## Final Report Summary - DURFEE (Geometry and Topology of Singularities (Durfee Conjectures))

1. Summary description of the project objectives.

I study the local geometry and topology of singular varieties. My main object of research is: the Isolated Complete Intersection Singularities (ICIS) of algebraic/analytic varieties, and their invariants.

Problem 1: Develop the method of induction on blow-ups.

Problem 2: Prove the bound (initiated by the conjecture of A.Durfee 1978) on the singularity genus in terms of the Milnor number for isolated complete intersection singularities.

2. The transfer of knowledge.

I gave numerous talks in the seminars and math conferences in Israeli universities. I have attended numerous international conferences (in France, Germany, Spain, Turkey) where I have presented my work on the project.

I visited several times my collaborator A.Nemethi (Renyi Institute of Math, Budapest, Hungary), with whom I work on some aspects of the project.

I have been supervising several postdocs (A.Gourevich S. Vishkautsan, Alberto Fernandez-Boix, Y.Ren) whom I taught the singularity theory and with whom I am working on some parts of my project.

Besides my research papers I have written (and published) 48 reviews in Algebraic Geometry and Singularities for MathSciNet of AMS.

3. The research and the main results.

3.1. The bound for ICIS resolvable by toric blowups with "large" Newton diagrams.

Jointly with A.Nemethi we have proved:

Theorem A. Suppose an ICIS is resolvable by a sequence of toric blowups whose centers are points. Suppose the Newton diagrams prescribing each of these blowups are large. Then the conjectured bound holds.

The cornerstone for this theorem is the case of Newton-non-degenerate ICIS (with large Newton diagrams). This later paper itself uses a (completely unexpected) extension of the Fortuin-Kasteleyn-Ginibre inequality (FKG-inequality of averages, 1971) in Combinatorics, Statistical Mechanics. This FKG theorem, being the state-of-art result for decades, extends the Chebyshev inequality of averages to distributive lattices. For our work we needed this inequality for the Young lattices of partitions/compositions. They appear frequently in Representation Theory and Algebraic Geometry (for Grassmanians or Flag Bundles) and are non-distributive. We proved:

Theorem B. The FKG inequality holds also for the lattices of Young type.

This extension is a result in Combinatorics and its "native" applications are far from our areas of expertise. But even an immediate application to the diagrams/polytopes answered an old question of M.Gromov on convexity of mixed (co-)volumes.

3.2 The discriminant of transversal singularity type.

Even if one starts from an isolated singularity, the resolution usually creates non-isolated singularities. Thus the induction on blowups involves the understanding of these intermediate types. As the singular locus is projective, the local topology along it changes: the transversal singularity type inevitably degenerates in codimension one. Thus, besides the generic points of the singular locus, we must study the "subscheme of degeneration".

Jointly with A.Nemethi we have developed the theory of this degeneration subscheme, whom we have named "the discriminant of transversal singularity type". As our results grew, this initial ``little auxiliary step" had split into two separate papers.

The first paper addresses the local properties of the discriminantal scheme. Its results are in the following spirit:

Theorem C. Suppose X is a locally complete intersection at each point, whose singular locus Z is also a locally complete intersection. Then the discriminant of transversal singularity type is an effective Cartier divisor in Z, with functorial pullback properties. This discriminant deforms flatly under those deformations of X that preserve its generic multiplicity along Z.

The second paper addresses the global properties of the discriminant, in particular we compute its equivalence class in the Picard group of Z. (The expression is involved.)

Here we have observed a rather pathological class of singularities: ICIS whose first blow-ups are not ICIS. The corresponding local rings have been noticed long ago in Commutative Algebra, but were not given much attention in Singularities. These types lie on the border between ICIS and not-necessarily-Gorenstein singularities. Thus they are the natural source of counterexamples to varius conjectures. Currently we are studying this class of singularities.

3.3 The proof of the Durfee bound for "most of the types".

With the developed theory (section 3.2) we have attacked the Durfee bound. We have established it for those complete intersections (of any dimension and co-dimension) that admit a resolution with ``not too many" subsequent blowups at centers of positive dimensions. The result can be roughly stated as follows:

Theorem D. Suppose an ICIS admits a resolution by blowups with the following condition: if one meets a chain of blowups with centers of positive dimensions, then a prescribed polynomial (in the Chern numbers of these centers and the Chern classes of their normal sheaves) is bounded by the singularity invariants of the isolated singularities at the borders of this chain. Then the corrected Durfee inequality holds for this ICIS.

The class of ICIS satisfying this condition is the extension of the class of Newton-non-degenerate singularities with ``large Newton diagrams" to the singularities not necessarily resolvable by toric blowups. ``Most" of the complete intersection singularities satisfy this condition. (In the same sense, as most divisors in the ample cone are sufficiently ample.)

