| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | April 11, 2011 |
| Latest Amendment Date: | June 17, 2013 |
| Award Number: | 1068190 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
Andrew Pollington
adpollin@nsf.gov (703)292-4878 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | July 1, 2011 |
| End Date: | June 30, 2015 (Estimated) |
| Total Intended Award Amount: | $290,649.00 |
| Total Awarded Amount to Date: | $290,649.00 |
| Funds Obligated to Date: |
FY 2012 = $96,873.00 FY 2013 = $97,959.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
1109 GEDDES AVE STE 3300 ANN ARBOR MI US 48109-1015 (734)763-6438 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
1109 GEDDES AVE STE 3300 ANN ARBOR MI US 48109-1015 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | ALGEBRA,NUMBER THEORY,AND COM |
| Primary Program Source: |
01001213DB NSF RESEARCH & RELATED ACTIVIT 01001314DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): | |
| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
The project concerns two questions about singularities. The first problem has two different incarnations: a first one in the algebro-geometric setting of graded sequences of ideals, and a second one, in the analytic setting of plurisubharmonic functions (in which context it was conjectured by Demailly and Koll\'{a}r, and it is known as the Openness Conjecture). The common point is that both versions reduce to understanding asymptotic versions of familiar invariants of singularities, such as the log canonical threshold, associated now to certain sequences of ideals. The first part of the project consists in the study of these asymptotic invariants, and of their connections to valuation theory. The second part will be devoted to a study of valuations from a point of view relevant to this problem. The second problem concerns connections between certain invariants of singularities in birational geometry (such as the log canonical threshold and multiplier ideals) and invariants introduced in commutative algebra, coming from tight closure theory (such as the F-pure threshold and the test ideals). There have been formulated precise conjectures regarding this correspondence via reduction to prime characteristic. This project concerns translating such conjectures into some more established questions regarding the Frobenius action on the cohomology of reductions of smooth projective varieties to positive characteristic, and then in trying to attack some special cases of these questions.
Singularities appear naturally in the study of algebraic varieties, and a good understanding of singularities is important, for example, in the classification of higher-dimensional algebraic varieties. This project addresses two important open problems related to singularities. By reducing them to questions in different settings, the PI hopes to bring into action tools from other areas, such as valuation theory and arithmetic geometry. The reduction itself would be of interest, by highlighting connections between these different settings.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
The work supported by this grant focussed on the study of singularities of algebraic varieties in positive characteristic. These were studied with two kind of applications in mind:
1) The connections between these invariants and similar invariants in characteristic zero.
2) Global results concerning algebraic varieties in positive characteristic.
Recall that the study of algebraic varieties can be done in two settings: in characteristic zero (which basically means over the complex numbers) and in positive characteristic (for example, over the algebraic closure of a finite field). The former setting has various tools at its disposal, such as complex analysis, while the former setting is closer to arihmetic. The global study of algebraic varieties in characteristic zero relies on the use of vanishing theorems, which are known to fail in positive characteristic. Some of the tools developed to study singularities in this latter setting (making use, in particular, of the Frobenius morphism) turn out to be very useful also for studying global problems. Several of the results obtained by the PI during this period pertain to this circle of ideas.
One interesting aspect in positive characteristic is that one can define invariants and classes of singularities using the Frobenius morphism. It turns out that these invariants are related in a subtle way with invariants defined in characteristic zero from completely different information. Some of the most intriguing questions in this area concern the precise relation and this is a topic studied by the PI during this period. For example, in joint work with Vasudevan Srinivas, he related a conjecture in this area with a conjecture of a very different flavor, that has been well-known to people working in arithmetic geometry (conjecture which is considered to be very hard).
Last Modified: 08/17/2015
Modified by: Mircea Mustata
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