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Infinitesimal presentations of the Torelli groups
 Ha2] [HL] [KM] [Jo83
, 1997
"... 2. Braid groups in positive genus 601 3. Relative completion of mapping class groups 603 4. Mixed Hodge structures on Torelli groups 608 ..."
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Cited by 63 (6 self)
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2. Braid groups in positive genus 601 3. Relative completion of mapping class groups 603 4. Mixed Hodge structures on Torelli groups 608
The construction problem in Kähler geometry
, 2004
"... One of the most surprising things in algebraic geometry is the fact that algebraic varieties over the complex numbers benefit from a collection of metric properties which strongly influence their topological and geometric shapes. The existence of a Kähler metric leads to all sorts of Hodge theoretic ..."
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Cited by 12 (2 self)
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One of the most surprising things in algebraic geometry is the fact that algebraic varieties over the complex numbers benefit from a collection of metric properties which strongly influence their topological and geometric shapes. The existence of a Kähler metric leads to all sorts of Hodge theoretical restrictions on the homotopy types of algebraic varieties. On the other hand, a sparse collection of examples shows that the remaining liberty is nontrivially large. Paradoxically, with all of this information, the research field remains as wide open as it was many decades ago, because the gap between the known restrictions, and the known examples of what can occur, only seems to grow wider and wider the more closely we look at it. In spite of the differentialgeometric nature of the questions and methods, the origins of the situation are very algebraic. We look at subvarieties of projective space over the complex numbers. The main overarching problem in algebraic geometry is to understand the classification of algebrogeometric objects. The topology of the usual complexvalued points of a variety plays
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"... Dedicated to Alexander Reznikov One of the most surprising things in algebraic geometry is the fact that algebraic varieties over the complex numbers benefit from a collection of metric properties which strongly influence their topological and geometric shapes. The existence of a K"ahler me ..."
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Dedicated to Alexander Reznikov One of the most surprising things in algebraic geometry is the fact that algebraic varieties over the complex numbers benefit from a collection of metric properties which strongly influence their topological and geometric shapes. The existence of a K&quot;ahler metric leads to all sorts of Hodge theoretical restrictions on the homotopy types of algebraic varieties. On the other hand, a sparse collection of examples shows that the remaining liberty is nontrivially large. Paradoxically, with all of this information, the research field remains as wide open as it was many decades ago, because the gap between the known restrictions, and the known examples of what can occur, only seems to grow wider and wider the more closely we look at it. In spite of the differentialgeometric nature of the questions and methods, the origins of the situation are very algebraic. We look at subvarieties of projective space over the complex numbers. The main overarching problem in algebraic geometry is to understand the classification of algebrogeometric objects. The topology of the usual complexvalued points of a variety plays an important role, because the topological type is a locally constant function on any classifying space. Thus the partitioning of the classification problem according to topological type provides a coarse, and more calculable, alternative to the partition by connected components. Furthermore, the topology of a variety strongly influences its geometric properties. A sociological observation is that the quest for understanding the topology of algebraic varieties has led to a rich set of techniques which found applications elsewhere, even in physics. Perhaps a word about the choice of ground field is appropriate. One could also look at the shapes of the real points of real algebraic varieties. However, by Weierstrass approximation, pretty much anything can arise if you let the degree get big enough. Thus the classical question in this case is &quot;which shapes can occur for a given degree? &quot; This is so much more difficult 1