The precise formulations are technical and will appear in: D.Kerner A.Nemethi "Around Durfee conjecture".

3.4 Computer program to list the Newton diagrams (jointly with Y.Ren)

As we do not have a proof of the Durfee bound in complete generality, one naturally turns to numerical checks and the hunt for a counterexample. These numerical checks mean: going over a huge class of cases, and in each case establishing the bound (by computer).

Strangely, there has been no developed algorithm to do this for Newton non degenerate singularities. Jointly with my postdoc Y.Ren we have filled this cavity, both constructing an efficient algorithm and realizing it in the computer systems "Singular"/"Sage". This alowed us to create huge databases of Newton diagrams. (We list all the diagrams satisfying a prescribed condition, e.g. all the diagrams under a given polytope, all the diagrams with a bounded number of faces, etc) The number of diagrams grows quickly, e.g. there are hundreds of thousands of diagrams under the diagram of the singularity x^7+y^7+z^7.

4 Potential Impact and applications

The current project is an excellent demonstration of how a deep conjecture motivates (and even forces) advances in both the nearby and distant areas.

4.1 The FKG inequality of averages is by now the everyday tool in the areas ranging from Statistical Mechanics to Combinatorics, Probability and Convexity. We hope that our extension of the FKG-inequality will both lead to the new results in these areas and (vice-versa) will bring some tools and ideas from these areas to the Singularities and Algebraic Geometry.

4.2 In the simplest case the degeneration of transversal singularity type has been studied since 80's (e.g. Siersma, Pellikaan, R.Piene T.de-Jong - J.de-Jong.) We extended this to complete intersections of arbitrary dimension and codimension. This brings immediate applications to both the local studies (Singularity Theory) and the global ones (Intersection Theory, Projective Geometry, Enumerative Geometry).

4.3 The established Durfee-type bound "for most of singularities" will be used far beyond the Singularity Theory. An immediate application to Algebraic Geometry of smooth(!) varieties is the strengthening of the classical Bogomolov-Miyaoka-Yau bound for varieties that map birationally onto complete intersections in P^n with isolated singularities.

4.4 Similarly, the databases of Newton diagrams are important far beyond the Durfee inequality and the area of Singularities (as the polygon for testing many conjectures). The are highly wanted in the areas as Combinatorics and Polytopes. We hope that our procedures and databases will be implemented in the subsequent versions of computer systems "Singular" and "Sage".

I study the local geometry and topology of singular varieties. My main object of research is: the Isolated Complete Intersection Singularities (ICIS) of algebraic/analytic varieties, and their invariants.

Problem 1: Develop the method of induction on blow-ups.

Problem 2: Prove the bound (initiated by the conjecture of A.Durfee 1978) on the singularity genus in terms of the Milnor number for isolated complete intersection singularities.

2. The transfer of knowledge.

I gave numerous talks in the seminars and math conferences in Israeli universities. I have attended numerous international conferences (in France, Germany, Spain, Turkey) where I have presented my work on the project.

I visited several times my collaborator A.Nemethi (Renyi Institute of Math, Budapest, Hungary), with whom I work on some aspects of the project.

I have been supervising several postdocs (A.Gourevich S. Vishkautsan, Alberto Fernandez-Boix, Y.Ren) whom I taught the singularity theory and with whom I am working on some parts of my project.

Besides my research papers I have written (and published) 48 reviews in Algebraic Geometry and Singularities for MathSciNet of AMS.

3. The research and the main results.

3.1. The bound for ICIS resolvable by toric blowups with "large" Newton diagrams.

Jointly with A.Nemethi we have proved:

Theorem A. Suppose an ICIS is resolvable by a sequence of toric blowups whose centers are points. Suppose the Newton diagrams prescribing each of these blowups are large. Then the conjectured bound holds.

The cornerstone for this theorem is the case of Newton-non-degenerate ICIS (with large Newton diagrams). This later paper itself uses a (completely unexpected) extension of the Fortuin-Kasteleyn-Ginibre inequality (FKG-inequality of averages, 1971) in Combinatorics, Statistical Mechanics. This FKG theorem, being the state-of-art result for decades, extends the Chebyshev inequality of averages to distributive lattices. For our work we needed this inequality for the Young lattices of partitions/compositions. They appear frequently in Representation Theory and Algebraic Geometry (for Grassmanians or Flag Bundles) and are non-distributive. We proved:

Theorem B. The FKG inequality holds also for the lattices of Young type.

This extension is a result in Combinatorics and its "native" applications are far from our areas of expertise. But even an immediate application to the diagrams/polytopes answered an old question of M.Gromov on convexity of mixed (co-)volumes.

