Cohomological descent of derived category and Fourier–Mukai to singular rational cohomology

Pluricanonical Systems of Algebraic Varieties of General Type of Dimension >/=5 (T. Ando). Uniqueness of Einstein Kahler Metrics Modulo Connected Group Actions (S. Bando, T. Mabuchi). Abelian Surfaces with (1, 2)-Polarization (W. Barth). Euler Characteristics and Swan Conductors (S.J. Bloch). Complete Intersections with Growth Conditions (O. Forster and T. Ohsawa). On the de Rham Cohomology Group of a Compact Kahler Symplectic Manifold (A. Fujiki). On Polarized Manifolds Whose Adjoint Bundles Are Not Semipositive (T. Fujita). Coverings of Algebraic Varieties (R.V. Gurjar). Stability of the Pluricanonical Maps of Threefolds (M. Hanamura). On p-adic Vanishing Cycles (Application of Ideas of Fontaine-Messing)(K. Kato). Supersingular Abelian Varieties of Dimension Two or Three and Class Numbers (T. Katsura, F. Oort). Introduction to the Minimal Model Problem (Y. Kawamata, K. Matsuda, K. Matsuki). Subadditivity of the Kodaira Dimension: Fibers of General Type (J. Kollar). The Rationality of the Moduli Spaces of Vector Bundles of Rank 2 on P 2 (M. Maruyama, with an Appendix by I. Naruki). Projective Degenerations of Surfaces According to S. Tsunoda (M. Miyanishi). The Chern Classes and Kodaira Dimension of a Minimal Variety (Y. Miyaoka). Cremona Transformations and Degrees of Period Maps for K3 Surfaces with Ordinary Double Points (D.R. Morrison, M.-H. Saito). Fourier Functor and Its Application to the Moduli of Bundles on an Abelian Variety (S. Mukai). The Lower Semi-Continuity of the Plurigenera of Complex Varieties (N. Nakayama). K. Saito's Period Map for Holomorphic Functions with Isolated Critical Points (T. Oda). Variation of Mixed Hodge Structure and the Torelli Problem (M.-H. Saito, Y. Shimizu, S. Usui). Mixed Hodge Structures on Cohomologies with Coefficients in a Polarized Variation of Hodge Structure (Y. Shimizu). Algebraic Cycles on Hypersurfaces in P N (T. Shioda). Constructible Sheaves Associated to the Whittaker Functions (T. Terasoma). Degenerations of Surfaces (S. Tsunoda). Compact Rigid Analytic Spaces, with Special Regard to Surfaces (K. Ueno).

Let V be a closed subscheme of a projective space P. We give algorithms to compute the Chern-Schwartz-MacPherson class, Euler characteristic and Segre class of V . These algorithms can be implemented using either symbolic or numerical methods. The basis for these algorithms is a method for calculating the projective degrees of a rational map defined by a homogeneous ideal. When combined with formula for the Chern-Schwartz-MacPherson class of a projective hypersurface and the Segre class of a projective variety in terms of the projective degrees of certain rational maps this gives us algorithms to compute the Chern-SchwartzMacPherson class and Segre class of a projective variety. Since the Euler characteristic of V is the degree of the zero dimensional component of the Chern-Schwartz-MacPherson class of V our algorithm also computes the Euler characteristic χ(V ). The algorithms are tested on several examples and are found to perform favourably compared to other algorithms for computing Chern-Schwartz-MacPherson classes, Segre classes and Euler characteristics.

Abstract: Algebraic geometry offers a powerful and elegant mathematical framework for the design and analysis of modern cryptographic protocols. This research paper investigates the application of algebraic geometry methods—such as elliptic curves, abelian varieties, and projective algebraic structures—in enhancing the security, efficiency, and scalability of cryptographic systems. By bridging advanced algebraic structures with cryptographic primitives, the study demonstrates how algebraic geometry enables the construction of secure public key protocols, zero-knowledge proofs, and post-quantum resilient schemes. Through theoretical modeling, performance benchmarking, and comparative analysis with classical cryptographic approaches, the paper illustrates the advantages of algebraic geometry in terms of computational hardness assumptions, structural integrity, and potential for innovation in secure communications. The findings contribute to the evolving landscape of cryptography by positioning algebraic geometry as a foundational tool in next-generation cryptographic protocol design. Keywords: algebraic geometry, cryptographic protocols, elliptic curves, public key cryptography, post-quantum cryptography, projective varieties, zero-knowledge proofs, secure communication, mathematical cryptography, abelian varieties

