produce elegant examples of hyperbolic surfaces of low degree in projective 3-space; thanks to a suitable concept of “partial projective projection” and the associated Wronskian operators, substantial improvements on Nadel’s degree estimate will be achieved here 2 J-P Demailly, Kobayashi hyperbolic projective varieties and jet differentials

Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials

A fake projective plane is a complex surface different from but has the same Betti numbers as the complex projective plane. It is a complex hyperbolic space form and has the smallest Euler Poincare characteristic among smooth surfaces of general type. The first example was constructed by Mumford. Later on two more examples were found by Ishida and Kato. A fourth possible one was recently proposed by Keum. In a recent joint work with Gopal Prasad, we showed that there are seventeen non-empty classes of fake projective planes and there can at most be four more specific classes. Higher dimensional analogues and examples were also obtained. The main purpose of the talk is to explain the joint work with Prasad and other related results such as arithmeticity of the lattices involved obtained earlier by Klingler and Yeung independently.

/pdf/fake-projective-planes-15vx3n2g5o.pdf

Fake projective planes

It was shown by Beauville that if the canonical map $\varphi_{|K_M|}$ of a complex smooth projective surface $M$ is generically finite, then $\operatorname{deg}(\varphi_{|K_M|}) \leq 36$. The first example of a surface with canonical degree $36$ was found by the second author. In this article, we show that for any surface which is a degree four Galois etale cover of a fake projective plane $X$ with the largest possible automorphism group $\operatorname{Aut}(X) = C_7:C_3$ (the unique non-abelian group of order $21$), the base locus of the canonical map is finite, and we verify that $35$ of these surfaces have maximal canonical degree $36$. We also classify all smooth degree four Galois etale covers of fake projective planes, which give possible candidates for surfaces of canonical degree $36$. Finally, we also confirm in this paper the optimal upper bound of the canonical degree of smooth threefolds of general type with sufficiently large geometric genus, related to earlier work of Hacon and Cai.

/pdf/examples-of-surfaces-with-canonical-map-of-maximal-degree-3zcnkxypou.pdf

Examples of Surfaces with Canonical Map of Maximal Degree

We prove that the defect vanishes for a holomorphic map f from the affine complex line to an abelian variety A and for an ample divisor D in A . The proof uses the translational invariance of the Zariski closure of the k -jet space of the image of f and the theorem of Riemann Roch to construct a nonidentically zero meromorphic k -jet differential whose pole divisor is dominated by a divisor equivalent to pD and which vanishes along the k -jet space of D to order q with p / q smaller than a prescribed small positive number. Then estimates involving the theta function with divisor D and the logarithmic derivative lemma are used. We also prove a pointwise Schwarz lemma which gives the vanishing of the pullback, by a holomorphic map from the affine complex line to a compact complex manifold, of a holomorphic jet differential vanishing on an ample divisor. This pointwise Schwarz lemma is a slight modification of a statement whose proof Green and Griffiths sketched in their alternative treatment of Bloch's theorem on entire curves in abelian varieties. The log-pole case of the pointwise Schwarz lemma is also given. We construct examples of hyperbolic hypersurface whose degree is only 16 times the square of its dimension.

Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees

Representations of semisimple lattices in mapping class groups

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Examples of Surfaces with Canonical Map of Maximal Degree

Defects for ample divisors of abelian varieties, Schwarz lemma, and hyperbolic hypersurfaces of low degrees

Representations of semisimple lattices in mapping class groups