3.2 The discriminant of transversal singularity type.

Even if one starts from an isolated singularity, the resolution usually creates non-isolated singularities. Thus the induction on blowups involves the understanding of these intermediate types. As the singular locus is projective, the local topology along it changes: the transversal singularity type inevitably degenerates in codimension one. Thus, besides the generic points of the singular locus, we must study the "subscheme of degeneration".

Jointly with A.Nemethi we have developed the theory of this degeneration subscheme, whom we have named "the discriminant of transversal singularity type". As our results grew, this initial ``little auxiliary step" had split into two separate papers.

The first paper addresses the local properties of the discriminantal scheme. Its results are in the following spirit:

Theorem C. Suppose X is a locally complete intersection at each point, whose singular locus Z is also a locally complete intersection. Then the discriminant of transversal singularity type is an effective Cartier divisor in Z, with functorial pullback properties. This discriminant deforms flatly under those deformations of X that preserve its generic multiplicity along Z.

The second paper addresses the global properties of the discriminant, in particular we compute its equivalence class in the Picard group of Z. (The expression is involved.)

Here we have observed a rather pathological class of singularities: ICIS whose first blow-ups are not ICIS. The corresponding local rings have been noticed long ago in Commutative Algebra, but were not given much attention in Singularities. These types lie on the border between ICIS and not-necessarily-Gorenstein singularities. Thus they are the natural source of counterexamples to varius conjectures. Currently we are studying this class of singularities.

3.3 The proof of the Durfee bound for "most of the types".

With the developed theory (section 3.2) we have attacked the Durfee bound. We have established it for those complete intersections (of any dimension and co-dimension) that admit a resolution with ``not too many" subsequent blowups at centers of positive dimensions. The result can be roughly stated as follows:

Theorem D. Suppose an ICIS admits a resolution by blowups with the following condition: if one meets a chain of blowups with centers of positive dimensions, then a prescribed polynomial (in the Chern numbers of these centers and the Chern classes of their normal sheaves) is bounded by the singularity invariants of the isolated singularities at the borders of this chain. Then the corrected Durfee inequality holds for this ICIS.

The class of ICIS satisfying this condition is the extension of the class of Newton-non-degenerate singularities with ``large Newton diagrams" to the singularities not necessarily resolvable by toric blowups. ``Most" of the complete intersection singularities satisfy this condition. (In the same sense, as most divisors in the ample cone are sufficiently ample.)

The precise formulations are technical and will appear in: D.Kerner A.Nemethi "Around Durfee conjecture".

3.4 Computer program to list the Newton diagrams (jointly with Y.Ren)

As we do not have a proof of the Durfee bound in complete generality, one naturally turns to numerical checks and the hunt for a counterexample. These numerical checks mean: going over a huge class of cases, and in each case establishing the bound (by computer).

Strangely, there has been no developed algorithm to do this for Newton non degenerate singularities. Jointly with my postdoc Y.Ren we have filled this cavity, both constructing an efficient algorithm and realizing it in the computer systems "Singular"/"Sage". This alowed us to create huge databases of Newton diagrams. (We list all the diagrams satisfying a prescribed condition, e.g. all the diagrams under a given polytope, all the diagrams with a bounded number of faces, etc) The number of diagrams grows quickly, e.g. there are hundreds of thousands of diagrams under the diagram of the singularity x^7+y^7+z^7.

4 Potential Impact and applications

The current project is an excellent demonstration of how a deep conjecture motivates (and even forces) advances in both the nearby and distant areas.

4.1 The FKG inequality of averages is by now the everyday tool in the areas ranging from Statistical Mechanics to Combinatorics, Probability and Convexity. We hope that our extension of the FKG-inequality will both lead to the new results in these areas and (vice-versa) will bring some tools and ideas from these areas to the Singularities and Algebraic Geometry.

4.2 In the simplest case the degeneration of transversal singularity type has been studied since 80's (e.g. Siersma, Pellikaan, R.Piene T.de-Jong - J.de-Jong.) We extended this to complete intersections of arbitrary dimension and codimension. This brings immediate applications to both the local studies (Singularity Theory) and the global ones (Intersection Theory, Projective Geometry, Enumerative Geometry).

4.3 The established Durfee-type bound "for most of singularities" will be used far beyond the Singularity Theory. An immediate application to Algebraic Geometry of smooth(!) varieties is the strengthening of the classical Bogomolov-Miyaoka-Yau bound for varieties that map birationally onto complete intersections in P^n with isolated singularities.

4.4 Similarly, the databases of Newton diagrams are important far beyond the Durfee inequality and the area of Singularities (as the polygon for testing many conjectures). The are highly wanted in the areas as Combinatorics and Polytopes. We hope that our procedures and databases will be implemented in the subsequent versions of computer systems "Singular" and "Sage".