Atiyah and Hirzebruch gave the first counterexamples to the Hodge conjecture with integral coefficients [3]. That conjecture predicted that every integral cohomology class of Hodge type (p, p) on a smooth projective variety should be the class of an algebraic cycle, but Atiyah and Hirzebruch found additional topological properties which must be satisfied by the integral cohomology class of an algebraic cycle. Here we provide a more systematic explanation for their results by showing that the classical cycle map, from algebraic cycles modulo algebraic equivalence to integral cohomology, factors naturally through a topologically defined ring which is richer than integral cohomology. The new ring is based on complex cobordism, a well-developed topological theory which has been used only rarely in algebraic geometry since Hirzebruch used it to prove the Riemann-Roch theorem [17]. This factorization of the classical cycle map implies the topological restrictions on algebraic cycles found by Atiyah and Hirzebruch. It goes beyond their work by giving a topological method to show that the classical cycle map can be noninjective, as well as nonsurjective. The kernel of the classical cycle map is called the Griffiths group, and the topological proof here that the Griffiths group can be nonzero is the first proof of this fact which does not use Hodge theory. (The proof here gives nonzero torsion elements in the Griffiths group, whereas Griffiths's Hodge-theoretic proof gives nontorsion elements [13].) This topological argument also gives examples of algebraic cycles in the kernel of various related cycle maps where few or no examples were known before, thus answering some questions posed by Colliot-Thelene and Schoen ([8], p. 14; [37], p. 13). Colliot-Thelene asked, in particular, whether the map CH2(X)/n -* H4(X, Z/n) is injective for all smooth complex projective varieties X. Here CH'X is the group of codimension i algebraic cycles modulo rational equivalence. The first examples where Colliot-Thelene's map is not injective were found by Kollar and van Geemen (see [4], p. 135); very recently, Bloch and Esnault found examples defined over number fields [7]. (Over nonalgebraically closed fields k there are other examples of smooth projective varieties Xk with CH2 (Xk)/n -* Hg4t(Xk, Z/n) not injective, due to Colliot-Thelene and Sansuc as reinterpreted by Salberger (see [9] and [8], Remark 7.6.1), and Parimala and Suresh [31]. These elements of CH2 (Xk)/n are not shown to remain nonzero in CH2(Xc)/n, however.) Here our topological method

This book, which grew out of lectures by E. Kunz for students with a background in algebra and algebraic geometry, develops local and global duality theory in the special case of (possibly singular) algebraic varieties over algebraically closed base fields. It describes duality and residue theorems in terms of Kahler differential forms and their residues. The properties of residues are introduced via local cohomology. Special emphasis is given to the relation between residues to classical results of algebraic geometry and their generalizations. The contribution by A. Dickenstein gives applications of residues and duality to polynomial solutions of constant coefficient partial differential equations and to problems in interpolation and ideal membership. D. A. Cox explains toric residues and relates them to the earlier text. The book is intended as an introduction to more advanced treatments and further applications of the subject, to which numerous bibliographical hints are given.

Now in paperback, this book provides a self-contained introduction to the cohomology theory of Lie groups and algebras and to some of its applications in physics. No previous knowledge of the mathematical theory is assumed beyond some notions of Cartan calculus and differential geometry (which are nevertheless reviewed in the book in detail). The examples, of current interest, are intended to clarify certain mathematical aspects and to show their usefulness in physical problems. The topics treated include the differential geometry of Lie groups, fibre bundles and connections, characteristic classes, index theorems, monopoles, instantons, extensions of Lie groups and algebras, some applications in supersymmetry, Chevalley-Eilenberg approach to Lie algebra cohomology, symplectic cohomology, jet-bundle approach to variational principles in mechanics, Wess-Zumino-Witten terms, infinite Lie algebras, the cohomological descent in mechanics and in gauge theories and anomalies. This book will be of interest to graduate students and researchers in theoretical physics and applied mathematics.

