Characteristic Classes Of Singular Varieties Research Articles

Enriched categories of correspondences and characteristic classes of singular varieties

For the category $\mathscr V$ of complex algebraic varieties, the Grothendieck group of the commutative monoid of the isomorphism classes of correspondences $X \xleftarrow f M \xrightarrow g Y$ with a proper morphism $f$ and a smooth morphism $g$ (such a

Motivic characteristic classes in cohomological Hall algebras

The equivariant Chern-Schwartz-MacPherson (CSM) class and the equivariant Motivic Chern (MC) class are important characteristic classes of singular varieties in cohomology and K theory—and their theory overlaps with the theory of Okounkov's stable envelopes. We study CSM and MC classes for the orbits of Dynkin quiver representations. We show that the problem of computing the CSM and MC classes of all these orbits can be reduced to some basic classes cβo, Cβo parameterized by positive roots β. We prove an identity in a deformed version of Kontsevich-Soibelman's Cohomological (and K-theoretic) Hall Algebra (CoHA, KHA), namely, that a product of exponentials of cβo (or Cβo) classes formally depending on a stability function Z, does not depend on Z. This identity—which encodes infinitely many identities among rational functions in growing number of variables—has the structure of Donaldson-Thomas type quantum dilogarithm identities. Using a wall-crossing argument we present the cβo, Cβo classes as certain commutators in the CoHA, KHA.

Homological congruence formulae for characteristic classes of singular varieties

For a pair $(f, g)$ of morphisms $f:X \to Z$ and $g:Y \to Z$ of (possibly singular) complex algebraic varieties $X,Y,Z$, we present congruence formulae for the difference $f_*T_{y*}(X) -g_*T_{y*}(Y)$ of pushforwards of the corresponding motivic Hirzebruch classes $T_{y*}$. If we consider the special pair of a fiber bundle $F \hookrightarrow E \to B$ and the projection $pr_2:F \times B \to B$ as such a pair $(f,g)$, then we get a congruence formula for the difference $f_*T_{y*}(E) -\chi_y(F)T_{y*}(B)$, which at degree level yields a congruence formula for $\chi_y(E) -\chi_y(F)\chi_y(B)$, expressed in terms of the Euler--Poincarv'e characteristic, Todd genus and signature in the case when $F, E, B$ are non-singular and compact. We also extend the finer congruence identities of Rovi--Yokura to the singular complex projective situation, by using the corresponding intersection (co)homology invariants.

“Counting” spaces and related topics

This is a short survey about the three well-known theories of characteristic classes of singular varieties and the so-called motivic Hirzebruch class, which in a sense “unifies” these three theories as an affirmative answer to a problem posed by Robert MacPherson in the early 1970s.

How many hypersurfaces does it take to cut out a Segre class?

We prove an identity of Segre classes for zero-schemes of compatible sections of two vector bundles. Applications include bounds on the number of equations needed to cut out a scheme with the same Segre class as a given subscheme of (for example) a projective variety, and a ‘Segre–Bertini’ theorem controlling the behavior of Segre classes of singularity subschemes of hypersurfaces under general hyperplane sections.These results interpolate between an observation of Samuel concerning multiplicities along components of a subscheme and facts concerning the integral closure of corresponding ideals. The Segre–Bertini theorem has applications to characteristic classes of singular varieties. The main results are motivated by the problem of computing Segre classes explicitly and applications of Segre classes to enumerative geometry.

ALGEBRO-GEOMETRIC FEYNMAN RULES

We give a general procedure to construct "algebro-geometric Feynman rules", that is, characters of the Connes–Kreimer Hopf algebra of Feynman graphs that factor through a Grothendieck ring of immersed conical varieties, via the class of the complement of the affine graph hypersurface. In particular, this maps to the usual Grothendieck ring of varieties, defining "motivic Feynman rules". We also construct an algebro-geometric Feynman rule with values in a polynomial ring, which does not factor through the usual Grothendieck ring, and which is defined in terms of characteristic classes of singular varieties. This invariant recovers, as a special value, the Euler characteristic of the projective graph hypersurface complement. The main result underlying the construction of this invariant is a formula for the characteristic classes of the join of two projective varieties. We discuss the BPHZ renormalization procedure in this algebro-geometric context and some motivic zeta functions arising from the partition functions associated to motivic Feynman rules.

Shadows of blow-up algebras

We study different notions of blow-up of a scheme X along a subscheme Y, depending on the datum of an embedding of X into an ambient scheme. The two extremes in this theory are the ordinary blow-up, corresponding to the identity, and the `quasi-symmetric blow-up', corresponding to an embedding into a nonsingular variety M. We prove that this latter blow-up is intrinsic of Y and X, and is universal with respect to the requirement of being embedded as a subscheme of the ordinary blow-up of some ambient space along Y. We consider these notions in the context of the theory of characteristic classes of singular varieties. We prove that if X is a hypersurface in a nonsingular variety and Y is its `singularity subscheme', these two extremes embody respectively the conormal and characteristic cycles of X. Consequently, the first carries the essential information computing Chern-Mather classes, and the second is likewise a carrier for Chern-Schwartz-MacPherson classes. In our approach, these classes are obtained from Segre class-like invariants, in precisely the same way as other intrinsic characteristic classes such as those proposed by William Fulton and by W. Fulton and Kent Johnson. We also identify a condition on the singularities of a hypersurface under which the quasi-symmetric blow-up is simply the linear fiber space associated with a coherent sheaf.