For many mathematicians, the lure of mathematics began with a gradual recognition of the beauty of Euclidean geometry. It attracts us with its simplicity and clarity. Later, with the discovery of Cartesian coordinates, we see the power and precision of algebra in describfng geometric ideas. Such seeds having been sown, I have found myself deeply enthralled for many years now with algebraic geometry. Recently, however, a yearning for a return to a more concrete geometry overcame me. It was a desire to understand the "geometry" in algebraic geometry which has led to a study of those classical examples which seem to be well understood by the old masters. With a background in algebraic groups, I found the symmetric spaces of Cartan to be a good starting point. Currently, the "geometry" in algebraic geometry is represented most effectively by the theory of complex manifolds, many of the most popular of which are projective algebraic varieties. (For one who thought theorems should be proved in a characteristic-free environment, this represents a significant shift in perspective.) The compact Hermitian symmetric spaces can be represented as homogeneous spaces for reductive algebraic groups (so are objects of study in algebraic geometry) and also as homogeneous spaces for compact Lie groups (so are objects of study in Riemannian geometry). Among these spaces are the complex projective spaces and the Grassmann manifolds which play a fundamental role in the study of algebraic manifolds and the geometry on them. This article is concerned with recent characterizations of the Hermitian symmetric spaces using the curvature of the natural K~ihler metrics on each. In particular, the only ones with everywhere positive biholmorphic sectional curvature are the complex projective spaces. A motivating problem for what follows is: Find a similar characterization of Grassmann manifolds in terms of their curvature.

Wolfram’s Elementary Cellular Automata (ECA) serve as fundamental models for studying discrete dynamical systems, yet their classification remains challenging under traditional statistical and heuristic methods. By leveraging tools from algebraic topology, homotopy theory and differential geometry, we establish a formal connection between topological invariants and ECA’s structural properties and evolution. We analyse the role of Betti numbers, Euler characteristics, edge complexity and persistent homology in achieving robust separation of the four ECA classes. Additionally, we apply coarse proximity theory and assessed the applicability of Poincaré duality, Nash embedding and Seifert–van Kampen theorems to quantify large-scale connectivity patterns. We find that Class 1 automata exhibit simple, contractible topological spaces, indicating minimal structural complexity, while Class 2 automata exhibit periodic fluctuations in their topological features, reflecting their cyclic structure and repeating patterns. Class 3 automata exhibit a higher variance in their structural properties with persistent topological features forming and dissolving across scales, a signature of chaotic evolution. Class 4 automata exhibit statistically significant increases in higher-dimensional topological voids, suggesting the appearance of stable formations. Edge complexity and fractal dimension emergd as the strongest predictors of increasing computational and topological complexity, confirming that self-similarity and structural complexity play a crucial role in distinguishing cellular automata classes. Further, we address the critical distinction between Class 3 and Class 4 automata, which holds paramount importance in practical applications. Our approach establishes a mathematical framework for automaton classification by identifying emergent structures, with potential applications in computational physics, artificial intelligence and theoretical biology.

These lectures are devoted to the study of various contemporary problems of algebraic geometry, using fundamental tools from complex potential theory, namely plurisubharmonic functions, positive currents and Monge-Ampère operators. Since their inception by Oka and Lelong in the mid 1940s, plurisubharmonic functions have been used extensively in many areas of algebraic and analytic geometry, as they are the function theoretic counterpart of pseudoconvexity, the complexified version of convexity. One such application is the theory of L 2 estimates via the Bochner-Kodaira-Hörmander technique, which provides very strong existence theorems for sections of holomorphic vector bundles with positive curvature. One can mention here the foundational work achieved by Bochner, Kodaira, Nakano, Morrey, Kohn, Andreotti-Vesentini, Grauert, Hörmander, Bombieri, Skoda and Ohsawa-Takegoshi in the course of more than four decades. Another development is the theory of holomorphic Morse inequalities (1985), which relate certain curvature integrals with the asymptotic cohomology of large tensor powers of line or vector bundles, and bring a useful complement to the Riemann-Roch formula.We describe here the main techniques involved in the proof of holomorphic Morse inequalities (Sect. 1) and their link with Monge-Ampère operators and intersection theory. Section 2, especially, gives a fundamental approximation theorem for closed (1, 1)-currents, using a Bergman kernel technique in combination with the Ohsawa-Takegoshi theorem. As an application, we study the geometric properties of positives cones of an algebraic variety (nef and pseudo-effective cone), and derive from there some results about asymptotic cohomology functionals in Sect. 3. The last Sect. 4 provides an application to the study of the Green-Griffiths-Lang conjecture. The latter conjecture asserts that every entire curve drawn on a projective variety of general type should satisfy a global algebraic equation; via a probabilistic curvature estimate, holomorphic Morse inequalities imply that entire curves must at least satisfy a global algebraic differential equation.

Algebraic geometry is, essentially, the study of the solution of equations and occupies a central position in pure mathematics. This short and readable introduction to algebraic geometry will be ideal for all undergraduate mathematicians coming to the subject for the first time. With the minimum of prerequisites, Dr Reid introduces the reader to the basic concepts of algebraic geometry including: plane conics, cubics and the group law, affine and projective varieties, and non-singularity and dimension. He is at pains to stress the connections the subject has with commutative algebra as well as its relation to topology, differential geometry, and number theory. The book arises from an undergraduate course given at the University of Warwick and contains numerous examples and exercises illustrating the theory.

We generalize a formula due to Macdonald that relates the singular Betti numbers of $X^{n}/G$ to those of $X$, where $X$ is a compact manifold and $G$ is any subgroup of the symmetric group $S_{n}$ acting on $X^{n}$ by permuting coordinates. Our result is completely axiomatic: in a general setting, given an endomorphism on the cohomology $H^{\bullet}(X)$, it explains how we can explicitly relate the Lefschetz series of the induced endomorphism on $H^{\bullet}(X^{n})^{G}$ to that of the given endomorphism on $H^{\bullet}(X)$ in the presence of the Kunneth formula with respect to a cup product. For example, when $X$ is a compact manifold, we take the Lefschetz series given by the singular cohomology with rational coefficients. On the other hand, when $X$ is a projective variety over a finite field $\mathbb{F}_{q}$, we use the $l$-adic etale cohomology with a suitable choice of prime number $l$. We also explain how our formula generalizes the Polya enumeration theorem, a classical theorem in combinatorics that counts colorings of a graph up to given symmetries, where $X$ is taken to be a finite set of colors. When $X$ is a smooth projective variety over $\mathbb{C}$, our formula also generalizes a result of Cheah that relates the Hodge numbers of $X^{n}/G$ to those of $X$. We will also see that our result generalizes the following facts: 1. the generating function of the Poincare polynomials of symmetric powers of a compact manifold $X$ is rational; 2. the generating function of the Hodge-Deligne polynomials of symmetric powers of a smooth projective variety $X$ over $\mathbb{C}$ is rational; 3. the zeta series of a projective variety $X$ over $\mathbb{F}_{q}$ is rational. We also prove analogous rationality results when we replace $S_{n}$ with $A_{n}$, alternating groups.

In this paper we construct Stiefel-Whitney and Euler classes in Chow cohomology for algebraic vector bundles with nondegenerate quadratic form. These classes are not in the algebra generated by the Chern classes of such bundles and are new characteristic classes in algebraic geometry. On complex varieties, they correspond to classes with the same name pulled back from the cohomology of the classifying space BSO(N,C). The classes we construct are the only new characteristic classes in algebraic geometry coming from the classical groups ([T2], [EG]). We begin by using the geometry of quadric bundles to study Chern classes of maximal isotropic subbundles. If V → X is a vector bundle with quadratic form, and if E and F are maximal isotropic subbundles of V then we prove (Theorem 1) that ci(E) and ci(F ) are equal mod 2. Moreover, if the rank of V is 2n, then cn(E) = ±cn(F ), proving a conjecture of Fulton. We define Stiefel-Whitney and Euler classes as Chow cohomology classes which pull back to Chern classes of maximal isotropic subbundles of the pullback bundle. Using the above theorem we show (Theorem 2) that these classes exist and are unique, even though V need not have a maximal isotropic subbundle. These constructions also make it possible to give “Schubert” presentations,

Roughly speaking, to any space $ M$ with perfect obstruction theory we associate a space $ N$ with symmetric perfect obstruction theory. It is a cone over $ M$ given by the dual of the obstruction sheaf of $ M$ and contains $ M$ as its zero section. It is locally the critical locus of a function. More precisely, in the language of derived algebraic geometry, to any quasi-smooth space $ M$ we associate its $ (\\!-\\!1)$-shifted cotangent bundle $ N$. By localising from $ N$ to its $ \\mathbb{C}^*$-fixed locus $ M$ this gives five notions of a virtual signed Euler characteristic of $ M$: The Ciocan-Fontanine-Kapranov/Fantechi-Göttsche signed virtual Euler characteristic of $ M$ defined using its own obstruction theory, Graber-Pandharipande's virtual Atiyah-Bott localisation of the virtual cycle of $ N$ to $ M$, Behrend's Kai-weighted Euler characteristic localisation of the virtual cycle of $ N$ to $ M$, Kiem-Li's cosection localisation of the virtual cycle of $ N$ to $ M$, $ (-1)^{\\textrm {vd}}$ times by the topological Euler characteristic of $ M$. Our main result is that (1)=(2) and (3)=(4)=(5). The first two are deformation invariant while the last three are not.

An explicit construction of closed, orientable, smooth, aspherical 4-manifolds with any odd Euler characteristic greater than 12 is presented. The constructed manifolds are all Haken manifolds in the sense of B. Foozwell and H. Rubinstein and can be systematically reduced to balls by suitably cutting them open along essential codimension-one submanifolds. Euler characteristics divisible by 3 are known to arise from complex algebraic geometry considerations. Examples with Euler characteristic 1, 5, 7, or 11 appear to be unknown.

The purpose of this note is to prove the following result: Theorem 1 Let X and Y be smooth projective complex varieties, and suppose that X = X1 ∪ ··· ∪ Xn and Y = Y1 ∪ ··· ∪ Yn are a disjoint union of quasiprojective subvarieties. Suppose that Xi is algebraically isomorphic to Yi, for all i. Then the Betti numbers of X and Y are equal, and in fact their Hodge numbers are equal. The result for the Betti numbers is part of the mathematical folklore; it has been used, for instance, in [1] and implicitly in [4]. Apparently it was originally noticed by Serre, who proved it by reducing the variety modulo p, counting points and using the solution to the Weil conjectures. The proof given in this paper uses mixed Hodge theory rather than the Weil conjectures; the general yoga of Hodge theory and the Weil conjectures suggest that such a proof should exist. The theorem is in fact a simple consequence of a sum formula for the Hodge numbers of the mixed Hodge structure of an arbitrary variety in terms of those of the pieces of a decomposition. (Throughout this paper, the term "variety" will mean "quasiprojective complex variety.") First, some definitions: According to [2], the rational cohomology groups of a complex algebraic variety have a weight filtration W and a Hodge filtration F, and so do its rational cohomology groups with compact support. The associated graded objects of these filtrations are denoted GrW and GrF, respectively. We refine the concept of Euler characteristic by using these filtrations: Let χ ( X ) = ∑ ( − 1 ) k dim H k ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq635.tif"/> χ m ( X ) = ∑ ( − 1 ) k dim G r m W H k ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq636.tif"/> χ p q ( X ) = ∑ ( − 1 ) k dim G r F p G r p + q W H k ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq637.tif"/> 100Similarly, define χc, χ m c https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq638.tif"/> and χ c pq https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq639.tif"/> using H c k ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq640.tif"/> (cohomology with compact supports) in place of Hk(X). These graded Euler characteristics satisfy χ ( X ) = ∑ m X m ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq641.tif"/> χ m ( X ) = ∑ p + q = m X p q ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq642.tif"/> and so forth. If X is smooth of dimension n, the Poincaré duality implies that χ c p q ( X ) = χ n − p , n − q ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq643.tif"/> If X is smooth and projective, then χ m ( X ) = ( − 1 ) m dim H m ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq644.tif"/>